TPTP Problem File: ITP240^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP240^1 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Pred 00667_037763
% 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 : 0069_VEBT_Pred_00667_037763 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 11327 (5634 unt;1082 typ; 0 def)
% Number of atoms : 28472 (12077 equ; 0 cnn)
% Maximal formula atoms : 71 ( 2 avg)
% Number of connectives : 115947 (2663 ~; 523 |;1755 &;100081 @)
% ( 0 <=>;10925 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 6 avg)
% Number of types : 99 ( 98 usr)
% Number of type conns : 5029 (5029 >; 0 *; 0 +; 0 <<)
% Number of symbols : 987 ( 984 usr; 64 con; 0-8 aty)
% Number of variables : 26680 (2440 ^;23449 !; 791 ?;26680 :)
% SPC : TH0_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 23:27:20.286
%------------------------------------------------------------------------------
% Could-be-implicit typings (98)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
produc5542196010084753463at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
set_Pr1281608226676607948nteger: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
produc1908205239877642774nteger: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
produc5491161045314408544at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
set_Pr9222295170931077689nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc2285326912895808259nt_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
set_Pr8056137968301705908nteger: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
produc8763457246119570046nteger: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
set_Pr1872883991513573699nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc7773217078559923341nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
produc1193250871479095198on_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
produc8306885398267862888on_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc6121120109295599847at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4471711990508489141at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
produc7036089656553540234on_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
produc2233624965454879586on_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
produc6241069584506657477e_term: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J_J,type,
set_Pr6308028481084910985omplex: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
produc8551481072490612790e_term: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_Pr2522554150109002629et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
produc8064648209034914857omplex: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
option6357759511663192854e_term: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
produc3447558737645232053on_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
produc4953844613479565601on_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
list_P7037539587688870467BT_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
list_P4547456442757143711BT_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
produc8243902056947475879T_VEBT: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
produc2115011035271226405et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
produc8923325533196201883nteger: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
option4927543243414619207at_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
filter1242075044329608583at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
produc9072475918466114483BT_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
produc4894624898956917775BT_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
produc8025551001238799321T_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
list_P7333126701944960589_nat_o: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
list_P6285523579766656935_o_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
list_P3795440434834930179_o_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
set_list_VEBT_VEBT: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
produc334124729049499915VEBT_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
produc2504756804600209347T_VEBT: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
produc6271795597528267376eger_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
product_prod_num_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
product_prod_nat_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
list_P4002435161011370285od_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
set_list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
list_VEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
product_prod_nat_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
product_prod_o_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
product_prod_o_int: $tType ).
thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
set_VEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
set_Code_integer: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
set_list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
option_num: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
set_char: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__VEBT____Definitions__OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Code____Numeral__Ointeger,type,
code_integer: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (984)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
archim3151403230148437115or_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
archim7778729529865785530nd_rat: rat > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
gbinomial_complex: complex > nat > complex ).
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
gbinomial_rat: rat > nat > rat ).
thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
gbinomial_real: real > nat > real ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Target__Int_Onegative,type,
code_Target_negative: num > int ).
thf(sy_c_Code__Target__Int_Opositive,type,
code_Target_positive: num > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
comple4887499456419720421f_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
complete_Sup_Sup_int: set_int > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Complex_Orcis,type,
rcis: real > real > complex ).
thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: product_prod_int_int > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
semiri5044797733671781792omplex: nat > complex ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
semiri1406184849735516958ct_int: nat > int ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
semiri2265585572941072030t_real: nat > real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
invers8013647133539491842omplex: complex > complex ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
inverse_inverse_rat: rat > rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
at_bot_real: filter_real ).
thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
at_top_nat: filter_nat ).
thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
at_top_real: filter_real ).
thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
filtermap_real_real: ( real > real ) > filter_real > filter_real ).
thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
thf(sy_c_Finite__Set_Ocard_001_Eo,type,
finite_card_o: set_o > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
finite_card_complex: set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
finite_card_char: set_char > nat ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
finite_finite_list_o: set_list_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
finite3922522038869484883st_int: set_list_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
finite_finite_num: set_num > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
finite_finite_rat: set_rat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(sy_c_Fun_Oid_001_Eo,type,
id_o: $o > $o ).
thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
id_nat: nat > nat ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
gcd_Gcd_int: set_int > int ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
gcd_gcd_int: int > int > int ).
thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
gcd_gcd_nat: nat > nat > nat ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
abs_abs_complex: complex > complex ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
abs_abs_rat: rat > rat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
minus_minus_rat: rat > rat > rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
one_one_Code_integer: code_integer ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
sgn_sgn_complex: complex > complex ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
sgn_sgn_int: int > int ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
sgn_sgn_rat: rat > rat ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
sgn_sgn_real: real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
uminus_uminus_rat: rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
zero_z3403309356797280102nteger: code_integer ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
groups6621422865394947399nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Rat__Orat,type,
groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Rat__Orat,type,
groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Complex__Ocomplex,type,
groups6381953495645901045omplex: ( product_prod_nat_nat > complex ) > set_Pr1261947904930325089at_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Int__Oint,type,
groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Nat__Onat,type,
groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
groups2316167850115554303t_real: ( int > real ) > set_int > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Complex__Ocomplex,type,
groups8110221916422527690omplex: ( product_prod_nat_nat > complex ) > set_Pr1261947904930325089at_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
groups6036352826371341000t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
groups4694064378042380927al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
groups1681761925125756287l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_HOL_OThe_001t__Int__Oint,type,
the_int: ( int > $o ) > int ).
thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
the_real: ( real > $o ) > real ).
thf(sy_c_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Extended____Nat__Oenat,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Num__Onum,type,
if_num: $o > num > num > num ).
thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_If_001t__Rat__Orat,type,
if_rat: $o > rat > rat > rat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: product_prod_nat_nat > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > product_prod_nat_nat ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(sy_c_Int_Ointrel,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
power_int_real: real > int > real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
ring_17405671764205052669omplex: int > complex ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
ring_1_of_int_rat: int > rat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
at_infinity_real: filter_real ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
fold_int_int: ( int > int > int ) > list_int > int > int ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(sy_c_List_Olist__update_001_Eo,type,
list_update_o: list_o > nat > $o > list_o ).
thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
list_update_complex: list_complex > nat > complex > list_complex ).
thf(sy_c_List_Olist__update_001t__Int__Oint,type,
list_update_int: list_int > nat > int > list_int ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
list_update_real: list_real > nat > real > list_real ).
thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(sy_c_List_Onth_001_Eo,type,
nth_o: list_o > nat > $o ).
thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
nth_complex: list_complex > nat > complex ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(sy_c_List_Oreplicate_001_Eo,type,
replicate_o: nat > $o > list_o ).
thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
replicate_complex: nat > complex > list_complex ).
thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
replicate_real: nat > real > list_real ).
thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
semiri681578069525770553at_rat: nat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
size_size_option_nat: option_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
size_size_option_num: option_num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: list_nat > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set_nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
neg_numeral_dbl_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
neg_numeral_dbl_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
neg_numeral_sub_int: num > num > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
numera6620942414471956472nteger: num > code_integer ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
numera6690914467698888265omplex: num > complex ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
numeral_numeral_rat: num > rat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
none_nat: option_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
none_num: option_num ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
some_num: num > option_num ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
size_option_nat: ( nat > nat ) > option_nat > nat ).
thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
size_option_num: ( num > nat ) > option_num > nat ).
thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
the_nat: option_nat > nat ).
thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
the_num: option_num > num ).
thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
bot_bot_int_int_o: int > int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
bot_bot_nat_nat_o: nat > nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
bot_bot_set_num: set_num ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
bot_bot_set_set_int: set_set_int ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
ord_Least_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_less_set_set_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
ord_le3636971675376928563eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
ord_le2124322318746777828_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
ord_le5643404153117327598_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_062_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J_J,type,
ord_le2556027599737686990_num_o: ( product_prod_num_num > product_prod_num_num > $o ) > ( product_prod_num_num > product_prod_num_num > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J,type,
ord_le2239182809043710856_num_o: ( product_prod_num_num > $o ) > ( product_prod_num_num > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
ord_max_num: num > num > num ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
ord_max_rat: rat > rat > rat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
ord_max_real: real > real > real ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
ord_max_set_int: set_int > set_int > set_int ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
order_9091379641038594480t_real: ( nat > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
top_top_set_char: set_char ).
thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
power_power_rat: rat > nat > rat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Product__Type_OPair_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
product_Pair_o_nat: $o > nat > product_prod_o_nat ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
product_Pair_nat_o: nat > $o > product_prod_nat_o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Product__Type_Omap__prod_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
produc8678311845419106900er_nat: ( code_integer > nat ) > ( code_integer > nat ) > produc8923325533196201883nteger > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
produc2592262431452330817omplex: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex ) > produc8763457246119570046nteger > set_complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Complex__Ocomplex_J,type,
produc8580519160106071146omplex: ( int > int > set_complex ) > product_prod_int_int > set_complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1656060378719767003at_nat: ( int > int > set_Pr1261947904930325089at_nat ) > product_prod_int_int > set_Pr1261947904930325089at_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: product_prod_int_int > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: product_prod_int_int > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > product_prod_int_int ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
field_5140801741446780682s_real: set_real ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
field_7254667332652039916t_real: rat > real ).
thf(sy_c_Rat_Onormalize,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod_int_int ).
thf(sy_c_Rat_Oratrel,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Real_OReal,type,
real2: ( nat > rat ) > real ).
thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Ocr__real,type,
cr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opcr__real,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $o ).
thf(sy_c_Real_Orealrel,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(sy_c_Real_Orep__real,type,
rep_real: real > nat > rat ).
thf(sy_c_Real_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
real_V2521375963428798218omplex: set_complex ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Relation_OField_001t__Nat__Onat,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
dvd_dvd_rat: rat > rat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
zero_n3304061248610475627l_real: $o > real ).
thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
suminf_complex: ( nat > complex ) > complex ).
thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
suminf_int: ( nat > int ) > int ).
thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
suminf_nat: ( nat > nat ) > nat ).
thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
summable_complex: ( nat > complex ) > $o ).
thf(sy_c_Series_Osummable_001t__Int__Oint,type,
summable_int: ( nat > int ) > $o ).
thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
summable_nat: ( nat > nat ) > $o ).
thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(sy_c_Series_Osums_001t__Int__Oint,type,
sums_int: ( nat > int ) > int > $o ).
thf(sy_c_Series_Osums_001t__Nat__Onat,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(sy_c_Series_Osums_001t__Real__Oreal,type,
sums_real: ( nat > real ) > real > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or370866239135849197et_int: set_int > set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
set_ord_atMost_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or58775011639299419et_int: set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
unique3096191561947761185of_nat: nat > char ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
cos_complex: complex > complex ).
thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
cos_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
cosh_real: real > real ).
thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
cot_real: real > real ).
thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
diffs_int: ( nat > int ) > nat > int ).
thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
diffs_rat: ( nat > rat ) > nat > rat ).
thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
diffs_real: ( nat > real ) > nat > real ).
thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
exp_complex: complex > complex ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
sin_complex: complex > complex ).
thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
sin_real: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
sinh_real: real > real ).
thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
tan_complex: complex > complex ).
thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
tan_real: real > real ).
thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
tanh_complex: complex > complex ).
thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
tanh_real: real > real ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
finite_psubset_int: set_Pr2522554150109002629et_int ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
finite_psubset_nat: set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_fChoice_001t__Real__Oreal,type,
fChoice_real: ( real > $o ) > real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(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 ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_minilow____,type,
minilow: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (10204)
thf(fact_0__092_060open_062low_Ax_A_Ideg_Adiv_A2_J_A_092_060le_062_Aminilow_092_060close_062,axiom,
ord_less_eq_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ minilow ).
% \<open>low x (deg div 2) \<le> minilow\<close>
thf(fact_1_bit__split__inv,axiom,
! [X: nat,D: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
= X ) ).
% bit_split_inv
thf(fact_2_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set_nat,X2: nat] :
( ( member_nat @ X2 @ Xs )
& ! [Y: nat] :
( ( member_nat @ Y @ Xs )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ) ).
% max_in_set_def
thf(fact_3__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% \<open>2 \<le> deg\<close>
thf(fact_4_True,axiom,
ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% True
thf(fact_5__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= na ) ).
% \<open>deg div 2 = n\<close>
thf(fact_6_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set_nat,X2: nat] :
( ( member_nat @ X2 @ Xs )
& ! [Y: nat] :
( ( member_nat @ Y @ Xs )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ).
% min_in_set_def
thf(fact_7_False,axiom,
( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat ) ).
% False
thf(fact_8_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_9_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_10_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_11_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_12_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_13_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_14__092_060open_062vebt__mint_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Aminilow_092_060close_062,axiom,
( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
= ( some_nat @ minilow ) ) ).
% \<open>vebt_mint (treeList ! high x (deg div 2)) = Some minilow\<close>
thf(fact_15_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_16_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_17_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N2: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_18_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X ) ) ).
% min_Null_member
thf(fact_19_divide__numeral__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_20_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_21_divide__numeral__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062minilow_O_Avebt__mint_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Aminilow_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Minilow: nat] :
( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= ( some_nat @ Minilow ) ) ).
% \<open>\<And>thesis. (\<And>minilow. vebt_mint (treeList ! high x (deg div 2)) = Some minilow \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23_minNullmin,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T )
=> ( ( vEBT_vebt_mint @ T )
= none_nat ) ) ).
% minNullmin
thf(fact_24_minminNull,axiom,
! [T: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T )
= none_nat )
=> ( vEBT_VEBT_minNull @ T ) ) ).
% minminNull
thf(fact_25_power__shift,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( power_power_nat @ X @ Y2 )
= Z )
= ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y2 ) )
= ( some_nat @ Z ) ) ) ).
% power_shift
thf(fact_26_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera6690914467698888265omplex @ M )
= ( numera6690914467698888265omplex @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_27_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_28_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_rat @ M )
= ( numeral_numeral_rat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_29_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_30_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_31_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_32_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_33_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_34_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_35_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_36_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_37_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_38__C5_Ohyps_C_I4_J,axiom,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% "5.hyps"(4)
thf(fact_39_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_40__C5_Ohyps_C_I10_J,axiom,
ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "5.hyps"(10)
thf(fact_41_less__shift,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% less_shift
thf(fact_42_lesseq__shift,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% lesseq_shift
thf(fact_43_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_44_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_45_i1,axiom,
( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
= none_nat )
| ~ ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% i1
thf(fact_46__092_060open_062high_Ax_An_A_060_A2_A_094_Am_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
& ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% \<open>high x n < 2 ^ m \<and> low x n < 2 ^ n\<close>
thf(fact_47__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
thf(fact_48_greater__shift,axiom,
( ord_less_nat
= ( ^ [Y: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% greater_shift
thf(fact_49_not__None__eq,axiom,
! [X: option_nat] :
( ( X != none_nat )
= ( ? [Y: nat] :
( X
= ( some_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_50_not__None__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( X != none_P5556105721700978146at_nat )
= ( ? [Y: product_prod_nat_nat] :
( X
= ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_51_not__None__eq,axiom,
! [X: option_num] :
( ( X != none_num )
= ( ? [Y: num] :
( X
= ( some_num @ Y ) ) ) ) ).
% not_None_eq
thf(fact_52_not__Some__eq,axiom,
! [X: option_nat] :
( ( ! [Y: nat] :
( X
!= ( some_nat @ Y ) ) )
= ( X = none_nat ) ) ).
% not_Some_eq
thf(fact_53_not__Some__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( ! [Y: product_prod_nat_nat] :
( X
!= ( some_P7363390416028606310at_nat @ Y ) ) )
= ( X = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
thf(fact_54_not__Some__eq,axiom,
! [X: option_num] :
( ( ! [Y: num] :
( X
!= ( some_num @ Y ) ) )
= ( X = none_num ) ) ).
% not_Some_eq
thf(fact_55_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_56_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_57_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_58_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_59_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_60_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_73_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X3: real] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_74_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_75_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_76_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_77_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_78__C5_Ohyps_C_I6_J,axiom,
( deg
= ( plus_plus_nat @ na @ m ) ) ).
% "5.hyps"(6)
thf(fact_79_mint__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some_nat @ Maxi ) )
=> ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% mint_member
thf(fact_80_option_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( some_nat @ X22 )
= ( some_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_81_option_Oinject,axiom,
! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
( ( ( some_P7363390416028606310at_nat @ X22 )
= ( some_P7363390416028606310at_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_82_option_Oinject,axiom,
! [X22: num,Y22: num] :
( ( ( some_num @ X22 )
= ( some_num @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_83__092_060open_062x_A_092_060le_062_Ama_092_060close_062,axiom,
ord_less_eq_nat @ xa @ ma ).
% \<open>x \<le> ma\<close>
thf(fact_84_member__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_vebt_member @ T @ X )
= ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% member_correct
thf(fact_85_mint__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some_nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T @ X )
=> ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% mint_corr_help
thf(fact_86_pow__sum,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% pow_sum
thf(fact_87__C5_Ohyps_C_I9_J,axiom,
ord_less_eq_nat @ mi @ ma ).
% "5.hyps"(9)
thf(fact_88_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_89__C5_Ohyps_C_I2_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "5.hyps"(2)
thf(fact_90_add__numeral__left,axiom,
! [V: num,W: num,Z: complex] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_91_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_92_add__numeral__left,axiom,
! [V: num,W: num,Z: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_93_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_94_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_95_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_96_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_97_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_98_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_99_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_100_misiz,axiom,
! [T: vEBT_VEBT,N: nat,M: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( some_nat @ M )
= ( vEBT_vebt_mint @ T ) )
=> ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% misiz
thf(fact_101_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_102_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_103_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_104_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_105__C5_Ohyps_C_I5_J,axiom,
( m
= ( suc @ na ) ) ).
% "5.hyps"(5)
thf(fact_106_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_107_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% local.power_def
thf(fact_108_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_109_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_110_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_111_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_112_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_113_numeral__Bit0,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_Bit0
thf(fact_114_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_115_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_Bit0
thf(fact_116_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_117_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_118_power__divide,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_divide
thf(fact_119_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_120_power__divide,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
= ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_divide
thf(fact_121_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_nat > $o,Y2: option_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_122_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_P5556105721700978146at_nat )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: nat,B2: product_prod_nat_nat] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y2
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_123_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_num > $o,Y2: option_num] :
( ( ( X = none_nat )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_num )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: nat,B2: num] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y2
= ( some_num @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_124_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y2: option_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: product_prod_nat_nat,B2: nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_125_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_P5556105721700978146at_nat )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y2
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_126_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y2: option_num] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_num )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: product_prod_nat_nat,B2: num] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y2
= ( some_num @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_127_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_nat > $o,Y2: option_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: num,B2: nat] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_128_combine__options__cases,axiom,
! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_P5556105721700978146at_nat )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: num,B2: product_prod_nat_nat] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y2
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_129_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_num > $o,Y2: option_num] :
( ( ( X = none_num )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2 = none_num )
=> ( P @ X @ Y2 ) )
=> ( ! [A3: num,B2: num] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y2
= ( some_num @ B2 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_130_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_131_split__option__all,axiom,
( ( ^ [P2: option4927543243414619207at_nat > $o] :
! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option4927543243414619207at_nat > $o] :
( ( P3 @ none_P5556105721700978146at_nat )
& ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_132_split__option__all,axiom,
( ( ^ [P2: option_num > $o] :
! [X4: option_num] : ( P2 @ X4 ) )
= ( ^ [P3: option_num > $o] :
( ( P3 @ none_num )
& ! [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_133_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_134_split__option__ex,axiom,
( ( ^ [P2: option4927543243414619207at_nat > $o] :
? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option4927543243414619207at_nat > $o] :
( ( P3 @ none_P5556105721700978146at_nat )
| ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_135_split__option__ex,axiom,
( ( ^ [P2: option_num > $o] :
? [X4: option_num] : ( P2 @ X4 ) )
= ( ^ [P3: option_num > $o] :
( ( P3 @ none_num )
| ? [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_136_option_Oexhaust,axiom,
! [Y2: option_nat] :
( ( Y2 != none_nat )
=> ~ ! [X23: nat] :
( Y2
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_137_option_Oexhaust,axiom,
! [Y2: option4927543243414619207at_nat] :
( ( Y2 != none_P5556105721700978146at_nat )
=> ~ ! [X23: product_prod_nat_nat] :
( Y2
!= ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_138_option_Oexhaust,axiom,
! [Y2: option_num] :
( ( Y2 != none_num )
=> ~ ! [X23: num] :
( Y2
!= ( some_num @ X23 ) ) ) ).
% option.exhaust
thf(fact_139_option_OdiscI,axiom,
! [Option: option_nat,X22: nat] :
( ( Option
= ( some_nat @ X22 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_140_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
( ( Option
= ( some_P7363390416028606310at_nat @ X22 ) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
thf(fact_141_option_OdiscI,axiom,
! [Option: option_num,X22: num] :
( ( Option
= ( some_num @ X22 ) )
=> ( Option != none_num ) ) ).
% option.discI
thf(fact_142_option_Odistinct_I1_J,axiom,
! [X22: nat] :
( none_nat
!= ( some_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_143_option_Odistinct_I1_J,axiom,
! [X22: product_prod_nat_nat] :
( none_P5556105721700978146at_nat
!= ( some_P7363390416028606310at_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_144_option_Odistinct_I1_J,axiom,
! [X22: num] :
( none_num
!= ( some_num @ X22 ) ) ).
% option.distinct(1)
thf(fact_145_mint__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
=> ( ( vEBT_vebt_mint @ T )
= ( some_nat @ X ) ) ) ) ).
% mint_sound
thf(fact_146_mint__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some_nat @ X ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% mint_corr
thf(fact_147_post__member__pre__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y2 )
=> ( ( vEBT_vebt_member @ T @ Y2 )
| ( X = Y2 ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_148_add__shift,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= Z )
= ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y2 ) )
= ( some_nat @ Z ) ) ) ).
% add_shift
thf(fact_149_maxt__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T @ X )
=> ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_150__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
( ( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
& ( vEBT_invar_vebt @ summary @ m ) ) ).
% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
thf(fact_151_maxt__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ Maxi ) )
=> ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% maxt_member
thf(fact_152__C5_Ohyps_C_I11_J,axiom,
( ( mi != ma )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ na )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
=> ( ( ord_less_nat @ mi @ X5 )
& ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% "5.hyps"(11)
thf(fact_153_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_154_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_155_field__less__half__sum,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_156_field__less__half__sum,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_157_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% bit_concat_def
thf(fact_158_low__inv,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_159_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus_nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_160_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% not_min_Null_member
thf(fact_161_valid__member__both__member__options,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_vebt_member @ T @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_162_both__member__options__equiv__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
= ( vEBT_vebt_member @ T @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_163_maxbmo,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% maxbmo
thf(fact_164_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% add_def
thf(fact_165_set__vebt__set__vebt_H__valid,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_set_vebt @ T )
= ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_166_maxt__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% maxt_corr
thf(fact_167_maxt__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
=> ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) ) ) ) ).
% maxt_sound
thf(fact_168_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_169_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_170_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_171_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_172_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_173_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_174_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_175_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_176_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_177_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_178_high__inv,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= Y2 ) ) ).
% high_inv
thf(fact_179_valid__insert__both__member__options__pres,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y2 ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_180_valid__insert__both__member__options__add,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_181_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_182_distrib__left__numeral,axiom,
! [V: num,B: complex,C: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_183_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_184_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_185_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_186_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_187_distrib__right__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_188_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_189_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_190_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_191_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_192_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_193_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_194_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_195_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_196_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_197_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_198_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_199_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_200_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_201_power__add__numeral,axiom,
! [A: complex,M: num,N: num] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_202_power__add__numeral,axiom,
! [A: real,M: num,N: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_203_power__add__numeral,axiom,
! [A: rat,M: num,N: num] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_204_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_205_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_206_power__add__numeral2,axiom,
! [A: complex,M: num,N: num,B: complex] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_207_power__add__numeral2,axiom,
! [A: real,M: num,N: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_208_power__add__numeral2,axiom,
! [A: rat,M: num,N: num,B: rat] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_209_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_210_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_211_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_212_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_213_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_214_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_215_pred__member,axiom,
! [T: vEBT_VEBT,X: nat,Y2: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y2 )
= ( ( vEBT_vebt_member @ T @ Y2 )
& ( ord_less_nat @ Y2 @ X )
& ! [Z2: nat] :
( ( ( vEBT_vebt_member @ T @ Z2 )
& ( ord_less_nat @ Z2 @ X ) )
=> ( ord_less_eq_nat @ Z2 @ Y2 ) ) ) ) ).
% pred_member
thf(fact_216__C5_Ohyps_C_I7_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% "5.hyps"(7)
thf(fact_217_power__Suc2,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_218_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_219_power__Suc2,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_220_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_221_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_222_power__Suc,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_Suc
thf(fact_223_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_224_power__Suc,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_Suc
thf(fact_225_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_226_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_227_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_228_power__commutes,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_commutes
thf(fact_229_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_230_power__commutes,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_commutes
thf(fact_231_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_232_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_233_power__mult__distrib,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_234_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_235_power__mult__distrib,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_236_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_237_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_238_power__commuting__commutes,axiom,
! [X: complex,Y2: complex,N: nat] :
( ( ( times_times_complex @ X @ Y2 )
= ( times_times_complex @ Y2 @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y2 )
= ( times_times_complex @ Y2 @ ( power_power_complex @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_239_power__commuting__commutes,axiom,
! [X: real,Y2: real,N: nat] :
( ( ( times_times_real @ X @ Y2 )
= ( times_times_real @ Y2 @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y2 )
= ( times_times_real @ Y2 @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_240_power__commuting__commutes,axiom,
! [X: rat,Y2: rat,N: nat] :
( ( ( times_times_rat @ X @ Y2 )
= ( times_times_rat @ Y2 @ X ) )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y2 )
= ( times_times_rat @ Y2 @ ( power_power_rat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_241_power__commuting__commutes,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= ( times_times_nat @ Y2 @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y2 )
= ( times_times_nat @ Y2 @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_242_power__commuting__commutes,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= ( times_times_int @ Y2 @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y2 )
= ( times_times_int @ Y2 @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_243_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_244_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_245_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_246_power__mult,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_247_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_248_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_249_power__odd__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_250_power__odd__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_251_power__odd__eq,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_252_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_253_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_254_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_255_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_256_mult__numeral__1__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_257_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_258_mult__numeral__1__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_259_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_260_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_261_mult__numeral__1,axiom,
! [A: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_262_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_263_mult__numeral__1,axiom,
! [A: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_264_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_265_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_266_power__add,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_add
thf(fact_267_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_268_power__add,axiom,
! [A: rat,M: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_269_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_270_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_271_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_272_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_273_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_274_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_275_left__add__twice,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_276_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_277_left__add__twice,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_278_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_279_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_280_mult__2__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ Z @ Z ) ) ).
% mult_2_right
thf(fact_281_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_282_mult__2__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_283_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_284_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_285_mult__2,axiom,
! [Z: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_complex @ Z @ Z ) ) ).
% mult_2
thf(fact_286_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_287_mult__2,axiom,
! [Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2
thf(fact_288_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_289_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_290_power2__eq__square,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_complex @ A @ A ) ) ).
% power2_eq_square
thf(fact_291_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_292_power2__eq__square,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ A @ A ) ) ).
% power2_eq_square
thf(fact_293_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_294_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_295_power4__eq__xxxx,axiom,
! [X: complex] :
( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_296_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_297_power4__eq__xxxx,axiom,
! [X: rat] :
( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_298_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_299_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_300_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_301_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_302_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_303_power__even__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_304_power2__sum,axiom,
! [X: complex,Y2: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_305_power2__sum,axiom,
! [X: real,Y2: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_306_power2__sum,axiom,
! [X: rat,Y2: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_307_power2__sum,axiom,
! [X: nat,Y2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_308_power2__sum,axiom,
! [X: int,Y2: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_309_field__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_310_field__sum__of__halves,axiom,
! [X: rat] :
( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_311_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_312_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% mul_def
thf(fact_313_mul__shift,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( times_times_nat @ X @ Y2 )
= Z )
= ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y2 ) )
= ( some_nat @ Z ) ) ) ).
% mul_shift
thf(fact_314_sum__squares__bound,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y2 ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_315_sum__squares__bound,axiom,
! [X: rat,Y2: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y2 ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_316_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_317_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_318_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_319_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_320_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_321_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_322_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_323_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_324_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_325_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S ) ) ) ).
% deg_SUcn_Node
thf(fact_326_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_327_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_328_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_329_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_330_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_331_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_332_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_333_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_334_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_335_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_336_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_337_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_338_power__mult__numeral,axiom,
! [A: complex,M: num,N: num] :
( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_339__C5_Ohyps_C_I8_J,axiom,
( ( mi = ma )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% "5.hyps"(8)
thf(fact_340_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_341_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_342_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L2: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_343_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L2: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_344_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_345_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_346_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_347_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_348_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_349_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_350_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_351_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_352_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_353_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_354_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_355_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_356_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_357_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_358_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_359_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_360_size__neq__size__imp__neq,axiom,
! [X: list_VEBT_VEBT,Y2: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ X )
!= ( size_s6755466524823107622T_VEBT @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_361_size__neq__size__imp__neq,axiom,
! [X: list_o,Y2: list_o] :
( ( ( size_size_list_o @ X )
!= ( size_size_list_o @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_362_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_363_size__neq__size__imp__neq,axiom,
! [X: list_int,Y2: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_364_size__neq__size__imp__neq,axiom,
! [X: num,Y2: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_365_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set_nat,X2: nat,Y: nat] :
( ( member_nat @ Y @ Xs )
& ( ord_less_nat @ Y @ X2 )
& ! [Z2: nat] :
( ( member_nat @ Z2 @ Xs )
=> ( ( ord_less_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_366_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_367_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_368_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_369_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_370_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_371_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_372_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_373_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_374_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_375_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_376_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_377_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_378_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_379_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_380_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_381_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_382_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_383_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_384_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_385_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_386_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_387_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_388_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_389_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_390_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_391_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_392_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_393_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_394_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_395_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_396_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
& ( M6 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_397_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_398_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
| ( M6 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_399_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_400_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_401_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_402_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_403_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_404_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_405_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_406_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_407_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_408_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_409_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_410_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_411_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_412_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_413_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_414_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_415_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_416_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_417_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_418_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_419_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_420_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_421_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_422_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_423_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_424_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_425_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_426_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_427_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_428_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_429_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_430_lift__Suc__mono__less,axiom,
! [F: nat > rat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_431_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_432_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_433_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_434_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_435_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_436_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_437_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_438_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_439_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_440_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_441_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_442_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_443_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_444_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_445_lift__Suc__mono__le,axiom,
! [F: nat > rat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_446_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_447_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_448_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_449_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_450_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_451_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_452_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_453_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_454_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_455_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_456_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_457_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_458_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_459_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_460_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_461_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_462_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_463_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_464_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_465_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_466_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_467_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less_nat @ X @ Ma )
& ( ord_less_nat @ Mi @ X )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_468_thisvalid,axiom,
vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% thisvalid
thf(fact_469__C5_Ohyps_C_I3_J,axiom,
! [X: nat,Px: nat] :
( ( ( vEBT_vebt_pred @ summary @ X )
= ( some_nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X @ Px ) ) ).
% "5.hyps"(3)
thf(fact_470_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
& ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_471_times__divide__eq__left,axiom,
! [B: complex,C: complex,A: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_472_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_473_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_474_divide__divide__eq__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_475_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_476_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_477_divide__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_478_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_479_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_480_times__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_481_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_482_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_483_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_484_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_485_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_486_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_487_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_488_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_489_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_490_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_491_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_492_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_493_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_494_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_495_inthall,axiom,
! [Xs2: list_real,P: real > $o,N: nat] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
=> ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_496_inthall,axiom,
! [Xs2: list_complex,P: complex > $o,N: nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_497_inthall,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_498_inthall,axiom,
! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_499_inthall,axiom,
! [Xs2: list_o,P: $o > $o,N: nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_500_inthall,axiom,
! [Xs2: list_nat,P: nat > $o,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_501_inthall,axiom,
! [Xs2: list_int,P: int > $o,N: nat] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_502_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_503_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_504_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_505_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_506_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_507_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_508_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_509_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_510_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_511__C5_Ohyps_C_I1_J,axiom,
! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ( ( vEBT_invar_vebt @ X5 @ na )
& ! [Xa: nat,Xb: nat] :
( ( ( vEBT_vebt_pred @ X5 @ Xa )
= ( some_nat @ Xb ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa @ Xb ) ) ) ) ).
% "5.hyps"(1)
thf(fact_512_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_513_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_514_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_515_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_516_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_517_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_518_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( some_nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_519_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma )
=> ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= none_nat ) ) ) ) ).
% pred_list_to_short
thf(fact_520_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_521_complete__real,axiom,
! [S3: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S3 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S3 )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_522_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
= none_nat ) ).
% vebt_pred.simps(4)
thf(fact_523_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_524_linordered__field__no__ub,axiom,
! [X5: rat] :
? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_525_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_526_linordered__field__no__lb,axiom,
! [X5: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_527_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_528_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_529_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_530_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_531_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_532_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_533_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_534_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_535_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_536_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A4: rat,B3: rat] : ( times_times_rat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_537_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_538_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_539_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_540_mult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_541_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_542_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_543_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_544_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_545_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_546_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_547_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_548_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_rat @ I2 @ K )
= ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_549_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_550_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_551_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_552_group__cancel_Oadd1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_553_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_554_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_555_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_556_group__cancel_Oadd2,axiom,
! [B4: rat,K: rat,B: rat,A: rat] :
( ( B4
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B4 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_557_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_558_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_559_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_560_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.assoc
thf(fact_561_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_562_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_563_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_564_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_565_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_566_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_567_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_568_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_569_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_570_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_571_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_572_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_573_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_574_add_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_575_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_576_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_577_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_578_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_579_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_580_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_581_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_582_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_583_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_584_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_585_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_586_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_587_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_588_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_589_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( I2 = J )
& ( ord_less_eq_real @ K @ L2 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_590_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( I2 = J )
& ( ord_less_eq_rat @ K @ L2 ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_591_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_592_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L2 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_593_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L2 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_594_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_eq_rat @ K @ L2 ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_595_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_596_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L2 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_597_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_598_add__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_599_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_600_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_601_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_602_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_603_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_604_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_605_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_606_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_607_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_608_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_609_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_610_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_611_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_612_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_613_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_614_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_615_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_616_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_617_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_618_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_619_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_620_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_621_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_622_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_623_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_624_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_625_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_626_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_627_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_628_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_629_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_630_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_631_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_632_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_633_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_634_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_635_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_636_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_637_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_638_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_639_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_640_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_641_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_642_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_643_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( I2 = J )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_644_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( I2 = J )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_645_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_646_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_647_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_648_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_649_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_650_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_651_divide__divide__eq__left_H,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_652_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_653_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_654_divide__divide__times__eq,axiom,
! [X: complex,Y2: complex,Z: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y2 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_655_divide__divide__times__eq,axiom,
! [X: real,Y2: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_656_divide__divide__times__eq,axiom,
! [X: rat,Y2: rat,Z: rat,W: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_657_times__divide__times__eq,axiom,
! [X: complex,Y2: complex,Z: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y2 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_658_times__divide__times__eq,axiom,
! [X: real,Y2: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_659_times__divide__times__eq,axiom,
! [X: rat,Y2: rat,Z: rat,W: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_660_add__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_661_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_662_add__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_663_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_664_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_665_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_666_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_667_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_668_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_669_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_670_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_671_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_672_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_eq_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_673_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_674_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_675_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_676_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_677_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_678_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_679__C2_C,axiom,
( ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
=> ( ( ( ord_less_nat @ mi @ xa )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( some_nat @ mi ) ) )
& ( ~ ( ord_less_nat @ mi @ xa )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= none_nat ) ) ) )
& ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
!= none_nat )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% "2"
thf(fact_680_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_681_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_682_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va ) ) )
=> ( ~ ( ord_less_nat @ Ma @ Mi )
=> ( ( Ma != Mi )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% nested_mint
thf(fact_683_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_684_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_685_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_686_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_687_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( Mi != Ma )
=> ( ( ord_less_nat @ Mi @ Ma )
& ? [M5: nat] :
( ( ( some_nat @ M5 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_688_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% summaxma
thf(fact_689_set__n__deg__not__0,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_690__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ one_one_nat @ na ).
% \<open>1 \<le> n\<close>
thf(fact_691_VEBT_Oinject_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_692_power__minus__is__div,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
= ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% power_minus_is_div
thf(fact_693_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_694_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_695_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_696_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_697_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_698_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_699_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_700_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_701_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_702_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_703_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_704_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_705_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_706_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_707_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_708_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_709_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_710_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_711_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_712_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_713_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_714_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_715_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_716_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_717_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_718_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_719_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_720_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_721_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_722_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_723_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_724_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_725_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_726_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_727_power__one,axiom,
! [N: nat] :
( ( power_power_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_one
thf(fact_728_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_729_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_730_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_731_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_732_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_733_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_734_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_735_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_736_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_737_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_738_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_739_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_740_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_741_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_742_right__diff__distrib__numeral,axiom,
! [V: num,B: complex,C: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_743_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_744_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_745_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_746_left__diff__distrib__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_747_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_748_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_749_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_750_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6690914467698888265omplex @ N )
= one_one_complex )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_751_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_752_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_rat @ N )
= one_one_rat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_753_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_754_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_755_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_756_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_757_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_758_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_759_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_760_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_761_power__inject__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ( power_power_rat @ A @ M )
= ( power_power_rat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_762_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_763_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_764_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_765_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_766_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_767_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_768_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_769_option_Ocollapse,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_770_option_Ocollapse,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( ( some_num @ ( the_num @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_771_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_772_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y2: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_773_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_774_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_775_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_776_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_777_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_778_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_779_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_780_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_781_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_782_power__increasing__iff,axiom,
! [B: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_783_power__increasing__iff,axiom,
! [B: rat,X: nat,Y2: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_784_power__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_785_power__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_786_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_787_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_788_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_789_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_790_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_791_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_792_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_793_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_794_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_795_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_796_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_797_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_798_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_799_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_800_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_801_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_802_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_803_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_804_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_805_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_806_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_807_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_808_diff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_809_diff__right__commute,axiom,
! [A: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_810_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_811_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_812_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_813_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_814_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_815_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_816_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_817_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_818_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_819_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_820_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_eq_rat @ A @ B )
= ( ord_less_eq_rat @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_821_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_822_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_823_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_824_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_825_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_826_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_827_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_828_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_829_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ D @ C )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_830_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_831_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_832_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ D @ C )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_833_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_834_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_835_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_rat @ A @ B )
= ( ord_less_rat @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_836_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_837_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_838_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_839_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_840_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_841_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_842_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_843_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_844_add__diff__add,axiom,
! [A: rat,C: rat,B: rat,D: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% add_diff_add
thf(fact_845_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_846_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_847_diff__diff__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_848_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_849_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_850_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_851_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( ( plus_plus_rat @ C @ B )
= A )
=> ( C
= ( minus_minus_rat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_852_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_853_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_854_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_855_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_856_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_857_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_858_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_859_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_860_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_861_diff__diff__eq2,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_862_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_863_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_864_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_865_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_866_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_867_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A
= ( minus_minus_rat @ C @ B ) )
= ( ( plus_plus_rat @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_868_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_869_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_870_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( minus_minus_rat @ A @ B )
= C )
= ( A
= ( plus_plus_rat @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_871_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_872_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_873_group__cancel_Osub1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( minus_minus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_874_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_875_diff__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_876_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_877_diff__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_878_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_879_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_880_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_881_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_882_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_883_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_884_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_885_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_886_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_887_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_888_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_889_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_890_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_891_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_892_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_893_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_894_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_895_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_896_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_897_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_898_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_899_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_900_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_901_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_902_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_903_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_904_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_905_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_906_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_907_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_908_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_909_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_910_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_911_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_912_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_913_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_914_option_Osel,axiom,
! [X22: nat] :
( ( the_nat @ ( some_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_915_option_Osel,axiom,
! [X22: product_prod_nat_nat] :
( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_916_option_Osel,axiom,
! [X22: num] :
( ( the_num @ ( some_num @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_917_option_Oexpand,axiom,
! [Option: option_nat,Option2: option_nat] :
( ( ( Option = none_nat )
= ( Option2 = none_nat ) )
=> ( ( ( Option != none_nat )
=> ( ( Option2 != none_nat )
=> ( ( the_nat @ Option )
= ( the_nat @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_918_option_Oexpand,axiom,
! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
( ( ( Option = none_P5556105721700978146at_nat )
= ( Option2 = none_P5556105721700978146at_nat ) )
=> ( ( ( Option != none_P5556105721700978146at_nat )
=> ( ( Option2 != none_P5556105721700978146at_nat )
=> ( ( the_Pr8591224930841456533at_nat @ Option )
= ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_919_option_Oexpand,axiom,
! [Option: option_num,Option2: option_num] :
( ( ( Option = none_num )
= ( Option2 = none_num ) )
=> ( ( ( Option != none_num )
=> ( ( Option2 != none_num )
=> ( ( the_num @ Option )
= ( the_num @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_920_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_921_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_922_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_923_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_924_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_925_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_926_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_927_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_928_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_929_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_930_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_931_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_932_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_933_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_934_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_935_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_936_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_937_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_938_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_939_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_940_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_941_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_942_mult__diff__mult,axiom,
! [X: real,Y2: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y2 ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y2 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_943_mult__diff__mult,axiom,
! [X: rat,Y2: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ Y2 ) @ ( times_times_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y2 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_944_mult__diff__mult,axiom,
! [X: int,Y2: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y2 ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y2 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_945_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_946_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_947_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_948_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_949_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_950_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_951_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_952_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_953_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_954_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_955_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_956_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_957_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_958_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_959_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_960_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_961_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_962_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_963_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_964_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_965_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% one_plus_numeral_commute
thf(fact_966_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_967_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% one_plus_numeral_commute
thf(fact_968_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_969_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_970_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_971_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_972_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_973_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_974_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_975_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_976_one__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% one_le_power
thf(fact_977_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_978_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_979_left__right__inverse__power,axiom,
! [X: complex,Y2: complex,N: nat] :
( ( ( times_times_complex @ X @ Y2 )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y2 @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_980_left__right__inverse__power,axiom,
! [X: real,Y2: real,N: nat] :
( ( ( times_times_real @ X @ Y2 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_981_left__right__inverse__power,axiom,
! [X: rat,Y2: rat,N: nat] :
( ( ( times_times_rat @ X @ Y2 )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y2 @ N ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_982_left__right__inverse__power,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_983_left__right__inverse__power,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_984_power__one__over,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% power_one_over
thf(fact_985_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_986_power__one__over,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_one_over
thf(fact_987_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_988_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_989_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_990_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_991_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_992_option_Oexhaust__sel,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( Option
= ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_993_option_Oexhaust__sel,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( Option
= ( some_num @ ( the_num @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_994_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_995_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_996_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_997_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_998_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_999_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1000_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1001_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_1002_gt__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% gt_half_sum
thf(fact_1003_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_1004_less__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_1005_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1006_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1007_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1008_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1009_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1010_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1011_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1012_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1013_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1014_power__gt1,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1015_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1016_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1017_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1018_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1019_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1020_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1021_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1022_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1023_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1024_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1025_power__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1026_power__increasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1027_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1028_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1029_power2__commute,axiom,
! [X: complex,Y2: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ ( minus_minus_complex @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1030_power2__commute,axiom,
! [X: real,Y2: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1031_power2__commute,axiom,
! [X: rat,Y2: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ ( minus_minus_rat @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1032_power2__commute,axiom,
! [X: int,Y2: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1033_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1034_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1035_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1036_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1037_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1038_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1039_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_1040_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_1041_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_1042_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_1043_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex ) ).
% one_power2
thf(fact_1044_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1045_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1046_power2__diff,axiom,
! [X: complex,Y2: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_diff
thf(fact_1047_power2__diff,axiom,
! [X: real,Y2: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_diff
thf(fact_1048_power2__diff,axiom,
! [X: rat,Y2: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_diff
thf(fact_1049_power2__diff,axiom,
! [X: int,Y2: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_diff
thf(fact_1050_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1051_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1052_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1053_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1054_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1055_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1056_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1057_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1058_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1059_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1060_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_1061__C1_C,axiom,
( ( ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
& ( ~ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( if_option_nat
@ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ mi @ xa ) @ ( some_nat @ mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% "1"
thf(fact_1062_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some_nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_1063_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some_nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_1064_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_1065_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1066_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_1067_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1068_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1069_vebt__maxt_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
= none_nat ) ).
% vebt_maxt.simps(2)
thf(fact_1070_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma )
=> ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_1071_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ).
% pred_lesseq_max
thf(fact_1072_vebt__mint_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
= none_nat ) ).
% vebt_mint.simps(2)
thf(fact_1073_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% set_vebt'_def
thf(fact_1074_lambda__one,axiom,
( ( ^ [X2: complex] : X2 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_1075_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_1076_lambda__one,axiom,
( ( ^ [X2: rat] : X2 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_1077_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1078_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_1079_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_code(2)
thf(fact_1080_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_1081_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_1082_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_1083_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_1084_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% set_vebt_def
thf(fact_1085_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y2: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z @ Y2 )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y2 @ Z ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y2 ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_1086_power__numeral__even,axiom,
! [Z: complex,W: num] :
( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_1087_power__numeral__even,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_1088_power__numeral__even,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_1089_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_1090_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_1091_real__arch__pow,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1092_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1093_linorder__neqE__linordered__idom,axiom,
! [X: rat,Y2: rat] :
( ( X != Y2 )
=> ( ~ ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1094_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1095_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1096_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1097_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1098_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1099_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1100_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1101_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1102_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1103_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1104_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1105_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1106_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_1107_distrib__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1108_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1109_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_1110_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_1111_distrib__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1112_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1113_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_1114_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1115_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1116_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1117_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1118_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1119_left__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1120_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1121_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1122_right__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1123_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1124_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1125_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
= ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1126_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1127_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1128_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1129_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1130_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1131_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1132_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
= ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1133_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: num > num > num,A: num,B: num] :
( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
= ( some_num @ ( F @ A @ B ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1134_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: nat > nat > nat,A: nat,B: nat] :
( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
= ( some_nat @ ( F @ A @ B ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1135_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
= none_P5556105721700978146at_nat ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1136_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu: num > num > num,Uv: option_num] :
( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
= none_num ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1137_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu: nat > nat > nat,Uv: option_nat] :
( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
= none_nat ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1138_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1139_less__1__mult,axiom,
! [M: rat,N: rat] :
( ( ord_less_rat @ one_one_rat @ M )
=> ( ( ord_less_rat @ one_one_rat @ N )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1140_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1141_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1142_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1143_add__le__add__imp__diff__le,axiom,
! [I2: rat,K: rat,N: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1144_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1145_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1146_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1147_add__le__imp__le__diff,axiom,
! [I2: rat,K: rat,N: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
=> ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1148_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1149_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1150_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1151_add__mono1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_1152_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1153_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1154_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1155_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_1156_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1157_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1158_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1159_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ ( ord_less_rat @ A @ B )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1160_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1161_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1162_square__diff__square__factored,axiom,
! [X: real,Y2: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y2 ) @ ( minus_minus_real @ X @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_1163_square__diff__square__factored,axiom,
! [X: rat,Y2: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y2 @ Y2 ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y2 ) @ ( minus_minus_rat @ X @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_1164_square__diff__square__factored,axiom,
! [X: int,Y2: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y2 ) @ ( minus_minus_int @ X @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_1165_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1166_eq__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( C
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1167_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1168_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1169_eq__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1170_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1171_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( some_nat @ Ma ) ) )
& ( ~ ( ord_less_nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_1172_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
= none_P5556105721700978146at_nat ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1173_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: num > num > num,V: num] :
( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
= none_num ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1174_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: nat > nat > nat,V: nat] :
( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
= none_nat ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1175_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y2: option4927543243414619207at_nat] :
( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
= Y2 )
=> ( ( ( Xa2 = none_P5556105721700978146at_nat )
=> ( Y2 != none_P5556105721700978146at_nat ) )
=> ( ( ? [V2: product_prod_nat_nat] :
( Xa2
= ( some_P7363390416028606310at_nat @ V2 ) )
=> ( ( Xb2 = none_P5556105721700978146at_nat )
=> ( Y2 != none_P5556105721700978146at_nat ) ) )
=> ~ ! [A3: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ! [B2: product_prod_nat_nat] :
( ( Xb2
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( Y2
!= ( some_P7363390416028606310at_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1176_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y2: option_num] :
( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
= Y2 )
=> ( ( ( Xa2 = none_num )
=> ( Y2 != none_num ) )
=> ( ( ? [V2: num] :
( Xa2
= ( some_num @ V2 ) )
=> ( ( Xb2 = none_num )
=> ( Y2 != none_num ) ) )
=> ~ ! [A3: num] :
( ( Xa2
= ( some_num @ A3 ) )
=> ! [B2: num] :
( ( Xb2
= ( some_num @ B2 ) )
=> ( Y2
!= ( some_num @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1177_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y2: option_nat] :
( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
= Y2 )
=> ( ( ( Xa2 = none_nat )
=> ( Y2 != none_nat ) )
=> ( ( ? [V2: nat] :
( Xa2
= ( some_nat @ V2 ) )
=> ( ( Xb2 = none_nat )
=> ( Y2 != none_nat ) ) )
=> ~ ! [A3: nat] :
( ( Xa2
= ( some_nat @ A3 ) )
=> ! [B2: nat] :
( ( Xb2
= ( some_nat @ B2 ) )
=> ( Y2
!= ( some_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1178_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1179_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1180_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1181_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1182_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1183_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1184_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1185_less__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1186_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1187_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1188_less__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1189_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1190_square__diff__one__factored,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_1191_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_1192_square__diff__one__factored,axiom,
! [X: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% square_diff_one_factored
thf(fact_1193_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1194_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less_nat @ X @ Mi )
& ( ~ ( ord_less_nat @ X @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ X )
& ( ~ ( ord_less_nat @ Ma @ X )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_1195_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_1196_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_1197_set__conv__nth,axiom,
( set_complex2
= ( ^ [Xs: list_complex] :
( collect_complex
@ ^ [Uu2: complex] :
? [I3: nat] :
( ( Uu2
= ( nth_complex @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1198_set__conv__nth,axiom,
( set_real2
= ( ^ [Xs: list_real] :
( collect_real
@ ^ [Uu2: real] :
? [I3: nat] :
( ( Uu2
= ( nth_real @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1199_set__conv__nth,axiom,
( set_list_nat2
= ( ^ [Xs: list_list_nat] :
( collect_list_nat
@ ^ [Uu2: list_nat] :
? [I3: nat] :
( ( Uu2
= ( nth_list_nat @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1200_set__conv__nth,axiom,
( set_VEBT_VEBT2
= ( ^ [Xs: list_VEBT_VEBT] :
( collect_VEBT_VEBT
@ ^ [Uu2: vEBT_VEBT] :
? [I3: nat] :
( ( Uu2
= ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1201_set__conv__nth,axiom,
( set_o2
= ( ^ [Xs: list_o] :
( collect_o
@ ^ [Uu2: $o] :
? [I3: nat] :
( ( Uu2
= ( nth_o @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1202_set__conv__nth,axiom,
( set_nat2
= ( ^ [Xs: list_nat] :
( collect_nat
@ ^ [Uu2: nat] :
? [I3: nat] :
( ( Uu2
= ( nth_nat @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1203_set__conv__nth,axiom,
( set_int2
= ( ^ [Xs: list_int] :
( collect_int
@ ^ [Uu2: int] :
? [I3: nat] :
( ( Uu2
= ( nth_int @ Xs @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1204_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1205_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S2 ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1206_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1207_divmod__step__eq,axiom,
! [L2: num,R2: nat,Q2: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_1208_divmod__step__eq,axiom,
! [L2: num,R2: int,Q2: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L2 ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_1209_divmod__step__eq,axiom,
! [L2: num,R2: code_integer,Q2: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_1210_maxt__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= none_nat )
=> ( ( vEBT_VEBT_set_vebt @ T )
= bot_bot_set_nat ) ) ) ).
% maxt_corr_help_empty
thf(fact_1211_mint__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= none_nat )
=> ( ( vEBT_VEBT_set_vebt @ T )
= bot_bot_set_nat ) ) ) ).
% mint_corr_help_empty
thf(fact_1212_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T2: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ T2 @ X2 )
| ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% both_member_options_def
thf(fact_1213_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1214_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_1215_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc8306885398267862888on_nat] :
( ! [Uu3: nat > nat > nat,Uv2: option_nat] :
( X
!= ( produc8929957630744042906on_nat @ Uu3 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
=> ( ! [Uw2: nat > nat > nat,V2: nat] :
( X
!= ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
=> ~ ! [F2: nat > nat > nat,A3: nat,B2: nat] :
( X
!= ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1216_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc5542196010084753463at_nat] :
( ! [Uu3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
( X
!= ( produc2899441246263362727at_nat @ Uu3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
=> ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
( X
!= ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
=> ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
( X
!= ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1217_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc1193250871479095198on_num] :
( ! [Uu3: num > num > num,Uv2: option_num] :
( X
!= ( produc5778274026573060048on_num @ Uu3 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
=> ( ! [Uw2: num > num > num,V2: num] :
( X
!= ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
=> ~ ! [F2: num > num > num,A3: num,B2: num] :
( X
!= ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1218_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc2233624965454879586on_nat] :
( ! [Uu3: nat > nat > $o,Uv2: option_nat] :
( X
!= ( produc4035269172776083154on_nat @ Uu3 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
=> ( ! [Uw2: nat > nat > $o,V2: nat] :
( X
!= ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
=> ~ ! [F2: nat > nat > $o,X3: nat,Y3: nat] :
( X
!= ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1219_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc5491161045314408544at_nat] :
( ! [Uu3: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
( X
!= ( produc3994169339658061776at_nat @ Uu3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
=> ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
( X
!= ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
=> ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( X
!= ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1220_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc7036089656553540234on_num] :
( ! [Uu3: num > num > $o,Uv2: option_num] :
( X
!= ( produc3576312749637752826on_num @ Uu3 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
=> ( ! [Uw2: num > num > $o,V2: num] :
( X
!= ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
=> ~ ! [F2: num > num > $o,X3: num,Y3: num] :
( X
!= ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1221_subset__code_I1_J,axiom,
! [Xs2: list_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B4 )
= ( ! [X2: real] :
( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
=> ( member_real @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1222_subset__code_I1_J,axiom,
! [Xs2: list_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B4 )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
=> ( member_complex @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1223_subset__code_I1_J,axiom,
! [Xs2: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B4 )
= ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1224_subset__code_I1_J,axiom,
! [Xs2: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B4 )
= ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( member_VEBT_VEBT @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1225_subset__code_I1_J,axiom,
! [Xs2: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1226_subset__code_I1_J,axiom,
! [Xs2: list_int,B4: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B4 )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
=> ( member_int @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1227_neq__if__length__neq,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
!= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1228_neq__if__length__neq,axiom,
! [Xs2: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs2 )
!= ( size_size_list_o @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1229_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1230_neq__if__length__neq,axiom,
! [Xs2: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs2 )
!= ( size_size_list_int @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1231_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1232_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_o] :
( ( size_size_list_o @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1233_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1234_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_int] :
( ( size_size_list_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1235_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_1236_mult__commute__abs,axiom,
! [C: rat] :
( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
= ( times_times_rat @ C ) ) ).
% mult_commute_abs
thf(fact_1237_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_1238_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_1239_length__induct,axiom,
! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
( ! [Xs3: list_VEBT_VEBT] :
( ! [Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1240_length__induct,axiom,
! [P: list_o > $o,Xs2: list_o] :
( ! [Xs3: list_o] :
( ! [Ys2: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1241_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1242_length__induct,axiom,
! [P: list_int > $o,Xs2: list_int] :
( ! [Xs3: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1243_nth__equalityI,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
= ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1244_nth__equalityI,axiom,
! [Xs2: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ Xs2 @ I4 )
= ( nth_o @ Ys @ I4 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1245_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I4 )
= ( nth_nat @ Ys @ I4 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1246_nth__equalityI,axiom,
! [Xs2: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I4 )
= ( nth_int @ Ys @ I4 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1247_Skolem__list__nth,axiom,
! [K: nat,P: nat > vEBT_VEBT > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1248_Skolem__list__nth,axiom,
! [K: nat,P: nat > $o > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: $o] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs: list_o] :
( ( ( size_size_list_o @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1249_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: nat] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1250_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: int] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1251_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : Y5 = Z5 )
= ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I3 )
= ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1252_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_o,Z5: list_o] : Y5 = Z5 )
= ( ^ [Xs: list_o,Ys3: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I3 )
= ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1253_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z5: list_nat] : Y5 = Z5 )
= ( ^ [Xs: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1254_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_int,Z5: list_int] : Y5 = Z5 )
= ( ^ [Xs: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I3 )
= ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1255_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_1256_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_1257_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_1258_discrete,axiom,
( ord_less_nat
= ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_1259_discrete,axiom,
( ord_less_int
= ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_1260_nth__mem,axiom,
! [N: nat,Xs2: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
=> ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1261_nth__mem,axiom,
! [N: nat,Xs2: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1262_nth__mem,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1263_nth__mem,axiom,
! [N: nat,Xs2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1264_nth__mem,axiom,
! [N: nat,Xs2: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1265_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1266_nth__mem,axiom,
! [N: nat,Xs2: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1267_list__ball__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1268_list__ball__nth,axiom,
! [N: nat,Xs2: list_o,P: $o > $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1269_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1270_list__ball__nth,axiom,
! [N: nat,Xs2: list_int,P: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1271_in__set__conv__nth,axiom,
! [X: real,Xs2: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
& ( ( nth_real @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1272_in__set__conv__nth,axiom,
! [X: complex,Xs2: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
& ( ( nth_complex @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1273_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1274_in__set__conv__nth,axiom,
! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
& ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1275_in__set__conv__nth,axiom,
! [X: $o,Xs2: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
& ( ( nth_o @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1276_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1277_in__set__conv__nth,axiom,
! [X: int,Xs2: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
& ( ( nth_int @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1278_all__nth__imp__all__set,axiom,
! [Xs2: list_real,P: real > $o,X: real] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
=> ( P @ ( nth_real @ Xs2 @ I4 ) ) )
=> ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1279_all__nth__imp__all__set,axiom,
! [Xs2: list_complex,P: complex > $o,X: complex] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P @ ( nth_complex @ Xs2 @ I4 ) ) )
=> ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1280_all__nth__imp__all__set,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1281_all__nth__imp__all__set,axiom,
! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) )
=> ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1282_all__nth__imp__all__set,axiom,
! [Xs2: list_o,P: $o > $o,X: $o] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
=> ( P @ ( nth_o @ Xs2 @ I4 ) ) )
=> ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1283_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I4 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1284_all__nth__imp__all__set,axiom,
! [Xs2: list_int,P: int > $o,X: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
=> ( P @ ( nth_int @ Xs2 @ I4 ) ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1285_all__set__conv__all__nth,axiom,
! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1286_all__set__conv__all__nth,axiom,
! [Xs2: list_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
=> ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1287_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1288_all__set__conv__all__nth,axiom,
! [Xs2: list_int,P: int > $o] :
( ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
=> ( P @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1289_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= bot_bot_set_nat ) ).
% buildup_gives_empty
thf(fact_1290_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1291_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: option_nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( Y2 != none_nat ) ) )
=> ( ! [A3: $o] :
( ? [Uw2: $o] :
( X
= ( vEBT_Leaf @ A3 @ Uw2 ) )
=> ( ( Xa2
= ( suc @ zero_zero_nat ) )
=> ~ ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y2 = none_nat ) ) ) ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ? [Va3: nat] :
( Xa2
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( B2
=> ( Y2
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y2 = none_nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y2 != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
=> ( Y2 != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
=> ( Y2 != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y2
= ( some_nat @ Ma2 ) ) )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y2
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_1292_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N2: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_1293_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_1294_obtain__set__pred,axiom,
! [Z: nat,X: nat,A2: set_nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
=> ( ( finite_finite_nat @ A2 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_1295_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_1296_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_nat @ X @ Mi )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_1297_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_nat @ Mi @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_1298_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_0_not
thf(fact_1299_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_tree_deg_neq_0
thf(fact_1300_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% deg_not_0
thf(fact_1301_Leaf__0__not,axiom,
! [A: $o,B: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_1302_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
= ( ? [A4: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).
% deg1Leaf
thf(fact_1303_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
=> ? [A3: $o,B2: $o] :
( T
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ).
% deg_1_Leaf
thf(fact_1304_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N = one_one_nat )
=> ? [A3: $o,B2: $o] :
( T
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).
% deg_1_Leafy
thf(fact_1305_set__vebt__finite,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_finite
thf(fact_1306_pred__none__empty,axiom,
! [Xs2: set_nat,A: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
=> ( ( finite_finite_nat @ Xs2 )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ Xs2 )
& ( ord_less_nat @ X5 @ A ) ) ) ) ).
% pred_none_empty
thf(fact_1307_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_1308_mod__mod__trivial,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_1309_mod__mod__trivial,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_1310_mod__mod__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_1311_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1312_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1313_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_1314_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1315_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_1316_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1317_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1318_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_1319_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1320_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_1321_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1322_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1323_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_1324_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1325_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_1326_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1327_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1328_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1329_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1330_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1331_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1332_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1333_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1334_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1335_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1336_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1337_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1338_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_1339_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_1340_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_1341_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1342_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1343_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_1344_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_1345_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1346_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_1347_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_1348_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_1349_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1350_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1351_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_1352_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_1353_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_1354_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1355_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1356_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_1357_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_1358_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_1359_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1360_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1361_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_1362_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_1363_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_1364_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1365_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1366_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1367_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1368_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_1369_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_1370_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_1371_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1372_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1373_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1374_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1375_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1376_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1377_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1378_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_1379_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_1380_diff__zero,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_zero
thf(fact_1381_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1382_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1383_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1384_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_1385_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_1386_diff__0__right,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_0_right
thf(fact_1387_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1388_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_1389_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_1390_diff__self,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% diff_self
thf(fact_1391_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1392_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_1393_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1394_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_1395_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1396_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1397_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_1398_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_1399_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_1400_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1401_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1402_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_1403_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_1404_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_1405_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1406_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1407_divide__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1408_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1409_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1410_divide__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( divide_divide_rat @ C @ A )
= ( divide_divide_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1411_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_1412_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1413_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_1414_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1415_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1416_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1417_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1418_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_1419_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_1420_bits__mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% bits_mod_0
thf(fact_1421_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1422_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1423_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1424_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1425_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1426_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_1427_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_1428_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self2
thf(fact_1429_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_1430_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_1431_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self1
thf(fact_1432_minus__mod__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mod_self2
thf(fact_1433_minus__mod__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mod_self2
thf(fact_1434_List_Ofinite__set,axiom,
! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_1435_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_1436_List_Ofinite__set,axiom,
! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_1437_List_Ofinite__set,axiom,
! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_1438_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1439_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1440_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1441_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1442_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1443_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1444_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1445_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1446_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_1447_length__list__update,axiom,
! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) )
= ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).
% length_list_update
thf(fact_1448_length__list__update,axiom,
! [Xs2: list_o,I2: nat,X: $o] :
( ( size_size_list_o @ ( list_update_o @ Xs2 @ I2 @ X ) )
= ( size_size_list_o @ Xs2 ) ) ).
% length_list_update
thf(fact_1449_length__list__update,axiom,
! [Xs2: list_nat,I2: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_1450_length__list__update,axiom,
! [Xs2: list_int,I2: nat,X: int] :
( ( size_size_list_int @ ( list_update_int @ Xs2 @ I2 @ X ) )
= ( size_size_list_int @ Xs2 ) ) ).
% length_list_update
thf(fact_1451_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_1452_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1453_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1454_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1455_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1456_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1457_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1458_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs2: list_int,X: int] :
( ( I2 != J )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J )
= ( nth_int @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1459_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs2: list_nat,X: nat] :
( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1460_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( I2 != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1461_list__update__id,axiom,
! [Xs2: list_int,I2: nat] :
( ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_1462_list__update__id,axiom,
! [Xs2: list_nat,I2: nat] :
( ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_1463_list__update__id,axiom,
! [Xs2: list_VEBT_VEBT,I2: nat] :
( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_1464_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1465_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1466_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1467_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1468_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1469_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1470_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_1471_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1472_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1473_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1474_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_1475_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1476_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1477_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1478_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_1479_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_1480_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1481_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1482_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_1483_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_1484_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1485_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1486_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1487_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1488_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1489_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1490_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1491_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1492_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1493_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1494_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1495_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_1496_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1497_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1498_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1499_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_1500_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1501_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1502_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1503_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1504_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_1505_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1506_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1507_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1508_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_1509_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1510_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1511_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1512_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_rat @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1513_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1514_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_1515_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1516_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ( times_times_rat @ A @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_1517_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1518_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_1519_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1520_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_1521_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1522_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_1523_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1524_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ( times_times_rat @ C @ A )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_1525_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1526_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_1527_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1528_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_1529_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1530_sum__squares__eq__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1531_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y2 @ Y2 ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y2 = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1532_sum__squares__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1533_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1534_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1535_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1536_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1537_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1538_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1539_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1540_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1541_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1542_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1543_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1544_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1545_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1546_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1547_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1548_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1549_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1550_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1551_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1552_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1553_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1554_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1555_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1556_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1557_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1558_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1559_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1560_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1561_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1562_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1563_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1564_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_1565_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1566_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_1567_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1568_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1569_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_1570_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_1571_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_1572_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1573_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1574_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_1575_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_1576_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_1577_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_1578_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1579_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1580_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1581_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1582_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_1583_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_1584_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_1585_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_1586_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_1587_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_1588_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1589_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1590_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1591_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1592_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1593_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1594_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_1595_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1596_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_1597_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_1598_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_1599_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_1600_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_1601_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_1602_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_1603_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_1604_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_1605_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_1606_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self2_is_0
thf(fact_1607_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_1608_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_1609_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self1_is_0
thf(fact_1610_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_1611_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_1612_bits__mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_1613_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_1614_bits__mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_1615_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% bits_mod_div_trivial
thf(fact_1616_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_1617_mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_1618_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_div_trivial
thf(fact_1619_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1620_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1621_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1622_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1623_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1624_mod__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1625_mod__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1626_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1627_mod__mult__self3,axiom,
! [C: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1628_mod__mult__self3,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1629_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1630_mod__mult__self2,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1631_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1632_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1633_mod__mult__self1,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1634_mod__mult__self1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1635_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1636_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1637_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ V ) ) )
& ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1638_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1639_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1640_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1641_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1642_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(4)
thf(fact_1643_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(4)
thf(fact_1644_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(4)
thf(fact_1645_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(4)
thf(fact_1646_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(4)
thf(fact_1647_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(3)
thf(fact_1648_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(3)
thf(fact_1649_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(3)
thf(fact_1650_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(3)
thf(fact_1651_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(3)
thf(fact_1652_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_1653_max__0__1_I1_J,axiom,
( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
= one_one_rat ) ).
% max_0_1(1)
thf(fact_1654_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_1655_max__0__1_I1_J,axiom,
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
= one_on7984719198319812577d_enat ) ).
% max_0_1(1)
thf(fact_1656_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_1657_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_1658_max__0__1_I2_J,axiom,
( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
= one_one_rat ) ).
% max_0_1(2)
thf(fact_1659_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_1660_max__0__1_I2_J,axiom,
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
= one_on7984719198319812577d_enat ) ).
% max_0_1(2)
thf(fact_1661_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_1662_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1663_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(6)
thf(fact_1664_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(6)
thf(fact_1665_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(6)
thf(fact_1666_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(6)
thf(fact_1667_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(6)
thf(fact_1668_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(5)
thf(fact_1669_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(5)
thf(fact_1670_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(5)
thf(fact_1671_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(5)
thf(fact_1672_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(5)
thf(fact_1673_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1674_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1675_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1676_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1677_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1678_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1679_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1680_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1681_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1682_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1683_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1684_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1685_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1686_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1687_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1688_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1689_list__update__beyond,axiom,
! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I2 )
=> ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1690_list__update__beyond,axiom,
! [Xs2: list_o,I2: nat,X: $o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I2 )
=> ( ( list_update_o @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1691_list__update__beyond,axiom,
! [Xs2: list_nat,I2: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I2 )
=> ( ( list_update_nat @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1692_list__update__beyond,axiom,
! [Xs2: list_int,I2: nat,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I2 )
=> ( ( list_update_int @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1693_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_1694_divide__le__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_1695_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_1696_zero__le__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_1697_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_1698_zero__less__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_1699_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1700_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1701_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1702_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1703_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1704_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1705_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1706_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1707_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_1708_divide__less__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_1709_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1710_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1711_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1712_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1713_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1714_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1715_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1716_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1717_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1718_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1719_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1720_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1721_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1722_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1723_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1724_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1725_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1726_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1727_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1728_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1729_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( power_power_rat @ A @ N )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1730_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1731_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1732_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1733_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1734_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1735_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1736_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1737_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1738_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1739_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1740_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1741_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1742_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1743_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1744_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1745_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_o,X: $o] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1746_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1747_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_int,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1748_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1749_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1750_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1751_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1752_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1753_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1754_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1755_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1756_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1757_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1758_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1759_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1760_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1761_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1762_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1763_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1764_zero__eq__power2,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% zero_eq_power2
thf(fact_1765_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_1766_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_1767_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_1768_zero__eq__power2,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_1769_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% bits_one_mod_two_eq_one
thf(fact_1770_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_one_mod_two_eq_one
thf(fact_1771_bits__one__mod__two__eq__one,axiom,
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% bits_one_mod_two_eq_one
thf(fact_1772_one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_mod_two_eq_one
thf(fact_1773_one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_mod_two_eq_one
thf(fact_1774_one__mod__two__eq__one,axiom,
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% one_mod_two_eq_one
thf(fact_1775_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_1776_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1777_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1778_set__swap,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,J: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
= ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1779_set__swap,axiom,
! [I2: nat,Xs2: list_o,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
=> ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I2 @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I2 ) ) )
= ( set_o2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1780_set__swap,axiom,
! [I2: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I2 ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1781_set__swap,axiom,
! [I2: nat,Xs2: list_int,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
=> ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I2 ) ) )
= ( set_int2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1782_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_1783_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_1784_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_1785_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_1786_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1787_power__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1788_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1789_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1790_power2__eq__iff__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1791_power2__eq__iff__nonneg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1792_power2__eq__iff__nonneg,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1793_power2__eq__iff__nonneg,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1794_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_1795_power2__less__eq__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% power2_less_eq_zero_iff
thf(fact_1796_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_1797_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_1798_zero__less__power2,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_power2
thf(fact_1799_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_1800_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y2: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y2 = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1801_sum__power2__eq__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1802_sum__power2__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1803_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1804_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1805_not__mod__2__eq__1__eq__0,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1806_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1807_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1808_not__mod__2__eq__0__eq__1,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1809_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_1810_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_1811_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1812_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1813_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1814_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1815_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1816_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1817_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1818_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size(4)
thf(fact_1819_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1820_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_1821_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1822_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_1823_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1824_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1825_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1826_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= A )
= ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1827_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= A )
= ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1828_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1829_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1830_mod__eq__0D,axiom,
! [M: nat,D: nat] :
( ( ( modulo_modulo_nat @ M @ D )
= zero_zero_nat )
=> ? [Q3: nat] :
( M
= ( times_times_nat @ D @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_1831_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1832_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1833_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1834_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1835_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1836_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( modulo364778990260209775nteger @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1837_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( modulo_modulo_nat @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1838_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( modulo_modulo_int @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1839_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1840_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1841_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1842_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1843_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1844_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1845_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_1846_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_1847_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(1)
thf(fact_1848_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B2: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1849_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1850_invar__vebt_Ointros_I1_J,axiom,
! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% invar_vebt.intros(1)
thf(fact_1851_max__add__distrib__left,axiom,
! [X: real,Y2: real,Z: real] :
( ( plus_plus_real @ ( ord_max_real @ X @ Y2 ) @ Z )
= ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y2 @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1852_max__add__distrib__left,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( plus_plus_rat @ ( ord_max_rat @ X @ Y2 ) @ Z )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y2 @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1853_max__add__distrib__left,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y2 ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y2 @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1854_max__add__distrib__left,axiom,
! [X: int,Y2: int,Z: int] :
( ( plus_plus_int @ ( ord_max_int @ X @ Y2 ) @ Z )
= ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y2 @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1855_max__add__distrib__right,axiom,
! [X: real,Y2: real,Z: real] :
( ( plus_plus_real @ X @ ( ord_max_real @ Y2 @ Z ) )
= ( ord_max_real @ ( plus_plus_real @ X @ Y2 ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1856_max__add__distrib__right,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( plus_plus_rat @ X @ ( ord_max_rat @ Y2 @ Z ) )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Y2 ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1857_max__add__distrib__right,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y2 @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y2 ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1858_max__add__distrib__right,axiom,
! [X: int,Y2: int,Z: int] :
( ( plus_plus_int @ X @ ( ord_max_int @ Y2 @ Z ) )
= ( ord_max_int @ ( plus_plus_int @ X @ Y2 ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1859_max__diff__distrib__left,axiom,
! [X: real,Y2: real,Z: real] :
( ( minus_minus_real @ ( ord_max_real @ X @ Y2 ) @ Z )
= ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y2 @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_1860_max__diff__distrib__left,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( minus_minus_rat @ ( ord_max_rat @ X @ Y2 ) @ Z )
= ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y2 @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_1861_max__diff__distrib__left,axiom,
! [X: int,Y2: int,Z: int] :
( ( minus_minus_int @ ( ord_max_int @ X @ Y2 ) @ Z )
= ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y2 @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_1862_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_1863_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1864_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_1865_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1866_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_1867_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ).
% zero_power
thf(fact_1868_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1869_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_1870_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1871_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_1872_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_1873_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_1874_mod__mult__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_1875_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_1876_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_1877_mod__mult__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_1878_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_1879_mult__mod__right,axiom,
! [C: int,A: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_1880_mult__mod__right,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_1881_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_1882_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_1883_mod__mult__mult2,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_1884_mod__mult__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_1885_mod__mult__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_1886_mod__mult__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B5 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_1887_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_1888_mod__mult__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_1889_mod__mult__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_1890_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1891_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1892_mod__add__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1893_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1894_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1895_mod__add__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1896_mod__add__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1897_mod__add__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1898_mod__add__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B5 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1899_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1900_mod__add__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1901_mod__add__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1902_mod__diff__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_1903_mod__diff__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_1904_mod__diff__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_1905_mod__diff__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_1906_mod__diff__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_1907_mod__diff__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B5 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_1908_mod__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_1909_mod__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_1910_power__mod,axiom,
! [A: nat,B: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
= ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_1911_power__mod,axiom,
! [A: int,B: int,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
= ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_1912_power__mod,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
= ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_1913_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_1914_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_1915_vebt__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
= ( ( ( X = zero_zero_nat )
=> A )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B )
& ( X = one_one_nat ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_1916_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_1917_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_1918_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_1919_VEBT_Oexhaust,axiom,
! [Y2: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
( Y2
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y2
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_1920_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu3: $o,Uv2: $o,D3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ D3 ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_1921_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1922_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1923_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A: $o,B: $o] :
( ( ( X = zero_zero_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ $true @ B ) ) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ A @ $true ) ) )
& ( ( X != one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_1924_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1925_vebt__pred_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
= none_nat ) ).
% vebt_pred.simps(1)
thf(fact_1926_set__update__subsetI,axiom,
! [Xs2: list_real,A2: set_real,X: real,I2: nat] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1927_set__update__subsetI,axiom,
! [Xs2: list_complex,A2: set_complex,X: complex,I2: nat] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
=> ( ( member_complex @ X @ A2 )
=> ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1928_set__update__subsetI,axiom,
! [Xs2: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I2: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
=> ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1929_set__update__subsetI,axiom,
! [Xs2: list_nat,A2: set_nat,X: nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1930_set__update__subsetI,axiom,
! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I2: nat] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
=> ( ( member_VEBT_VEBT @ X @ A2 )
=> ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1931_set__update__subsetI,axiom,
! [Xs2: list_int,A2: set_int,X: int,I2: nat] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1932_finite__list,axiom,
! [A2: set_VEBT_VEBT] :
( ( finite5795047828879050333T_VEBT @ A2 )
=> ? [Xs3: list_VEBT_VEBT] :
( ( set_VEBT_VEBT2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1933_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1934_finite__list,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ? [Xs3: list_int] :
( ( set_int2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1935_finite__list,axiom,
! [A2: set_complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ? [Xs3: list_complex] :
( ( set_complex2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1936_finite__lists__length__eq,axiom,
! [A2: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
& ( ( size_s3451745648224563538omplex @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1937_finite__lists__length__eq,axiom,
! [A2: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A2 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
& ( ( size_s6755466524823107622T_VEBT @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1938_finite__lists__length__eq,axiom,
! [A2: set_o,N: nat] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
& ( ( size_size_list_o @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1939_finite__lists__length__eq,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
& ( ( size_size_list_nat @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1940_finite__lists__length__eq,axiom,
! [A2: set_int,N: nat] :
( ( finite_finite_int @ A2 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
& ( ( size_size_list_int @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1941_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1942_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1943_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_1944_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1945_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1946_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1947_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1948_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1949_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1950_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1951_field__lbound__gt__zero,axiom,
! [D1: rat,D22: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D22 )
=> ? [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
& ( ord_less_rat @ E2 @ D1 )
& ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1952_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1953_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_1954_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1955_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1956_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N ) ) ).
% zero_neq_numeral
thf(fact_1957_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_1958_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_1959_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1960_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_1961_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_1962_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1963_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_1964_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1965_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1966_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_1967_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1968_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_1969_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1970_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1971_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_1972_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1973_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_1974_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1975_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1976_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1977_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1978_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1979_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1980_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1981_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1982_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1983_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1984_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1985_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1986_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1987_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1988_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1989_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1990_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1991_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_1992_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_1993_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_1994_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1995_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1996_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1997_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1998_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1999_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_2000_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: complex,Z5: complex] : Y5 = Z5 )
= ( ^ [A4: complex,B3: complex] :
( ( minus_minus_complex @ A4 @ B3 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2001_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z5: real] : Y5 = Z5 )
= ( ^ [A4: real,B3: real] :
( ( minus_minus_real @ A4 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2002_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
= ( ^ [A4: rat,B3: rat] :
( ( minus_minus_rat @ A4 @ B3 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2003_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2004_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_2005_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_2006_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_2007_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_2008_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_2009_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_2010_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_2011_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_2012_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_2013_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_2014_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_2015_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_2016_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_2017_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_2018_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_2019_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_2020_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_2021_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_2022_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [Va3: nat] :
( X
!= ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_2023_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_2024_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_2025_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_2026_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_2027_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_2028_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_2029_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_2030_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_2031_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_2032_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_2033_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_2034_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_2035_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_2036_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_2037_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_2038_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_2039_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_2040_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_2041_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_2042_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_2043_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2044_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2045_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2046_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2047_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2048_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2049_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2050_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2051_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2052_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_2053_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_2054_lambda__zero,axiom,
( ( ^ [H: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_2055_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_2056_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_2057_finite__lists__length__le,axiom,
! [A2: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2058_finite__lists__length__le,axiom,
! [A2: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A2 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2059_finite__lists__length__le,axiom,
! [A2: set_o,N: nat] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2060_finite__lists__length__le,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2061_finite__lists__length__le,axiom,
! [A2: set_int,N: nat] :
( ( finite_finite_int @ A2 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2062_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2063_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2064_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2065_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_2066_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_2067_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_2068_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_2069_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_2070_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_2071_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D3: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% mod_eqE
thf(fact_2072_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ B @ C ) )
=> ~ ! [D3: code_integer] :
( B
!= ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% mod_eqE
thf(fact_2073_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_2074_div__add1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_2075_div__add1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_2076_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_2077_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_2078_mod__induct,axiom,
! [P: nat > $o,N: nat,P4: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P4 )
=> ( ( ord_less_nat @ M @ P4 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P4 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2079_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_2080_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2081_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2082_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2083_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2084_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N2 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_2085_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_2086_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_2087_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y2 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y2 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_2088_vebt__pred_Osimps_I2_J,axiom,
! [A: $o,Uw: $o] :
( ( A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
= none_nat ) ) ) ).
% vebt_pred.simps(2)
thf(fact_2089_vebt__mint_Osimps_I1_J,axiom,
! [A: $o,B: $o] :
( ( A
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( B
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
= none_nat ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_2090_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_2091_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_2092_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_2093_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_2094_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_2095_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_le_numeral
thf(fact_2096_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_2097_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_2098_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2099_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2100_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2101_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2102_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2103_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2104_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2105_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2106_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_2107_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_2108_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_2109_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2110_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2111_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2112_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2113_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2114_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2115_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2116_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2117_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2118_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2119_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2120_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2121_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2122_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2123_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2124_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2125_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2126_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2127_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2128_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2129_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2130_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2131_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2132_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_2133_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_2134_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_2135_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_2136_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2137_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2138_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2139_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2140_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2141_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2142_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2143_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2144_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2145_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2146_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2147_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2148_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2149_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2150_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2151_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2152_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2153_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2154_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2155_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2156_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2157_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2158_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2159_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_2160_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_2161_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_2162_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_2163_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_2164_zero__less__numeral,axiom,
! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_less_numeral
thf(fact_2165_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_2166_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_2167_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_2168_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_2169_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_2170_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_2171_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2172_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2173_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2174_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2175_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_2176_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_2177_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_2178_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_2179_add__nonpos__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2180_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y2 )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y2 = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2181_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2182_add__nonpos__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2183_add__nonneg__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2184_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ( plus_plus_rat @ X @ Y2 )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y2 = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2185_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2186_add__nonneg__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2187_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_2188_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_2189_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_2190_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_2191_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2192_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2193_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2194_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2195_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2196_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2197_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2198_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2199_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2200_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2201_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2202_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2203_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2204_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2205_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2206_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2207_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2208_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2209_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2210_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2211_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2212_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2213_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2214_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_2215_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_2216_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_2217_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2218_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2219_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2220_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_2221_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_2222_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_2223_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_2224_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_2225_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_2226_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_2227_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_2228_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2229_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2230_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2231_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2232_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_2233_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_2234_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_2235_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_2236_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2237_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2238_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2239_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2240_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2241_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2242_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2243_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2244_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2245_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2246_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2247_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2248_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2249_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2250_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2251_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2252_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2253_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2254_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2255_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2256_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2257_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2258_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2259_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2260_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2261_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2262_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2263_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2264_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2265_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2266_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2267_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2268_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2269_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2270_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2271_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2272_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2273_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2274_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2275_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2276_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2277_vebt__maxt_Osimps_I1_J,axiom,
! [B: $o,A: $o] :
( ( B
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B
=> ( ( A
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
= none_nat ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_2278_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_2279_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_2280_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_2281_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_2282_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_2283_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_2284_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_2285_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_2286_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_2287_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_2288_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_2289_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_2290_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_2291_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_2292_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_2293_add__less__zeroD,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_2294_add__less__zeroD,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X @ Y2 ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X @ zero_zero_rat )
| ( ord_less_rat @ Y2 @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_2295_add__less__zeroD,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_2296_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2297_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2298_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2299_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2300_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_2301_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2302_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2303_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2304_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2305_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_2306_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_2307_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_2308_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_2309_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2310_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2311_divide__nonpos__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2312_divide__nonpos__nonpos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2313_divide__nonpos__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2314_divide__nonpos__nonneg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2315_divide__nonneg__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2316_divide__nonneg__nonpos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2317_divide__nonneg__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2318_divide__nonneg__nonneg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2319_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2320_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2321_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_2322_divide__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_2323_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2324_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2325_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_2326_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A4: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_2327_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_2328_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2329_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2330_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2331_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2332_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2333_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2334_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_2335_divide__less__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
thf(fact_2336_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2337_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2338_divide__pos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_2339_divide__pos__pos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_2340_divide__pos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_2341_divide__pos__neg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_2342_divide__neg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_2343_divide__neg__pos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_2344_divide__neg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_2345_divide__neg__neg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_2346_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2347_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2348_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2349_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2350_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2351_zero__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2352_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2353_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2354_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2355_zero__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2356_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2357_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2358_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( times_times_complex @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2359_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2360_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( times_times_rat @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2361_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( B
= ( times_times_complex @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2362_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2363_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C )
= A )
= ( B
= ( times_times_rat @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2364_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2365_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2366_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= B )
=> ( A
= ( divide_divide_rat @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2367_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C ) )
=> ( ( divide1717551699836669952omplex @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2368_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2369_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C ) )
=> ( ( divide_divide_rat @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2370_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_2371_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_2372_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_2373_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_2374_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_2375_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C )
= A )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_2376_frac__eq__eq,axiom,
! [Y2: complex,Z: complex,X: complex,W: complex] :
( ( Y2 != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y2 )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X @ Z )
= ( times_times_complex @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2377_frac__eq__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y2 )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2378_frac__eq__eq,axiom,
! [Y2: rat,Z: rat,X: rat,W: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y2 )
= ( divide_divide_rat @ W @ Z ) )
= ( ( times_times_rat @ X @ Z )
= ( times_times_rat @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2379_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2380_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2381_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2382_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_2383_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_2384_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_2385_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_2386_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_2387_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_2388_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_2389_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_2390_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_2391_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_2392_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_2393_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_2394_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_2395_option_Osize_I4_J,axiom,
! [X22: nat] :
( ( size_size_option_nat @ ( some_nat @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_2396_option_Osize_I4_J,axiom,
! [X22: product_prod_nat_nat] :
( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_2397_option_Osize_I4_J,axiom,
! [X22: num] :
( ( size_size_option_num @ ( some_num @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_2398_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_2399_option_Osize_I3_J,axiom,
( ( size_size_option_nat @ none_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_2400_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_2401_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_2402_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_2403_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_2404_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_2405_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_2406_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_2407_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_2408_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2409_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2410_divmod__digit__0_I2_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2411_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_2412_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_2413_bits__stable__imp__add__self,axiom,
! [A: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_nat ) ) ).
% bits_stable_imp_add_self
thf(fact_2414_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= zero_zero_int ) ) ).
% bits_stable_imp_add_self
thf(fact_2415_bits__stable__imp__add__self,axiom,
! [A: code_integer] :
( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= zero_z3403309356797280102nteger ) ) ).
% bits_stable_imp_add_self
thf(fact_2416_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_2417_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_2418_vebt__insert_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B2: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ X3 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X3 ) )
=> ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_2419_vebt__pred_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu3: $o,Uv2: $o] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) )
=> ( ! [A3: $o,Uw2: $o] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
=> ( ! [A3: $o,B2: $o,Va3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
=> ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% vebt_pred.cases
thf(fact_2420_VEBT__internal_Omembermima_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu3: $o,Uv2: $o,Uw2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X3 ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_2421_vebt__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B2: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_2422_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X )
= ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) ) ).
% vebt_insert.simps(2)
thf(fact_2423_vebt__pred_Osimps_I3_J,axiom,
! [B: $o,A: $o,Va: nat] :
( ( B
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B
=> ( ( A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
= none_nat ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_2424_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_2425_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2426_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2427_divmod__digit__0_I1_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2428_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_2429_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_2430_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_2431_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_2432_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_2433_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_2434_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_2435_mult__div__mod__eq,axiom,
! [B: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_2436_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_2437_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_2438_mod__mult__div__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_2439_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_2440_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_2441_mod__div__mult__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_2442_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_2443_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_2444_div__mult__mod__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_2445_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_2446_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_2447_mod__div__decomp,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_2448_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] :
( A
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_2449_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2450_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2451_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
= ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2452_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2453_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2454_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
= ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2455_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_2456_div__mult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_2457_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_2458_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_2459_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_2460_minus__mult__div__eq__mod,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_2461_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2462_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2463_minus__mod__eq__mult__div,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2464_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_2465_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_2466_minus__mod__eq__div__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_2467_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_2468_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_2469_minus__div__mult__eq__mod,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_2470_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y2 @ N ) )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus_nat @ Y2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2471_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2472_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2473_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_2474_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M6: nat,N2: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_2475_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu3: $o] :
( X
!= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_2476_set__update__memI,axiom,
! [N: nat,Xs2: list_real,X: real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
=> ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2477_set__update__memI,axiom,
! [N: nat,Xs2: list_complex,X: complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2478_set__update__memI,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2479_set__update__memI,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2480_set__update__memI,axiom,
! [N: nat,Xs2: list_o,X: $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2481_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2482_set__update__memI,axiom,
! [N: nat,Xs2: list_int,X: int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_2483_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_2484_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_o,X: $o] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( ( list_update_o @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_o @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_2485_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_2486_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_int,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( ( list_update_int @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_int @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_2487_nth__list__update,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( ( I2 = J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_2488_nth__list__update,axiom,
! [I2: nat,Xs2: list_o,X: $o,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X ) @ J )
= ( ( ( I2 = J )
=> X )
& ( ( I2 != J )
=> ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_2489_nth__list__update,axiom,
! [I2: nat,Xs2: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I2 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_2490_nth__list__update,axiom,
! [I2: nat,Xs2: list_int,J: nat,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( ( I2 = J )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J )
= ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_2491_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2492_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2493_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2494_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2495_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2496_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2497_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2498_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2499_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2500_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2501_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2502_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2503_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2504_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2505_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2506_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2507_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2508_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2509_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2510_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2511_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2512_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2513_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2514_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2515_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2516_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2517_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2518_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2519_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2520_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2521_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2522_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2523_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2524_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2525_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2526_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2527_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2528_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2529_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2530_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2531_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2532_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2533_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2534_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2535_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2536_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2537_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2538_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2539_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2540_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2541_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu3: $o] :
( X
!= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_2542_field__le__epsilon,axiom,
! [X: real,Y2: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y2 @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_epsilon
thf(fact_2543_field__le__epsilon,axiom,
! [X: rat,Y2: rat] :
( ! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y2 @ E2 ) ) )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ).
% field_le_epsilon
thf(fact_2544_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_2545_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_2546_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_2547_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_2548_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2549_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2550_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2551_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2552_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_2553_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_2554_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_2555_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_2556_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2557_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2558_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2559_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2560_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2561_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2562_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2563_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2564_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2565_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2566_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2567_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2568_divide__nonpos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_2569_divide__nonpos__pos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_2570_divide__nonpos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2571_divide__nonpos__neg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2572_divide__nonneg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2573_divide__nonneg__pos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2574_divide__nonneg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_2575_divide__nonneg__neg,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_2576_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2577_divide__le__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2578_frac__less2,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_2579_frac__less2,axiom,
! [X: rat,Y2: rat,W: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_rat @ zero_zero_rat @ W )
=> ( ( ord_less_rat @ W @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_2580_frac__less,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_2581_frac__less,axiom,
! [X: rat,Y2: rat,W: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_rat @ zero_zero_rat @ W )
=> ( ( ord_less_eq_rat @ W @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_2582_frac__le,axiom,
! [Y2: real,X: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_2583_frac__le,axiom,
! [Y2: rat,X: rat,W: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_rat @ zero_zero_rat @ W )
=> ( ( ord_less_eq_rat @ W @ Z )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_2584_div__positive,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_positive
thf(fact_2585_div__positive,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_positive
thf(fact_2586_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2587_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2588_mult__left__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2589_mult__left__le__one__le,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2590_mult__left__le__one__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2591_mult__right__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2592_mult__right__le__one__le,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2593_mult__right__le__one__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2594_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_2595_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_2596_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_2597_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_2598_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2599_mult__left__le,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2600_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2601_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2602_sum__squares__ge__zero,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ).
% sum_squares_ge_zero
thf(fact_2603_sum__squares__ge__zero,axiom,
! [X: rat,Y2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y2 @ Y2 ) ) ) ).
% sum_squares_ge_zero
thf(fact_2604_sum__squares__ge__zero,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) ) ).
% sum_squares_ge_zero
thf(fact_2605_sum__squares__le__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2606_sum__squares__le__zero__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y2 = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2607_sum__squares__le__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2608_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2609_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2610_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2611_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2612_not__sum__squares__lt__zero,axiom,
! [X: real,Y2: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_2613_not__sum__squares__lt__zero,axiom,
! [X: rat,Y2: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_2614_not__sum__squares__lt__zero,axiom,
! [X: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_2615_sum__squares__gt__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) )
= ( ( X != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2616_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y2 @ Y2 ) ) )
= ( ( X != zero_zero_rat )
| ( Y2 != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2617_sum__squares__gt__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) )
= ( ( X != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2618_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2619_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2620_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_2621_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_2622_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_2623_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_2624_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2625_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2626_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2627_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2628_mult__imp__less__div__pos,axiom,
! [Y2: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y2 ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2629_mult__imp__less__div__pos,axiom,
! [Y2: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_rat @ ( times_times_rat @ Z @ Y2 ) @ X )
=> ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2630_mult__imp__div__pos__less,axiom,
! [Y2: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2631_mult__imp__div__pos__less,axiom,
! [Y2: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y2 ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2632_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_2633_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_2634_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2635_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2636_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2637_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2638_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_2639_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_2640_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2641_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2642_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2643_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2644_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2645_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2646_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_2647_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_2648_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_2649_power__le__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_2650_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_2651_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_2652_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: complex,C: complex] :
( ( ( numera6690914467698888265omplex @ W )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2653_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2654_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( numeral_numeral_rat @ W )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2655_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( numera6690914467698888265omplex @ W ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2656_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2657_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( numeral_numeral_rat @ W ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2658_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_2659_divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y2: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y2 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_2660_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_2661_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y2 )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_2662_add__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y2: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y2 @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_2663_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y2 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_2664_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y2 @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_2665_add__num__frac,axiom,
! [Y2: complex,Z: complex,X: complex] :
( ( Y2 != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_2666_add__num__frac,axiom,
! [Y2: real,Z: real,X: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_2667_add__num__frac,axiom,
! [Y2: rat,Z: rat,X: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_2668_add__frac__num,axiom,
! [Y2: complex,X: complex,Z: complex] :
( ( Y2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y2 ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_2669_add__frac__num,axiom,
! [Y2: real,X: real,Z: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_2670_add__frac__num,axiom,
! [Y2: rat,X: rat,Z: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y2 ) @ Z )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_2671_add__frac__eq,axiom,
! [Y2: complex,Z: complex,X: complex,W: complex] :
( ( Y2 != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y2 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y2 ) ) @ ( times_times_complex @ Y2 @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_2672_add__frac__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_2673_add__frac__eq,axiom,
! [Y2: rat,Z: rat,X: rat,W: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_2674_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2675_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2676_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2677_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2678_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2679_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2680_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2681_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ ( suc @ N ) )
= ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2682_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2683_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2684_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2685_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2686_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2687_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2688_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_2689_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_2690_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_2691_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_2692_divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y2: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y2 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_2693_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y2 )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_2694_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y2 )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_2695_diff__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y2: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y2 @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_2696_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y2 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_2697_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y2 @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_2698_diff__frac__eq,axiom,
! [Y2: complex,Z: complex,X: complex,W: complex] :
( ( Y2 != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y2 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y2 ) ) @ ( times_times_complex @ Y2 @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2699_diff__frac__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2700_diff__frac__eq,axiom,
! [Y2: rat,Z: rat,X: rat,W: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2701_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2702_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2703_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2704_mod__double__modulus,axiom,
! [M: code_integer,X: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( modulo364778990260209775nteger @ X @ M ) )
| ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2705_mod__double__modulus,axiom,
! [M: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_nat @ X @ M ) )
| ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2706_mod__double__modulus,axiom,
! [M: int,X: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_int @ X @ M ) )
| ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2707_divmod__digit__1_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2708_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2709_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2710_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_2711_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2712_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_2713_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_2714_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_2715_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_2716_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_2717_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_2718_length__pos__if__in__set,axiom,
! [X: real,Xs2: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2719_length__pos__if__in__set,axiom,
! [X: complex,Xs2: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2720_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2721_length__pos__if__in__set,axiom,
! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2722_length__pos__if__in__set,axiom,
! [X: $o,Xs2: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2723_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2724_length__pos__if__in__set,axiom,
! [X: int,Xs2: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2725_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_2726_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_2727_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_2728_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_2729_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_2730_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_2731_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_2732_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_2733_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2734_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_2735_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_2736_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_2737_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X )
= ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).
% vebt_insert.simps(3)
thf(fact_2738_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_2739_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_2740_vebt__mint_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A3: $o,B2: $o] :
( X
!= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_mint.cases
thf(fact_2741_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2742_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2743_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2744_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2745_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2746_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2747_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2748_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2749_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2750_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2751_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2752_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2753_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2754_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2755_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2756_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2757_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2758_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2759_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2760_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2761_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2762_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2763_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2764_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2765_field__le__mult__one__interval,axiom,
! [X: real,Y2: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_2766_field__le__mult__one__interval,axiom,
! [X: rat,Y2: rat] :
( ! [Z3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z3 )
=> ( ( ord_less_rat @ Z3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y2 ) ) )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_2767_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y2: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y2 )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y2 )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y2 )
=> ( ( ? [Uu3: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ $true ) )
=> Y2 )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y2 )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> Y2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_2768_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2769_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2770_mult__imp__le__div__pos,axiom,
! [Y2: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y2 ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2771_mult__imp__le__div__pos,axiom,
! [Y2: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y2 ) @ X )
=> ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2772_mult__imp__div__pos__le,axiom,
! [Y2: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2773_mult__imp__div__pos__le,axiom,
! [Y2: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y2 ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2774_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_2775_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_2776_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2777_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2778_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2779_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2780_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_2781_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_2782_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_2783_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_2784_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2785_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2786_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2787_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2788_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2789_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2790_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_2791_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_2792_convex__bound__le,axiom,
! [X: real,A: real,Y2: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y2 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2793_convex__bound__le,axiom,
! [X: rat,A: rat,Y2: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ( ord_less_eq_rat @ Y2 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2794_convex__bound__le,axiom,
! [X: int,A: int,Y2: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y2 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2795_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2796_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2797_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2798_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2799_frac__le__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_2800_frac__le__eq,axiom,
! [Y2: rat,Z: rat,X: rat,W: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_2801_divmod__digit__1_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2802_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
= ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2803_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
= ( divide_divide_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2804_frac__less__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_2805_frac__less__eq,axiom,
! [Y2: rat,Z: rat,X: rat,W: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_2806_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2807_power__Suc__less,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2808_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2809_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2810_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2811_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2812_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2813_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2814_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_2815_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_2816_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_2817_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_2818_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2819_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2820_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2821_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2822_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y2: option_nat] :
( ( ( vEBT_vebt_mint @ X )
= Y2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( B2
=> ( Y2
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( Y2 = none_nat ) ) ) ) ) )
=> ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( Y2 != none_nat ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( some_nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_2823_power__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2824_power__decreasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2825_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2826_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2827_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_2828_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_2829_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_2830_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_2831_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_2832_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y2: option_nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( B2
=> ( Y2
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y2 = none_nat ) ) ) ) ) )
=> ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( Y2 != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( some_nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_2833_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2834_self__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2835_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2836_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2837_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> ( Y2
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y2
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y2
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) )
=> ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y2
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y2
!= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_2838_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2839_one__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2840_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2841_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2842_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_2843_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_2844_power__diff,axiom,
! [A: complex,N: nat,M: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_2845_power__diff,axiom,
! [A: real,N: nat,M: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_2846_power__diff,axiom,
! [A: rat,N: nat,M: nat] :
( ( A != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_2847_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_2848_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_2849_div__if,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M6 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_2850_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_2851_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_2852_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_2853_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_2854_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_2855_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_2856_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_2857_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_2858_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_2859_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_2860_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_2861_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
= none_nat ) ).
% vebt_pred.simps(5)
thf(fact_2862_convex__bound__lt,axiom,
! [X: real,A: real,Y2: real,U: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y2 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2863_convex__bound__lt,axiom,
! [X: rat,A: rat,Y2: rat,U: rat,V: rat] :
( ( ord_less_rat @ X @ A )
=> ( ( ord_less_rat @ Y2 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2864_convex__bound__lt,axiom,
! [X: int,A: int,Y2: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y2 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2865_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2866_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2867_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2868_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2869_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_2870_half__gt__zero__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% half_gt_zero_iff
thf(fact_2871_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_2872_half__gt__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_2873_scaling__mono,axiom,
! [U: real,V: real,R2: real,S2: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_2874_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S2: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ( ord_less_eq_rat @ R2 @ S2 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_2875_power2__le__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_2876_power2__le__imp__le,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_2877_power2__le__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_2878_power2__le__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_2879_power2__eq__imp__eq,axiom,
! [X: real,Y2: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( X = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2880_power2__eq__imp__eq,axiom,
! [X: rat,Y2: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( X = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2881_power2__eq__imp__eq,axiom,
! [X: nat,Y2: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( X = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2882_power2__eq__imp__eq,axiom,
! [X: int,Y2: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( X = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2883_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2884_zero__le__power2,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2885_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2886_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_2887_power2__less__0,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% power2_less_0
thf(fact_2888_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_2889_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2890_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2891_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2892_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2893_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2894_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2895_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_2896_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_2897_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_2898_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_2899_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_2900_power__eq__if,axiom,
( power_power_complex
= ( ^ [P5: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_2901_power__eq__if,axiom,
( power_power_real
= ( ^ [P5: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_2902_power__eq__if,axiom,
( power_power_rat
= ( ^ [P5: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_2903_power__eq__if,axiom,
( power_power_nat
= ( ^ [P5: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_2904_power__eq__if,axiom,
( power_power_int
= ( ^ [P5: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_2905_power__minus__mult,axiom,
! [N: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_2906_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_2907_power__minus__mult,axiom,
! [N: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_2908_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_2909_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_2910_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_2911_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_2912_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2913_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2914_div__exp__mod__exp__eq,axiom,
! [A: code_integer,N: nat,M: nat] :
( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2915_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
= none_nat ) ).
% vebt_pred.simps(6)
thf(fact_2916_power2__less__imp__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ X @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_2917_power2__less__imp__less,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_rat @ X @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_2918_power2__less__imp__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ord_less_nat @ X @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_2919_power2__less__imp__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_int @ X @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_2920_sum__power2__le__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2921_sum__power2__le__zero__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y2 = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2922_sum__power2__le__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2923_sum__power2__ge__zero,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2924_sum__power2__ge__zero,axiom,
! [X: rat,Y2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2925_sum__power2__ge__zero,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2926_sum__power2__gt__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2927_sum__power2__gt__zero__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_rat )
| ( Y2 != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2928_sum__power2__gt__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2929_not__sum__power2__lt__zero,axiom,
! [X: real,Y2: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_2930_not__sum__power2__lt__zero,axiom,
! [X: rat,Y2: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% not_sum_power2_lt_zero
thf(fact_2931_not__sum__power2__lt__zero,axiom,
! [X: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_2932_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_2933_zero__le__even__power_H,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_2934_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_2935_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2936_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2937_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2938_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2939_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2940_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2941_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( Y2
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> Y2 )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( Y2
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_2942_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_2943_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_2944_odd__0__le__power__imp__0__le,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2945_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2946_odd__0__le__power__imp__0__le,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2947_odd__power__less__zero,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_2948_odd__power__less__zero,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_2949_odd__power__less__zero,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_2950_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_2951_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_2952_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_2953_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_2954_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> Y2 )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> Y2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( Y2
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( Y2
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( Y2
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_2955_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu3: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_2956_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( Y2
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> Y2 )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y2 )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> Y2 )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y2
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_2957_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_2958_arith__geo__mean,axiom,
! [U: real,X: real,Y2: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_2959_arith__geo__mean,axiom,
! [U: rat,X: rat,Y2: rat] :
( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ X @ Y2 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_2960_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A3: $o,B2: $o] :
( A1
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( A22
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5 = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M5 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M5 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5 = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M5 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M5 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_2961_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A4: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A4 @ B3 ) )
& ( A23
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
& ( A23
= ( plus_plus_nat @ N2 @ N2 ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
& ( A23
= ( plus_plus_nat @ N2 @ N2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_2962_verit__le__mono__div,axiom,
! [A2: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A2 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B4 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2963_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% inrange
thf(fact_2964_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2965_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2966_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2967_finite__Collect__subsets,axiom,
! [A2: set_complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2968_finite__Collect__subsets,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2969_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: real > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2970_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: nat > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2971_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: int > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2972_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: complex > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2973_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: real > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2974_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2975_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: int > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2976_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: complex > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2977_finite__Collect__bounded__ex,axiom,
! [P: int > $o,Q: real > int > $o] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
? [Y: int] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: int] :
( ( P @ Y )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2978_finite__Collect__bounded__ex,axiom,
! [P: int > $o,Q: nat > int > $o] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y: int] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: int] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2979_set__bit__0,axiom,
! [A: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2980_set__bit__0,axiom,
! [A: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2981_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z2: real] :
( ( power_power_real @ Z2 @ N )
= one_one_real ) ) ) ) ).
% finite_roots_unity
thf(fact_2982_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= one_one_complex ) ) ) ) ).
% finite_roots_unity
thf(fact_2983_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: option_nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( ( Y2 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
=> ( ! [A3: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ Uw2 ) )
=> ( ( Xa2
= ( suc @ zero_zero_nat ) )
=> ( ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y2 = none_nat ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ! [Va3: nat] :
( ( Xa2
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( B2
=> ( Y2
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y2 = none_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y2 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
=> ( ( Y2 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
=> ( ( Y2 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y2
= ( some_nat @ Ma2 ) ) )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y2
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_2984_max__less__iff__conj,axiom,
! [X: extended_enat,Y2: extended_enat,Z: extended_enat] :
( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y2 ) @ Z )
= ( ( ord_le72135733267957522d_enat @ X @ Z )
& ( ord_le72135733267957522d_enat @ Y2 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_2985_max__less__iff__conj,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y2 ) @ Z )
= ( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Y2 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_2986_max__less__iff__conj,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ ( ord_max_rat @ X @ Y2 ) @ Z )
= ( ( ord_less_rat @ X @ Z )
& ( ord_less_rat @ Y2 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_2987_max__less__iff__conj,axiom,
! [X: num,Y2: num,Z: num] :
( ( ord_less_num @ ( ord_max_num @ X @ Y2 ) @ Z )
= ( ( ord_less_num @ X @ Z )
& ( ord_less_num @ Y2 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_2988_max__less__iff__conj,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y2 ) @ Z )
= ( ( ord_less_nat @ X @ Z )
& ( ord_less_nat @ Y2 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_2989_max__less__iff__conj,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y2 ) @ Z )
= ( ( ord_less_int @ X @ Z )
& ( ord_less_int @ Y2 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_2990_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_2991_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_2992_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_2993_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_2994_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_2995_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_2996_atLeastAtMost__iff,axiom,
! [I2: set_int,L2: set_int,U: set_int] :
( ( member_set_int @ I2 @ ( set_or370866239135849197et_int @ L2 @ U ) )
= ( ( ord_less_eq_set_int @ L2 @ I2 )
& ( ord_less_eq_set_int @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_2997_atLeastAtMost__iff,axiom,
! [I2: rat,L2: rat,U: rat] :
( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L2 @ U ) )
= ( ( ord_less_eq_rat @ L2 @ I2 )
& ( ord_less_eq_rat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_2998_atLeastAtMost__iff,axiom,
! [I2: num,L2: num,U: num] :
( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L2 @ U ) )
= ( ( ord_less_eq_num @ L2 @ I2 )
& ( ord_less_eq_num @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_2999_atLeastAtMost__iff,axiom,
! [I2: nat,L2: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
= ( ( ord_less_eq_nat @ L2 @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3000_atLeastAtMost__iff,axiom,
! [I2: int,L2: int,U: int] :
( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L2 @ U ) )
= ( ( ord_less_eq_int @ L2 @ I2 )
& ( ord_less_eq_int @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3001_atLeastAtMost__iff,axiom,
! [I2: real,L2: real,U: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L2 @ U ) )
= ( ( ord_less_eq_real @ L2 @ I2 )
& ( ord_less_eq_real @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3002_Icc__eq__Icc,axiom,
! [L2: set_int,H2: set_int,L3: set_int,H3: set_int] :
( ( ( set_or370866239135849197et_int @ L2 @ H2 )
= ( set_or370866239135849197et_int @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
& ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3003_Icc__eq__Icc,axiom,
! [L2: rat,H2: rat,L3: rat,H3: rat] :
( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
= ( set_or633870826150836451st_rat @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_rat @ L2 @ H2 )
& ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3004_Icc__eq__Icc,axiom,
! [L2: num,H2: num,L3: num,H3: num] :
( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
= ( set_or7049704709247886629st_num @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_num @ L2 @ H2 )
& ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3005_Icc__eq__Icc,axiom,
! [L2: nat,H2: nat,L3: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
= ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L2 @ H2 )
& ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3006_Icc__eq__Icc,axiom,
! [L2: int,H2: int,L3: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
= ( set_or1266510415728281911st_int @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L2 @ H2 )
& ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3007_Icc__eq__Icc,axiom,
! [L2: real,H2: real,L3: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
= ( set_or1222579329274155063t_real @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L2 @ H2 )
& ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3008_max_Obounded__iff,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_3009_max_Obounded__iff,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
= ( ( ord_less_eq_rat @ B @ A )
& ( ord_less_eq_rat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_3010_max_Obounded__iff,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
= ( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_3011_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_3012_max_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_3013_max_Oabsorb2,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3014_max_Oabsorb2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3015_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3016_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3017_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3018_max_Oabsorb1,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3019_max_Oabsorb1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3020_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3021_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3022_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3023_max_Oabsorb3,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_3024_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_3025_max_Oabsorb3,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_3026_max_Oabsorb3,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_3027_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_3028_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_3029_max_Oabsorb4,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_3030_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_3031_max_Oabsorb4,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_3032_max_Oabsorb4,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_3033_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_3034_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_3035_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_3036_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_3037_atLeastatMost__empty__iff2,axiom,
! [A: set_int,B: set_int] :
( ( bot_bot_set_set_int
= ( set_or370866239135849197et_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3038_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3039_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num
= ( set_or7049704709247886629st_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3040_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3041_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3042_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3043_atLeastatMost__empty__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( set_or370866239135849197et_int @ A @ B )
= bot_bot_set_set_int )
= ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3044_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3045_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3046_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3047_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3048_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3049_atLeastatMost__subset__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( ( ord_less_eq_set_int @ C @ A )
& ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3050_atLeastatMost__subset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C @ A )
& ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3051_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3052_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3053_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3054_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3055_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_3056_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_3057_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_3058_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_3059_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_3060_infinite__Icc__iff,axiom,
! [A: rat,B: rat] :
( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
= ( ord_less_rat @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_3061_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
= ( ord_less_real @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_3062_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_3063_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_3064_verit__le__mono__div__int,axiom,
! [A2: int,B4: int,N: int] :
( ( ord_less_int @ A2 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B4 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B4 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_3065_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_3066_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_3067_zmod__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_3068_div__mod__decomp__int,axiom,
! [A2: int,N: int] :
( A2
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_3069_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_3070_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_3071_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_3072_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_3073_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_3074_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_3075_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_3076_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
= ( ord_less_eq_int @ K @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_3077_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_3078_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_3079_div__positive__int,axiom,
! [L2: int,K: int] :
( ( ord_less_eq_int @ L2 @ K )
=> ( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% div_positive_int
thf(fact_3080_div__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
= ( ( K = zero_zero_int )
| ( L2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L2 ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_3081_zdiv__mono2__neg,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_3082_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_3083_zdiv__eq__0__iff,axiom,
! [I2: int,K: int] :
( ( ( divide_divide_int @ I2 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_3084_zdiv__mono2,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_3085_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_3086_div__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ord_less_eq_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_3087_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_3088_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% neg_zmod_mult_2
thf(fact_3089_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_3090_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_3091_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_3092_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_3093_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_3094_atLeastatMost__psubset__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D: set_int] :
( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_int @ A @ B )
| ( ( ord_less_eq_set_int @ C @ A )
& ( ord_less_eq_set_int @ B @ D )
& ( ( ord_less_set_int @ C @ A )
| ( ord_less_set_int @ B @ D ) ) ) )
& ( ord_less_eq_set_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3095_atLeastatMost__psubset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C @ A )
& ( ord_less_eq_rat @ B @ D )
& ( ( ord_less_rat @ C @ A )
| ( ord_less_rat @ B @ D ) ) ) )
& ( ord_less_eq_rat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3096_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D )
& ( ( ord_less_num @ C @ A )
| ( ord_less_num @ B @ D ) ) ) )
& ( ord_less_eq_num @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3097_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3098_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3099_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3100_infinite__Icc,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_3101_infinite__Icc,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_3102_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_3103_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_3104_finite__maxlen,axiom,
! [M7: set_list_VEBT_VEBT] :
( ( finite3004134309566078307T_VEBT @ M7 )
=> ? [N3: nat] :
! [X5: list_VEBT_VEBT] :
( ( member2936631157270082147T_VEBT @ X5 @ M7 )
=> ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3105_finite__maxlen,axiom,
! [M7: set_list_o] :
( ( finite_finite_list_o @ M7 )
=> ? [N3: nat] :
! [X5: list_o] :
( ( member_list_o @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3106_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N3: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3107_finite__maxlen,axiom,
! [M7: set_list_int] :
( ( finite3922522038869484883st_int @ M7 )
=> ? [N3: nat] :
! [X5: list_int] :
( ( member_list_int @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3108_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_3109_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ ( ord_less_eq_rat @ A @ B )
| ~ ( ord_less_eq_rat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_3110_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_3111_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_3112_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_3113_verit__comp__simplify1_I2_J,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3114_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3115_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3116_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3117_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3118_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3119_verit__comp__simplify1_I1_J,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3120_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3121_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3122_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3123_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_3124_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_3125_real__arch__pow__inv,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_3126_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_3127_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_3128_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_3129_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_3130_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_3131_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3132_verit__comp__simplify1_I3_J,axiom,
! [B5: rat,A5: rat] :
( ( ~ ( ord_less_eq_rat @ B5 @ A5 ) )
= ( ord_less_rat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3133_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3134_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3135_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3136_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_3137_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_3138_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_3139_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_3140_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_3141_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3142_finite__has__minimal2,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( member_set_int @ A @ A2 )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
& ( ord_less_eq_set_int @ X3 @ A )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3143_finite__has__minimal2,axiom,
! [A2: set_rat,A: rat] :
( ( finite_finite_rat @ A2 )
=> ( ( member_rat @ A @ A2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( ord_less_eq_rat @ X3 @ A )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3144_finite__has__minimal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ( ord_less_eq_num @ X3 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3145_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3146_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3147_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3148_finite__has__maximal2,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( member_set_int @ A @ A2 )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
& ( ord_less_eq_set_int @ A @ X3 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3149_finite__has__maximal2,axiom,
! [A2: set_rat,A: rat] :
( ( finite_finite_rat @ A2 )
=> ( ( member_rat @ A @ A2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( ord_less_eq_rat @ A @ X3 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3150_finite__has__maximal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ( ord_less_eq_num @ A @ X3 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3151_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3152_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3153_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_3154_rev__finite__subset,axiom,
! [B4: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_3155_rev__finite__subset,axiom,
! [B4: set_complex,A2: set_complex] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( finite3207457112153483333omplex @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_3156_rev__finite__subset,axiom,
! [B4: set_int,A2: set_int] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( finite_finite_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_3157_infinite__super,axiom,
! [S3: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_3158_infinite__super,axiom,
! [S3: set_complex,T3: set_complex] :
( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ~ ( finite3207457112153483333omplex @ S3 )
=> ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% infinite_super
thf(fact_3159_infinite__super,axiom,
! [S3: set_int,T3: set_int] :
( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ~ ( finite_finite_int @ S3 )
=> ~ ( finite_finite_int @ T3 ) ) ) ).
% infinite_super
thf(fact_3160_finite__subset,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_3161_finite__subset,axiom,
! [A2: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( finite3207457112153483333omplex @ B4 )
=> ( finite3207457112153483333omplex @ A2 ) ) ) ).
% finite_subset
thf(fact_3162_finite__subset,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( finite_finite_int @ B4 )
=> ( finite_finite_int @ A2 ) ) ) ).
% finite_subset
thf(fact_3163_max_OcoboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3164_max_OcoboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3165_max_OcoboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3166_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3167_max_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3168_max_OcoboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3169_max_OcoboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ A )
=> ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3170_max_OcoboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3171_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3172_max_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3173_max_Oabsorb__iff2,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_3174_max_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_max_rat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_3175_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_max_num @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_3176_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_max_nat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_3177_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_max_int @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_3178_max_Oabsorb__iff1,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_3179_max_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_max_rat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_3180_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_max_num @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_3181_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_max_nat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_3182_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_max_int @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_3183_le__max__iff__disj,axiom,
! [Z: extended_enat,X: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y2 ) )
= ( ( ord_le2932123472753598470d_enat @ Z @ X )
| ( ord_le2932123472753598470d_enat @ Z @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_3184_le__max__iff__disj,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y2 ) )
= ( ( ord_less_eq_rat @ Z @ X )
| ( ord_less_eq_rat @ Z @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_3185_le__max__iff__disj,axiom,
! [Z: num,X: num,Y2: num] :
( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y2 ) )
= ( ( ord_less_eq_num @ Z @ X )
| ( ord_less_eq_num @ Z @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_3186_le__max__iff__disj,axiom,
! [Z: nat,X: nat,Y2: nat] :
( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y2 ) )
= ( ( ord_less_eq_nat @ Z @ X )
| ( ord_less_eq_nat @ Z @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_3187_le__max__iff__disj,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y2 ) )
= ( ( ord_less_eq_int @ Z @ X )
| ( ord_less_eq_int @ Z @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_3188_max_Ocobounded2,axiom,
! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded2
thf(fact_3189_max_Ocobounded2,axiom,
! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded2
thf(fact_3190_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_3191_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_3192_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_3193_max_Ocobounded1,axiom,
! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded1
thf(fact_3194_max_Ocobounded1,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded1
thf(fact_3195_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_3196_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_3197_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_3198_max_Oorder__iff,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( A4
= ( ord_ma741700101516333627d_enat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_3199_max_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A4: rat] :
( A4
= ( ord_max_rat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_3200_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( A4
= ( ord_max_num @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_3201_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( ord_max_nat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_3202_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( A4
= ( ord_max_int @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_3203_max_OboundedI,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_3204_max_OboundedI,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ A )
=> ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_3205_max_OboundedI,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_3206_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_3207_max_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_3208_max_OboundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
=> ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_3209_max_OboundedE,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_rat @ B @ A )
=> ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_3210_max_OboundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_num @ B @ A )
=> ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% max.boundedE
thf(fact_3211_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_3212_max_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% max.boundedE
thf(fact_3213_max_OorderI,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% max.orderI
thf(fact_3214_max_OorderI,axiom,
! [A: rat,B: rat] :
( ( A
= ( ord_max_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% max.orderI
thf(fact_3215_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_3216_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_3217_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_3218_max_OorderE,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.orderE
thf(fact_3219_max_OorderE,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( A
= ( ord_max_rat @ A @ B ) ) ) ).
% max.orderE
thf(fact_3220_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_3221_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_3222_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_3223_max_Omono,axiom,
! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ( ord_le2932123472753598470d_enat @ D @ B )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3224_max_Omono,axiom,
! [C: rat,A: rat,D: rat,B: rat] :
( ( ord_less_eq_rat @ C @ A )
=> ( ( ord_less_eq_rat @ D @ B )
=> ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3225_max_Omono,axiom,
! [C: num,A: num,D: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ( ord_less_eq_num @ D @ B )
=> ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3226_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3227_max_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3228_less__max__iff__disj,axiom,
! [Z: extended_enat,X: extended_enat,Y2: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y2 ) )
= ( ( ord_le72135733267957522d_enat @ Z @ X )
| ( ord_le72135733267957522d_enat @ Z @ Y2 ) ) ) ).
% less_max_iff_disj
thf(fact_3229_less__max__iff__disj,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y2 ) )
= ( ( ord_less_real @ Z @ X )
| ( ord_less_real @ Z @ Y2 ) ) ) ).
% less_max_iff_disj
thf(fact_3230_less__max__iff__disj,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y2 ) )
= ( ( ord_less_rat @ Z @ X )
| ( ord_less_rat @ Z @ Y2 ) ) ) ).
% less_max_iff_disj
thf(fact_3231_less__max__iff__disj,axiom,
! [Z: num,X: num,Y2: num] :
( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y2 ) )
= ( ( ord_less_num @ Z @ X )
| ( ord_less_num @ Z @ Y2 ) ) ) ).
% less_max_iff_disj
thf(fact_3232_less__max__iff__disj,axiom,
! [Z: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Z @ X )
| ( ord_less_nat @ Z @ Y2 ) ) ) ).
% less_max_iff_disj
thf(fact_3233_less__max__iff__disj,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y2 ) )
= ( ( ord_less_int @ Z @ X )
| ( ord_less_int @ Z @ Y2 ) ) ) ).
% less_max_iff_disj
thf(fact_3234_max_Ostrict__boundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
=> ~ ( ( ord_le72135733267957522d_enat @ B @ A )
=> ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_3235_max_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_3236_max_Ostrict__boundedE,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
=> ~ ( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_3237_max_Ostrict__boundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_3238_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_3239_max_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_3240_max_Ostrict__order__iff,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( A4
= ( ord_ma741700101516333627d_enat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_3241_max_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( A4
= ( ord_max_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_3242_max_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [B3: rat,A4: rat] :
( ( A4
= ( ord_max_rat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_3243_max_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( A4
= ( ord_max_num @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_3244_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( ord_max_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_3245_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( A4
= ( ord_max_int @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_3246_max_Ostrict__coboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ C @ A )
=> ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_3247_max_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_3248_max_Ostrict__coboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ A )
=> ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_3249_max_Ostrict__coboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_num @ C @ A )
=> ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_3250_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_3251_max_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_3252_max_Ostrict__coboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_3253_max_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_3254_max_Ostrict__coboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_3255_max_Ostrict__coboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_3256_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_3257_max_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_3258_finite__image__set,axiom,
! [P: real > $o,F: real > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3259_finite__image__set,axiom,
! [P: real > $o,F: real > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3260_finite__image__set,axiom,
! [P: real > $o,F: real > int] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [Uu2: int] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3261_finite__image__set,axiom,
! [P: real > $o,F: real > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3262_finite__image__set,axiom,
! [P: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3263_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3264_finite__image__set,axiom,
! [P: nat > $o,F: nat > int] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [Uu2: int] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3265_finite__image__set,axiom,
! [P: nat > $o,F: nat > complex] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3266_finite__image__set,axiom,
! [P: int > $o,F: int > real] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: int] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3267_finite__image__set,axiom,
! [P: int > $o,F: int > nat] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: int] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3268_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3269_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3270_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > int] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [Uu2: int] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3271_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3272_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3273_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3274_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > int] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [Uu2: int] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3275_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3276_finite__image__set2,axiom,
! [P: real > $o,Q: int > $o,F: real > int > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_int @ ( collect_int @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real,Y: int] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3277_finite__image__set2,axiom,
! [P: real > $o,Q: int > $o,F: real > int > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_int @ ( collect_int @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real,Y: int] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3278_max__def__raw,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_3279_max__def__raw,axiom,
( ord_max_set_int
= ( ^ [A4: set_int,B3: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_3280_max__def__raw,axiom,
( ord_max_rat
= ( ^ [A4: rat,B3: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_3281_max__def__raw,axiom,
( ord_max_num
= ( ^ [A4: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_3282_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_3283_max__def__raw,axiom,
( ord_max_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_3284_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3285_finite__has__minimal,axiom,
! [A2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( A2 != bot_bot_set_set_int )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3286_finite__has__minimal,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( A2 != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3287_finite__has__minimal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3288_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3289_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3290_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3291_finite__has__maximal,axiom,
! [A2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( A2 != bot_bot_set_set_int )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3292_finite__has__maximal,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( A2 != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3293_finite__has__maximal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3294_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3295_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3296_div__less__mono,axiom,
! [A2: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A2 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A2 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B4 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_3297_set__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3298_set__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3299_set__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3300_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_3301_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y2
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y2
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y2
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
=> ( ( Y2
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
=> ( ( Y2
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ Xa2 ) ) ) )
=> ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y2
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y2
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_3302_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3303_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( Y2
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y2
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ( ~ Y2
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y2
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3304_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_3305_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_3306_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( Y2
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( Y2
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_3307_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_3308_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_3309_max__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
= Q2 ) ).
% max_enat_simps(2)
thf(fact_3310_max__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
= Q2 ) ).
% max_enat_simps(3)
thf(fact_3311_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_3312_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_3313_mod__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( modulo_modulo_int @ K @ L2 )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_3314_mod__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( modulo_modulo_int @ K @ L2 )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_3315_mod__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ord_less_eq_int @ L2 @ K )
=> ( ( modulo_modulo_int @ K @ L2 )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% mod_pos_geq
thf(fact_3316_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_3317_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
=> ? [Q3: int] :
( M
= ( times_times_int @ D @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_3318_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
= ( ? [Q4: int] :
( M
= ( times_times_int @ D @ Q4 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_3319_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_3320_neg__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% neg_mod_conj
thf(fact_3321_pos__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
& ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% pos_mod_conj
thf(fact_3322_zmod__trivial__iff,axiom,
! [I2: int,K: int] :
( ( ( modulo_modulo_int @ I2 @ K )
= I2 )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K @ I2 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_3323_mod__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L2 )
= ( plus_plus_int @ K @ L2 ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_3324_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( ord_less_int @ B @ R4 )
=> ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_3325_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( ord_less_int @ R2 @ B )
=> ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_3326_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q2: int,R2: int,B5: int,Q5: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_3327_zdiv__mono2__lemma,axiom,
! [B: int,Q2: int,R2: int,B5: int,Q5: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_3328_int__mod__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_3329_int__mod__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_3330_q__pos__lemma,axiom,
! [B5: int,Q5: int,R4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_3331_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_3332_Euclidean__Division_Opos__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_3333_neg__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_3334_Euclidean__Division_Opos__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_3335_neg__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% neg_mod_bound
thf(fact_3336_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_3337_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_3338_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_3339_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_3340_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ~ Y2
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ~ Y2
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( Y2
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( Y2
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( Y2
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_3341_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_3342_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_3343_cpmi,axiom,
! [D4: int,P: int > $o,P6: int > $o,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ( P6 @ X2 ) )
| ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ? [Y: int] :
( ( member_int @ Y @ B4 )
& ( P @ ( plus_plus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_3344_cppi,axiom,
! [D4: int,P: int > $o,P6: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ( P6 @ X2 ) )
| ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ? [Y: int] :
( ( member_int @ Y @ A2 )
& ( P @ ( minus_minus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_3345_bset_I6_J,axiom,
! [D4: int,B4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_3346_bset_I8_J,axiom,
! [D4: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_3347_aset_I6_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_3348_aset_I8_J,axiom,
! [D4: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_3349_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_3350_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_3351_double__eq__0__iff,axiom,
! [A: rat] :
( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_3352_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_3353_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_3354_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_3355_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_3356_minf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ~ ( ord_less_rat @ T @ X5 ) ) ).
% minf(7)
thf(fact_3357_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ~ ( ord_less_num @ T @ X5 ) ) ).
% minf(7)
thf(fact_3358_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_3359_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_3360_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_3361_minf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ord_less_rat @ X5 @ T ) ) ).
% minf(5)
thf(fact_3362_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ord_less_num @ X5 @ T ) ) ).
% minf(5)
thf(fact_3363_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_3364_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_3365_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3366_minf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3367_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3368_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3369_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3370_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3371_minf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3372_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3373_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3374_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3375_minf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3376_minf_I2_J,axiom,
! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3377_minf_I2_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3378_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3379_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3380_minf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3381_minf_I1_J,axiom,
! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3382_minf_I1_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3383_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3384_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3385_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3386_pinf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ord_less_rat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3387_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ord_less_num @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3388_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3389_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3390_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3391_pinf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ~ ( ord_less_rat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3392_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ~ ( ord_less_num @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3393_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3394_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3395_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3396_pinf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3397_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3398_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3399_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3400_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3401_pinf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3402_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3403_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3404_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3405_pinf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3406_pinf_I2_J,axiom,
! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3407_pinf_I2_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3408_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3409_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3410_pinf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3411_pinf_I1_J,axiom,
! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3412_pinf_I1_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3413_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3414_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3415_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_3416_minf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% minf(8)
thf(fact_3417_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_3418_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_3419_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_3420_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_3421_minf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ord_less_eq_rat @ X5 @ T ) ) ).
% minf(6)
thf(fact_3422_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_3423_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_3424_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_3425_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3426_pinf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ord_less_eq_rat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3427_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3428_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3429_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3430_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3431_pinf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3432_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3433_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3434_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3435_inf__period_I1_J,axiom,
! [P: real > $o,D4: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ! [X5: real,K4: real] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_3436_inf__period_I1_J,axiom,
! [P: rat > $o,D4: rat,Q: rat > $o] :
( ! [X3: rat,K2: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ( ! [X3: rat,K2: rat] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ! [X5: rat,K4: rat] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_3437_inf__period_I1_J,axiom,
! [P: int > $o,D4: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_3438_inf__period_I2_J,axiom,
! [P: real > $o,D4: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ! [X5: real,K4: real] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_3439_inf__period_I2_J,axiom,
! [P: rat > $o,D4: rat,Q: rat > $o] :
( ! [X3: rat,K2: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ( ! [X3: rat,K2: rat] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ! [X5: rat,K4: rat] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_3440_inf__period_I2_J,axiom,
! [P: int > $o,D4: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_3441_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P6: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_3442_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P6: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_3443_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_3444_times__int__code_I2_J,axiom,
! [L2: int] :
( ( times_times_int @ zero_zero_int @ L2 )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_3445_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_3446_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_3447_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus_int @ zero_zero_int @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_3448_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_3449_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_3450_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_3451_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_3452_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_3453_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_3454_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_3455_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_3456_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_3457_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_3458_aset_I2_J,axiom,
! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
| ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_3459_aset_I1_J,axiom,
! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
& ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_3460_bset_I2_J,axiom,
! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
| ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_3461_bset_I1_J,axiom,
! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
& ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_3462_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_3463_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_3464_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_3465_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_3466_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_3467_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_3468_plusinfinity,axiom,
! [D: int,P6: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [X_12: int] : ( P6 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_3469_int__induct,axiom,
! [P: int > $o,K: int,I2: int] :
( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
thf(fact_3470_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_3471_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_3472_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_3473_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_3474_aset_I7_J,axiom,
! [D4: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_3475_aset_I5_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_3476_aset_I4_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D4 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_3477_aset_I3_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D4 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_3478_bset_I7_J,axiom,
! [D4: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_3479_bset_I5_J,axiom,
! [D4: int,B4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_3480_bset_I4_J,axiom,
! [D4: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D4 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_3481_bset_I3_J,axiom,
! [D4: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D4 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_3482_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_3483_psubsetI,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less_set_int @ A2 @ B4 ) ) ) ).
% psubsetI
thf(fact_3484_Diff__eq__empty__iff,axiom,
! [A2: set_real,B4: set_real] :
( ( ( minus_minus_set_real @ A2 @ B4 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_3485_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_3486_Diff__eq__empty__iff,axiom,
! [A2: set_int,B4: set_int] :
( ( ( minus_minus_set_int @ A2 @ B4 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_3487_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_3488_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_3489_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_3490_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_3491_subset__empty,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_3492_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_3493_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3494_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3495_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3496_flip__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3497_flip__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3498_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3499_unset__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3500_unset__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3501_subsetI,axiom,
! [A2: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% subsetI
thf(fact_3502_subsetI,axiom,
! [A2: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% subsetI
thf(fact_3503_subsetI,axiom,
! [A2: set_complex,B4: set_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_complex @ X3 @ B4 ) )
=> ( ord_le211207098394363844omplex @ A2 @ B4 ) ) ).
% subsetI
thf(fact_3504_subsetI,axiom,
! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( member8440522571783428010at_nat @ X3 @ B4 ) )
=> ( ord_le3146513528884898305at_nat @ A2 @ B4 ) ) ).
% subsetI
thf(fact_3505_subsetI,axiom,
! [A2: set_int,B4: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B4 ) )
=> ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% subsetI
thf(fact_3506_subset__antisym,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_3507_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_3508_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_3509_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_3510_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_3511_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_3512_double__diff,axiom,
! [A2: set_nat,B4: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C4 )
=> ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_3513_double__diff,axiom,
! [A2: set_int,B4: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ C4 )
=> ( ( minus_minus_set_int @ B4 @ ( minus_minus_set_int @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_3514_Diff__subset,axiom,
! [A2: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_3515_Diff__subset,axiom,
! [A2: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_3516_Diff__mono,axiom,
! [A2: set_nat,C4: set_nat,D4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ D4 @ B4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_3517_Diff__mono,axiom,
! [A2: set_int,C4: set_int,D4: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A2 @ C4 )
=> ( ( ord_less_eq_set_int @ D4 @ B4 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B4 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_3518_in__mono,axiom,
! [A2: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_3519_in__mono,axiom,
! [A2: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_3520_in__mono,axiom,
! [A2: set_complex,B4: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( member_complex @ X @ A2 )
=> ( member_complex @ X @ B4 ) ) ) ).
% in_mono
thf(fact_3521_in__mono,axiom,
! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B4 )
=> ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( member8440522571783428010at_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_3522_in__mono,axiom,
! [A2: set_int,B4: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( member_int @ X @ A2 )
=> ( member_int @ X @ B4 ) ) ) ).
% in_mono
thf(fact_3523_subsetD,axiom,
! [A2: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_3524_subsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_3525_subsetD,axiom,
! [A2: set_complex,B4: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B4 ) ) ) ).
% subsetD
thf(fact_3526_subsetD,axiom,
! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B4 )
=> ( ( member8440522571783428010at_nat @ C @ A2 )
=> ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_3527_subsetD,axiom,
! [A2: set_int,B4: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_3528_equalityE,axiom,
! [A2: set_int,B4: set_int] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ~ ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_3529_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_3530_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ( member_real @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_3531_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [X2: complex] :
( ( member_complex @ X2 @ A6 )
=> ( member_complex @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_3532_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A6 )
=> ( member8440522571783428010at_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_3533_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A6 )
=> ( member_int @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_3534_equalityD1,axiom,
! [A2: set_int,B4: set_int] :
( ( A2 = B4 )
=> ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_3535_equalityD2,axiom,
! [A2: set_int,B4: set_int] :
( ( A2 = B4 )
=> ( ord_less_eq_set_int @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_3536_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_3537_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_3538_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [T2: complex] :
( ( member_complex @ T2 @ A6 )
=> ( member_complex @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_3539_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T2 @ A6 )
=> ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_3540_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_3541_subset__refl,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_3542_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_3543_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X3: real] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_3544_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_3545_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_3546_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_3547_subset__trans,axiom,
! [A2: set_int,B4: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ C4 )
=> ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_3548_set__eq__subset,axiom,
( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_3549_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X2: complex] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_3550_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_3551_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X2: list_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_3552_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_3553_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_3554_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_3555_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A6 )
@ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_3556_less__eq__set__def,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
( ord_le4573692005234683329plex_o
@ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
@ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_3557_less__eq__set__def,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ord_le704812498762024988_nat_o
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A6 )
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_3558_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_3559_Collect__subset,axiom,
! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_3560_Collect__subset,axiom,
! [A2: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_3561_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_3562_Collect__subset,axiom,
! [A2: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_3563_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_3564_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_3565_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_set_int @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_3566_subset__psubset__trans,axiom,
! [A2: set_int,B4: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( ord_less_set_int @ B4 @ C4 )
=> ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_3567_subset__not__subset__eq,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_3568_psubset__subset__trans,axiom,
! [A2: set_int,B4: set_int,C4: set_int] :
( ( ord_less_set_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ C4 )
=> ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_3569_psubset__imp__subset,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_set_int @ A2 @ B4 )
=> ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_3570_psubset__eq,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_3571_psubsetE,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_set_int @ A2 @ B4 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ).
% psubsetE
thf(fact_3572_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B2: real,C2: real] :
( ( P @ A3 @ B2 )
=> ( ( P @ B2 @ C2 )
=> ( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( P @ A3 @ C2 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [A3: real,B2: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D5 ) )
=> ( P @ A3 @ B2 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_3573_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ Z ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3574_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y2 @ Z ) )
= ( ord_less_eq_rat @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3575_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y2 @ Z ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3576_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3577_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y2 ) )
= ( ord_less_eq_rat @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3578_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3579_divides__aux__eq,axiom,
! [Q2: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_3580_divides__aux__eq,axiom,
! [Q2: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_3581_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3582_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3583_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3584_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3585_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
= ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3586_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N )
= ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3587_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N )
= ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3588_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
= ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3589_product__nth,axiom,
! [N: nat,Xs2: list_nat,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) @ N )
= ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3590_product__nth,axiom,
! [N: nat,Xs2: list_nat,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs2 @ Ys ) @ N )
= ( product_Pair_nat_o @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3591_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_3592_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > complex,Y2: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3593_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > complex,Y2: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3594_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > complex,Y2: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3595_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > complex,Y2: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3596_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y2: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3597_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y2: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3598_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y2: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3599_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > real,Y2: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3600_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > rat,Y2: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( times_times_rat @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3601_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > rat,Y2: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( times_times_rat @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3602_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3603_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_o] :
( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3604_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_3605_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_int] :
( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_3606_length__product,axiom,
! [Xs2: list_o,Ys: list_VEBT_VEBT] :
( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3607_length__product,axiom,
! [Xs2: list_o,Ys: list_o] :
( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3608_length__product,axiom,
! [Xs2: list_o,Ys: list_nat] :
( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_3609_length__product,axiom,
! [Xs2: list_o,Ys: list_int] :
( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_3610_length__product,axiom,
! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3611_length__product,axiom,
! [Xs2: list_nat,Ys: list_o] :
( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3612_unique__remainder,axiom,
! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_3613_unique__quotient,axiom,
! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
=> ( Q2 = Q5 ) ) ) ).
% unique_quotient
thf(fact_3614_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% eucl_rel_int_by0
thf(fact_3615_div__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( divide_divide_int @ K @ L2 )
= Q2 ) ) ).
% div_int_unique
thf(fact_3616_mod__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( modulo_modulo_int @ K @ L2 )
= R2 ) ) ).
% mod_int_unique
thf(fact_3617_eucl__rel__int__dividesI,axiom,
! [L2: int,K: int,Q2: int] :
( ( L2 != zero_zero_int )
=> ( ( K
= ( times_times_int @ Q2 @ L2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_3618_eucl__rel__int,axiom,
! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% eucl_rel_int
thf(fact_3619_eucl__rel__int__iff,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( ( K
= ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L2 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L2 @ zero_zero_int )
=> ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3620_mult__less__iff1,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ Z ) )
= ( ord_less_real @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_3621_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y2 @ Z ) )
= ( ord_less_rat @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_3622_mult__less__iff1,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y2 @ Z ) )
= ( ord_less_int @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_3623_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3624_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > complex,Y2: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3625_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > complex,Y2: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3626_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > complex,Y2: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3627_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > complex,Y2: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3628_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y2: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3629_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y2: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3630_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y2: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3631_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > real,Y2: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3632_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > rat,Y2: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3633_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > rat,Y2: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y2 @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3634_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y22: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_3635_prod_Oinject,axiom,
! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ X1 @ X22 )
= ( produc6137756002093451184nteger @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_3636_prod_Oinject,axiom,
! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ X1 @ X22 )
= ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_3637_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
= ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_3638_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
= ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_3639_old_Oprod_Oinject,axiom,
! [A: int,B: int,A5: int,B5: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_3640_old_Oprod_Oinject,axiom,
! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ A @ B )
= ( produc6137756002093451184nteger @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_3641_old_Oprod_Oinject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_3642_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_3643_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_3644_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_int_int] :
~ ! [A3: int,B2: int] :
( Y2
!= ( product_Pair_int_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3645_old_Oprod_Oexhaust,axiom,
! [Y2: produc8763457246119570046nteger] :
~ ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( Y2
!= ( produc6137756002093451184nteger @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3646_old_Oprod_Oexhaust,axiom,
! [Y2: produc1908205239877642774nteger] :
~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( Y2
!= ( produc8603105652947943368nteger @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3647_old_Oprod_Oexhaust,axiom,
! [Y2: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( Y2
!= ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3648_old_Oprod_Oexhaust,axiom,
! [Y2: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( Y2
!= ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3649_surj__pair,axiom,
! [P4: product_prod_int_int] :
? [X3: int,Y3: int] :
( P4
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3650_surj__pair,axiom,
! [P4: produc8763457246119570046nteger] :
? [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( P4
= ( produc6137756002093451184nteger @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3651_surj__pair,axiom,
! [P4: produc1908205239877642774nteger] :
? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( P4
= ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3652_surj__pair,axiom,
! [P4: produc2285326912895808259nt_int] :
? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P4
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3653_surj__pair,axiom,
! [P4: produc7773217078559923341nt_int] :
? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P4
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3654_prod__cases,axiom,
! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
( ! [A3: int,B2: int] : ( P @ ( product_Pair_int_int @ A3 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_3655_prod__cases,axiom,
! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_3656_prod__cases,axiom,
! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_3657_prod__cases,axiom,
! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_3658_prod__cases,axiom,
! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_3659_Pair__inject,axiom,
! [A: int,B: int,A5: int,B5: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_3660_Pair__inject,axiom,
! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ A @ B )
= ( produc6137756002093451184nteger @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_3661_Pair__inject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_3662_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_3663_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_3664_prod__cases3,axiom,
! [Y2: produc8763457246119570046nteger] :
~ ! [A3: code_integer > option6357759511663192854e_term,B2: code_integer,C2: code_integer] :
( Y2
!= ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_3665_prod__cases3,axiom,
! [Y2: produc1908205239877642774nteger] :
~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C2: code_integer] :
( Y2
!= ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_3666_prod__cases3,axiom,
! [Y2: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C2: int] :
( Y2
!= ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_3667_prod__cases3,axiom,
! [Y2: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B2: int,C2: int] :
( Y2
!= ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_3668_prod__induct3,axiom,
! [P: produc8763457246119570046nteger > $o,X: produc8763457246119570046nteger] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3669_prod__induct3,axiom,
! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3670_prod__induct3,axiom,
! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3671_prod__induct3,axiom,
! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B2: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3672_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
=> ( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
=> ( P @ M5 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_3673_concat__bit__Suc,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
= ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).
% concat_bit_Suc
thf(fact_3674_dbl__simps_I3_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3675_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3676_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3677_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3678_dual__order_Orefl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% dual_order.refl
thf(fact_3679_dual__order_Orefl,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% dual_order.refl
thf(fact_3680_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_3681_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_3682_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_3683_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_3684_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_3685_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_3686_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_3687_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_3688_concat__bit__0,axiom,
! [K: int,L2: int] :
( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
= L2 ) ).
% concat_bit_0
thf(fact_3689_dbl__simps_I2_J,axiom,
( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% dbl_simps(2)
thf(fact_3690_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_3691_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% dbl_simps(2)
thf(fact_3692_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_3693_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
= ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% concat_bit_nonnegative_iff
thf(fact_3694_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L2: int] :
( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
= ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% concat_bit_negative_iff
thf(fact_3695_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_3696_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_3697_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
= ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_3698_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_3699_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L2: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
= ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_3700_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_3701_dbl__def,axiom,
( neg_numeral_dbl_rat
= ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_3702_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_3703_order__antisym__conv,axiom,
! [Y2: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y2 @ X )
=> ( ( ord_less_eq_set_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_3704_order__antisym__conv,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_eq_rat @ Y2 @ X )
=> ( ( ord_less_eq_rat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_3705_order__antisym__conv,axiom,
! [Y2: num,X: num] :
( ( ord_less_eq_num @ Y2 @ X )
=> ( ( ord_less_eq_num @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_3706_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_3707_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_3708_linorder__le__cases,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_3709_linorder__le__cases,axiom,
! [X: num,Y2: num] :
( ~ ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_3710_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_3711_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_3712_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3713_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3714_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3715_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3716_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3717_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3718_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3719_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3720_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3721_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3722_ord__eq__le__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3723_ord__eq__le__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3724_ord__eq__le__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3725_ord__eq__le__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3726_ord__eq__le__subst,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3727_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3728_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3729_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3730_ord__eq__le__subst,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3731_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3732_linorder__linear,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
| ( ord_less_eq_rat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_3733_linorder__linear,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
| ( ord_less_eq_num @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_3734_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_3735_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_3736_order__eq__refl,axiom,
! [X: set_int,Y2: set_int] :
( ( X = Y2 )
=> ( ord_less_eq_set_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_3737_order__eq__refl,axiom,
! [X: rat,Y2: rat] :
( ( X = Y2 )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_3738_order__eq__refl,axiom,
! [X: num,Y2: num] :
( ( X = Y2 )
=> ( ord_less_eq_num @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_3739_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_3740_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_3741_order__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3742_order__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3743_order__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3744_order__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3745_order__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3746_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3747_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3748_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3749_order__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3750_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_3751_order__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3752_order__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3753_order__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3754_order__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3755_order__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3756_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3757_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3758_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3759_order__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3760_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_3761_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
= ( ^ [A4: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A4 @ B3 )
& ( ord_less_eq_set_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_3762_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ A4 @ B3 )
& ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_3763_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : Y5 = Z5 )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_3764_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_3765_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_3766_antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_3767_antisym,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_3768_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_3769_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_3770_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_3771_dual__order_Otrans,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C @ B )
=> ( ord_less_eq_set_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_3772_dual__order_Otrans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_3773_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_3774_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_3775_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_3776_dual__order_Oantisym,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_3777_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_3778_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_3779_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_3780_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_3781_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
= ( ^ [A4: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ B3 @ A4 )
& ( ord_less_eq_set_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_3782_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ B3 @ A4 )
& ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_3783_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : Y5 = Z5 )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_3784_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_3785_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_3786_linorder__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B2: rat] :
( ( ord_less_eq_rat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: rat,B2: rat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_3787_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_3788_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_3789_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_3790_order__trans,axiom,
! [X: set_int,Y2: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ( ord_less_eq_set_int @ Y2 @ Z )
=> ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_3791_order__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_eq_rat @ X @ Z ) ) ) ).
% order_trans
thf(fact_3792_order__trans,axiom,
! [X: num,Y2: num,Z: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_3793_order__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_3794_order__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_3795_order_Otrans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% order.trans
thf(fact_3796_order_Otrans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% order.trans
thf(fact_3797_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_3798_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_3799_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_3800_order__antisym,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ( ord_less_eq_set_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_3801_order__antisym,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_3802_order__antisym,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_3803_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_3804_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_3805_ord__le__eq__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_3806_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_3807_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_3808_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_3809_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_3810_ord__eq__le__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( A = B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_3811_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_3812_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_3813_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_3814_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_3815_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
= ( ^ [X2: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y )
& ( ord_less_eq_set_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_3816_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
& ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_3817_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : Y5 = Z5 )
= ( ^ [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
& ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_3818_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_3819_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_3820_le__cases3,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ X )
=> ~ ( ord_less_eq_rat @ X @ Z ) )
=> ( ( ( ord_less_eq_rat @ X @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X )
=> ~ ( ord_less_eq_rat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_3821_le__cases3,axiom,
! [X: num,Y2: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_num @ Y2 @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_num @ Z @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ X ) )
=> ( ( ( ord_less_eq_num @ Y2 @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_3822_le__cases3,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_3823_le__cases3,axiom,
! [X: int,Y2: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_3824_nle__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A @ B ) )
= ( ( ord_less_eq_rat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_3825_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_3826_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_3827_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_3828_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_3829_lt__ex,axiom,
! [X: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% lt_ex
thf(fact_3830_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_3831_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_3832_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_3833_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_3834_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_3835_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_3836_dense,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ? [Z3: rat] :
( ( ord_less_rat @ X @ Z3 )
& ( ord_less_rat @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_3837_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_3838_less__imp__neq,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_3839_less__imp__neq,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_3840_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_3841_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_3842_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_3843_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_3844_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_3845_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_3846_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_3847_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3848_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3849_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3850_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3851_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3852_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3853_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3854_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3855_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3856_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3857_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_3858_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_3859_antisym__conv3,axiom,
! [Y2: rat,X: rat] :
( ~ ( ord_less_rat @ Y2 @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_3860_antisym__conv3,axiom,
! [Y2: num,X: num] :
( ~ ( ord_less_num @ Y2 @ X )
=> ( ( ~ ( ord_less_num @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_3861_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_3862_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_3863_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_3864_linorder__cases,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_rat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_3865_linorder__cases,axiom,
! [X: num,Y2: num] :
( ~ ( ord_less_num @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_num @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_3866_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_3867_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_3868_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_3869_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_3870_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_3871_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_3872_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_3873_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_3874_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_3875_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_3876_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_3877_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_3878_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ~ ( P3 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_3879_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3880_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B2: rat] :
( ( ord_less_rat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: rat] : ( P @ A3 @ A3 )
=> ( ! [A3: rat,B2: rat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3881_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num] : ( P @ A3 @ A3 )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3882_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3883_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3884_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_3885_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_3886_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_3887_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_3888_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_3889_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3890_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( ( ord_less_rat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3891_not__less__iff__gr__or__eq,axiom,
! [X: num,Y2: num] :
( ( ~ ( ord_less_num @ X @ Y2 ) )
= ( ( ord_less_num @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3892_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3893_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3894_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_3895_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_3896_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_3897_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_3898_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_3899_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3900_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3901_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3902_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3903_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3904_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3905_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3906_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3907_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3908_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3909_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_3910_linorder__neqE,axiom,
! [X: rat,Y2: rat] :
( ( X != Y2 )
=> ( ~ ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_3911_linorder__neqE,axiom,
! [X: num,Y2: num] :
( ( X != Y2 )
=> ( ~ ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_3912_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_3913_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_3914_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_3915_order__less__asym,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_3916_order__less__asym,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_3917_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_3918_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_3919_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3920_linorder__neq__iff,axiom,
! [X: rat,Y2: rat] :
( ( X != Y2 )
= ( ( ord_less_rat @ X @ Y2 )
| ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3921_linorder__neq__iff,axiom,
! [X: num,Y2: num] :
( ( X != Y2 )
= ( ( ord_less_num @ X @ Y2 )
| ( ord_less_num @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3922_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3923_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3924_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_3925_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_3926_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_3927_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_3928_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_3929_order__less__trans,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3930_order__less__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3931_order__less__trans,axiom,
! [X: num,Y2: num,Z: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3932_order__less__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3933_order__less__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3934_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3935_ord__eq__less__subst,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3936_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3937_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3938_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3939_ord__eq__less__subst,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3940_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3941_ord__eq__less__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3942_ord__eq__less__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3943_ord__eq__less__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3944_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3945_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3946_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3947_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3948_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3949_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3950_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3951_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3952_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3953_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3954_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_3955_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_3956_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_3957_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_3958_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_3959_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3960_order__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3961_order__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3962_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3963_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3964_order__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3965_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3966_order__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3967_order__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3968_order__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3969_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3970_order__less__subst2,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3971_order__less__subst2,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3972_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3973_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3974_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3975_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3976_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3977_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3978_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3979_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_3980_order__less__not__sym,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_3981_order__less__not__sym,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_3982_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_3983_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_3984_order__less__imp__triv,axiom,
! [X: real,Y2: real,P: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3985_order__less__imp__triv,axiom,
! [X: rat,Y2: rat,P: $o] :
( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_rat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3986_order__less__imp__triv,axiom,
! [X: num,Y2: num,P: $o] :
( ( ord_less_num @ X @ Y2 )
=> ( ( ord_less_num @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3987_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3988_order__less__imp__triv,axiom,
! [X: int,Y2: int,P: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3989_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_3990_linorder__less__linear,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_rat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_3991_linorder__less__linear,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_num @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_3992_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_3993_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_3994_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_3995_order__less__imp__not__eq,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_3996_order__less__imp__not__eq,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_3997_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_3998_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_3999_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_4000_order__less__imp__not__eq2,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_4001_order__less__imp__not__eq2,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_4002_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_4003_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_4004_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_4005_order__less__imp__not__less,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_4006_order__less__imp__not__less,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_4007_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_4008_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_4009_full__exhaustive__int_H_Ocases,axiom,
! [X: produc2285326912895808259nt_int] :
~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I4: int] :
( X
!= ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I4 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_4010_exhaustive__int_H_Ocases,axiom,
! [X: produc7773217078559923341nt_int] :
~ ! [F2: int > option6357759511663192854e_term,D3: int,I4: int] :
( X
!= ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I4 ) ) ) ).
% exhaustive_int'.cases
thf(fact_4011_small__lazy_H_Ocases,axiom,
! [X: product_prod_int_int] :
~ ! [D3: int,I4: int] :
( X
!= ( product_Pair_int_int @ D3 @ I4 ) ) ).
% small_lazy'.cases
thf(fact_4012_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_4013_leD,axiom,
! [Y2: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y2 @ X )
=> ~ ( ord_less_set_int @ X @ Y2 ) ) ).
% leD
thf(fact_4014_leD,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_eq_rat @ Y2 @ X )
=> ~ ( ord_less_rat @ X @ Y2 ) ) ).
% leD
thf(fact_4015_leD,axiom,
! [Y2: num,X: num] :
( ( ord_less_eq_num @ Y2 @ X )
=> ~ ( ord_less_num @ X @ Y2 ) ) ).
% leD
thf(fact_4016_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_4017_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_4018_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_4019_leI,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X ) ) ).
% leI
thf(fact_4020_leI,axiom,
! [X: num,Y2: num] :
( ~ ( ord_less_num @ X @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X ) ) ).
% leI
thf(fact_4021_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_4022_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_4023_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_4024_nless__le,axiom,
! [A: set_int,B: set_int] :
( ( ~ ( ord_less_set_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_4025_nless__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_4026_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_4027_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_4028_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_4029_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_4030_antisym__conv1,axiom,
! [X: set_int,Y2: set_int] :
( ~ ( ord_less_set_int @ X @ Y2 )
=> ( ( ord_less_eq_set_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_4031_antisym__conv1,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_4032_antisym__conv1,axiom,
! [X: num,Y2: num] :
( ~ ( ord_less_num @ X @ Y2 )
=> ( ( ord_less_eq_num @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_4033_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_4034_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_4035_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_4036_antisym__conv2,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ( ~ ( ord_less_set_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_4037_antisym__conv2,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_4038_antisym__conv2,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ( ~ ( ord_less_num @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_4039_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_4040_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_4041_dense__ge,axiom,
! [Z: real,Y2: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_4042_dense__ge,axiom,
! [Z: rat,Y2: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ord_less_eq_rat @ Y2 @ X3 ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_4043_dense__le,axiom,
! [Y2: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ).
% dense_le
thf(fact_4044_dense__le,axiom,
! [Y2: rat,Z: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ X3 @ Z ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ).
% dense_le
thf(fact_4045_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_4046_less__le__not__le,axiom,
( ord_less_set_int
= ( ^ [X2: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y )
& ~ ( ord_less_eq_set_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_4047_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
& ~ ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_4048_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
& ~ ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_4049_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_4050_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_4051_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_4052_not__le__imp__less,axiom,
! [Y2: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y2 @ X )
=> ( ord_less_rat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_4053_not__le__imp__less,axiom,
! [Y2: num,X: num] :
( ~ ( ord_less_eq_num @ Y2 @ X )
=> ( ord_less_num @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_4054_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_4055_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_4056_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_4057_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B3: set_int] :
( ( ord_less_set_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_4058_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_rat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_4059_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_4060_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_4061_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_4062_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_4063_order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [A4: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_4064_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_4065_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_4066_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_4067_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_4068_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_4069_order_Ostrict__trans1,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_set_int @ B @ C )
=> ( ord_less_set_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_4070_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_4071_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_4072_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_4073_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_4074_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_4075_order_Ostrict__trans2,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_set_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_4076_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_4077_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_4078_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_4079_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_4080_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_4081_order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [A4: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A4 @ B3 )
& ~ ( ord_less_eq_set_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_4082_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ A4 @ B3 )
& ~ ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_4083_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_4084_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_4085_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_4086_dense__ge__bounded,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y2 @ W2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_4087_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ! [W2: rat] :
( ( ord_less_rat @ Z @ W2 )
=> ( ( ord_less_rat @ W2 @ X )
=> ( ord_less_eq_rat @ Y2 @ W2 ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_4088_dense__le__bounded,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y2 )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_4089_dense__le__bounded,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ! [W2: rat] :
( ( ord_less_rat @ X @ W2 )
=> ( ( ord_less_rat @ W2 @ Y2 )
=> ( ord_less_eq_rat @ W2 @ Z ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_4090_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_4091_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [B3: set_int,A4: set_int] :
( ( ord_less_set_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_4092_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_less_rat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_4093_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_num @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_4094_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_4095_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_4096_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_4097_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [B3: set_int,A4: set_int] :
( ( ord_less_eq_set_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_4098_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_less_eq_rat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_4099_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_4100_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_4101_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_4102_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_4103_dual__order_Ostrict__trans1,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_set_int @ C @ B )
=> ( ord_less_set_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_4104_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_4105_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_4106_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_4107_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_4108_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_4109_dual__order_Ostrict__trans2,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C @ B )
=> ( ord_less_set_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_4110_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_4111_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_4112_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_4113_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_4114_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_4115_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [B3: set_int,A4: set_int] :
( ( ord_less_eq_set_int @ B3 @ A4 )
& ~ ( ord_less_eq_set_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_4116_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_less_eq_rat @ B3 @ A4 )
& ~ ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_4117_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_4118_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_4119_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_4120_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_4121_order_Ostrict__implies__order,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_4122_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_4123_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_4124_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_4125_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_4126_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_4127_dual__order_Ostrict__implies__order,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ord_less_eq_set_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_4128_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_4129_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_4130_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_4131_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_4132_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_4133_order__le__less,axiom,
( ord_less_eq_set_int
= ( ^ [X2: set_int,Y: set_int] :
( ( ord_less_set_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_4134_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_4135_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_4136_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_4137_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_4138_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_4139_order__less__le,axiom,
( ord_less_set_int
= ( ^ [X2: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_4140_order__less__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_4141_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_4142_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_4143_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_4144_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_4145_linorder__not__le,axiom,
! [X: rat,Y2: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y2 ) )
= ( ord_less_rat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_4146_linorder__not__le,axiom,
! [X: num,Y2: num] :
( ( ~ ( ord_less_eq_num @ X @ Y2 ) )
= ( ord_less_num @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_4147_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_4148_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_4149_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_4150_linorder__not__less,axiom,
! [X: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( ord_less_eq_rat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_4151_linorder__not__less,axiom,
! [X: num,Y2: num] :
( ( ~ ( ord_less_num @ X @ Y2 ) )
= ( ord_less_eq_num @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_4152_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_4153_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_4154_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_4155_order__less__imp__le,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_set_int @ X @ Y2 )
=> ( ord_less_eq_set_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_4156_order__less__imp__le,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_4157_order__less__imp__le,axiom,
! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_eq_num @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_4158_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_4159_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_4160_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_4161_order__le__neq__trans,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_4162_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_4163_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_4164_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_4165_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_4166_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_4167_order__neq__le__trans,axiom,
! [A: set_int,B: set_int] :
( ( A != B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_4168_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_4169_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_4170_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_4171_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_4172_order__le__less__trans,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_4173_order__le__less__trans,axiom,
! [X: set_int,Y2: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ( ord_less_set_int @ Y2 @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_4174_order__le__less__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_4175_order__le__less__trans,axiom,
! [X: num,Y2: num,Z: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_4176_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_4177_order__le__less__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_4178_order__less__le__trans,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_4179_order__less__le__trans,axiom,
! [X: set_int,Y2: set_int,Z: set_int] :
( ( ord_less_set_int @ X @ Y2 )
=> ( ( ord_less_eq_set_int @ Y2 @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_4180_order__less__le__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_4181_order__less__le__trans,axiom,
! [X: num,Y2: num,Z: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_4182_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_4183_order__less__le__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_4184_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4185_order__le__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4186_order__le__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4187_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4188_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4189_order__le__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4190_order__le__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4191_order__le__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4192_order__le__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4193_order__le__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_4194_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4195_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4196_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4197_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4198_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4199_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4200_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4201_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4202_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4203_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_4204_order__less__le__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4205_order__less__le__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4206_order__less__le__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4207_order__less__le__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4208_order__less__le__subst1,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4209_order__less__le__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4210_order__less__le__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4211_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4212_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4213_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4214_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4215_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4216_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4217_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4218_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4219_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4220_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4221_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4222_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4223_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > rat,C: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4224_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_4225_linorder__le__less__linear,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
| ( ord_less_rat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_4226_linorder__le__less__linear,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
| ( ord_less_num @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_4227_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_4228_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_4229_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4230_order__le__imp__less__or__eq,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ( ord_less_set_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4231_order__le__imp__less__or__eq,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_rat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4232_order__le__imp__less__or__eq,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ( ord_less_num @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4233_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4234_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4235_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_4236_bot_Oextremum__uniqueI,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
=> ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
thf(fact_4237_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_4238_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_4239_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_4240_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_4241_bot_Oextremum__unique,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
= ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
thf(fact_4242_bot_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_4243_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_4244_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_4245_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_4246_bot_Oextremum,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% bot.extremum
thf(fact_4247_bot_Oextremum,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% bot.extremum
thf(fact_4248_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_4249_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_4250_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_4251_bot_Oextremum__strict,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% bot.extremum_strict
thf(fact_4252_bot_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_4253_bot_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_4254_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_4255_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_4256_bot_Onot__eq__extremum,axiom,
! [A: extended_enat] :
( ( A != bot_bo4199563552545308370d_enat )
= ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_4257_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% bot.not_eq_extremum
thf(fact_4258_bot_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% bot.not_eq_extremum
thf(fact_4259_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_4260_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
= ( P @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_4261_max__absorb2,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y2 )
=> ( ( ord_ma741700101516333627d_enat @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_4262_max__absorb2,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ( ord_max_set_int @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_4263_max__absorb2,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_max_rat @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_4264_max__absorb2,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ( ord_max_num @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_4265_max__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_max_nat @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_4266_max__absorb2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_max_int @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_4267_max__absorb1,axiom,
! [Y2: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y2 @ X )
=> ( ( ord_ma741700101516333627d_enat @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_4268_max__absorb1,axiom,
! [Y2: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y2 @ X )
=> ( ( ord_max_set_int @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_4269_max__absorb1,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_eq_rat @ Y2 @ X )
=> ( ( ord_max_rat @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_4270_max__absorb1,axiom,
! [Y2: num,X: num] :
( ( ord_less_eq_num @ Y2 @ X )
=> ( ( ord_max_num @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_4271_max__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_max_nat @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_4272_max__absorb1,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_max_int @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_4273_max__def,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4274_max__def,axiom,
( ord_max_set_int
= ( ^ [A4: set_int,B3: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4275_max__def,axiom,
( ord_max_rat
= ( ^ [A4: rat,B3: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4276_max__def,axiom,
( ord_max_num
= ( ^ [A4: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4277_max__def,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4278_max__def,axiom,
( ord_max_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4279_infinite__growing,axiom,
! [X8: set_real] :
( ( X8 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X8 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X8 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X8 ) ) ) ).
% infinite_growing
thf(fact_4280_infinite__growing,axiom,
! [X8: set_rat] :
( ( X8 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ X8 )
=> ? [Xa: rat] :
( ( member_rat @ Xa @ X8 )
& ( ord_less_rat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_rat @ X8 ) ) ) ).
% infinite_growing
thf(fact_4281_infinite__growing,axiom,
! [X8: set_num] :
( ( X8 != bot_bot_set_num )
=> ( ! [X3: num] :
( ( member_num @ X3 @ X8 )
=> ? [Xa: num] :
( ( member_num @ Xa @ X8 )
& ( ord_less_num @ X3 @ Xa ) ) )
=> ~ ( finite_finite_num @ X8 ) ) ) ).
% infinite_growing
thf(fact_4282_infinite__growing,axiom,
! [X8: set_nat] :
( ( X8 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X8 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X8 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_4283_infinite__growing,axiom,
! [X8: set_int] :
( ( X8 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X8 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X8 )
& ( ord_less_int @ X3 @ Xa ) ) )
=> ~ ( finite_finite_int @ X8 ) ) ) ).
% infinite_growing
thf(fact_4284_ex__min__if__finite,axiom,
! [S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ( S3 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S3 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S3 )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4285_ex__min__if__finite,axiom,
! [S3: set_rat] :
( ( finite_finite_rat @ S3 )
=> ( ( S3 != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ S3 )
& ~ ? [Xa: rat] :
( ( member_rat @ Xa @ S3 )
& ( ord_less_rat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4286_ex__min__if__finite,axiom,
! [S3: set_num] :
( ( finite_finite_num @ S3 )
=> ( ( S3 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ S3 )
& ~ ? [Xa: num] :
( ( member_num @ Xa @ S3 )
& ( ord_less_num @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4287_ex__min__if__finite,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S3 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S3 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4288_ex__min__if__finite,axiom,
! [S3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S3 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S3 )
& ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4289_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_4290_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_4291_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_4292_option_Osize__gen_I2_J,axiom,
! [X: nat > nat,X22: nat] :
( ( size_option_nat @ X @ ( some_nat @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_4293_option_Osize__gen_I2_J,axiom,
! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_4294_option_Osize__gen_I2_J,axiom,
! [X: num > nat,X22: num] :
( ( size_option_num @ X @ ( some_num @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_4295_even__succ__div__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_4296_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_4297_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_4298_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_4299_signed__take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_4300_diff__shunt__var,axiom,
! [X: set_real,Y2: set_real] :
( ( ( minus_minus_set_real @ X @ Y2 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_4301_diff__shunt__var,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_4302_diff__shunt__var,axiom,
! [X: set_int,Y2: set_int] :
( ( ( minus_minus_set_int @ X @ Y2 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_4303_add__scale__eq__noteq,axiom,
! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4304_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4305_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4306_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4307_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4308_artanh__def,axiom,
( artanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_4309_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_4310_dvd__add__triv__left__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4311_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4312_dvd__add__triv__left__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4313_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4314_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4315_dvd__add__triv__right__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4316_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4317_dvd__add__triv__right__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4318_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4319_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4320_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_4321_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_4322_div__dvd__div,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ C )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_4323_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_4324_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_4325_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_4326_signed__take__bit__of__0,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_4327_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4328_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4329_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4330_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4331_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4332_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4333_dvd__mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4334_dvd__mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4335_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4336_dvd__mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4337_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4338_dvd__mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4339_dvd__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4340_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4341_dvd__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4342_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4343_unit__prod,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% unit_prod
thf(fact_4344_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_4345_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_4346_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4347_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4348_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4349_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4350_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4351_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4352_dvd__add__times__triv__left__iff,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4353_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4354_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4355_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4356_dvd__mult__div__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_4357_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_4358_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_4359_dvd__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_4360_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_4361_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_4362_unit__div__1__div__1,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_4363_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_4364_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_4365_unit__div__1__unit,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% unit_div_1_unit
thf(fact_4366_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_4367_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_4368_unit__div,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% unit_div
thf(fact_4369_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_4370_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_4371_div__add,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_4372_div__add,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_4373_div__add,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_4374_div__diff,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_4375_div__diff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_4376_ln__le__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_4377_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_4378_signed__take__bit__numeral__of__1,axiom,
! [K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_numeral_of_1
thf(fact_4379_unit__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_4380_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_4381_unit__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_4382_unit__mult__div__div,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
= ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_4383_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_4384_unit__mult__div__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_4385_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_4386_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_4387_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_4388_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_4389_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_4390_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_4391_even__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_4392_even__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_4393_even__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_4394_even__add,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_4395_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_4396_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_4397_odd__add,axiom,
! [A: code_integer,B: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_4398_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_4399_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_4400_even__mod__2__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_4401_even__mod__2__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_4402_even__mod__2__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_4403_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_4404_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_4405_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_4406_zero__le__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_4407_zero__le__power__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_4408_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_4409_power__less__zero__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_4410_power__less__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_4411_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_4412_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_4413_power__less__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_4414_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_4415_even__plus__one__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_4416_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_4417_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_4418_even__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% even_diff
thf(fact_4419_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_4420_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_4421_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_4422_zero__less__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_4423_zero__less__power__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_4424_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_4425_even__succ__div__2,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_4426_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_4427_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_4428_even__succ__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_4429_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_4430_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_4431_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% odd_succ_div_two
thf(fact_4432_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_4433_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_4434_even__power,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_4435_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_4436_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_4437_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_4438_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_4439_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_4440_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_4441_power__le__zero__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_4442_power__le__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_4443_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_4444_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4445_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4446_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4447_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_4448_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A4: complex,B3: complex] :
( ( A4 = zero_zero_complex )
=> ( B3 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_4449_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A4: real,B3: real] :
( ( A4 = zero_zero_real )
=> ( B3 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_4450_dvd__field__iff,axiom,
( dvd_dvd_rat
= ( ^ [A4: rat,B3: rat] :
( ( A4 = zero_zero_rat )
=> ( B3 = zero_zero_rat ) ) ) ) ).
% dvd_field_iff
thf(fact_4451_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ~ ! [K2: code_integer] :
( A
!= ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4452_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K2: real] :
( A
!= ( times_times_real @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4453_dvdE,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ~ ! [K2: rat] :
( A
!= ( times_times_rat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4454_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K2: nat] :
( A
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4455_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K2: int] :
( A
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4456_dvdI,axiom,
! [A: code_integer,B: code_integer,K: code_integer] :
( ( A
= ( times_3573771949741848930nteger @ B @ K ) )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% dvdI
thf(fact_4457_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_4458_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A
= ( times_times_rat @ B @ K ) )
=> ( dvd_dvd_rat @ B @ A ) ) ).
% dvdI
thf(fact_4459_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_4460_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_4461_dvd__def,axiom,
( dvd_dvd_Code_integer
= ( ^ [B3: code_integer,A4: code_integer] :
? [K3: code_integer] :
( A4
= ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4462_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B3: real,A4: real] :
? [K3: real] :
( A4
= ( times_times_real @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4463_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B3: rat,A4: rat] :
? [K3: rat] :
( A4
= ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4464_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A4: nat] :
? [K3: nat] :
( A4
= ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4465_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A4: int] :
? [K3: int] :
( A4
= ( times_times_int @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4466_dvd__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ C )
=> ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4467_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4468_dvd__mult,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4469_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4470_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4471_dvd__mult2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4472_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4473_dvd__mult2,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4474_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4475_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4476_dvd__mult__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
=> ( dvd_dvd_Code_integer @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4477_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4478_dvd__mult__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
=> ( dvd_dvd_rat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4479_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4480_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4481_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4482_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4483_dvd__triv__left,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4484_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4485_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4486_mult__dvd__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C @ D )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4487_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4488_mult__dvd__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ C @ D )
=> ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4489_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4490_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4491_dvd__mult__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
=> ( dvd_dvd_Code_integer @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4492_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4493_dvd__mult__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
=> ( dvd_dvd_rat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4494_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4495_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4496_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4497_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4498_dvd__triv__right,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4499_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4500_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4501_dvd__productE,axiom,
! [P4: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P4
= ( times_times_nat @ X3 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_4502_dvd__productE,axiom,
! [P4: int,A: int,B: int] :
( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( times_times_int @ X3 @ Y3 ) )
=> ( ( dvd_dvd_int @ X3 @ A )
=> ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_4503_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B7: nat,C5: nat] :
( ( A
= ( times_times_nat @ B7 @ C5 ) )
& ( dvd_dvd_nat @ B7 @ B )
& ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% division_decomp
thf(fact_4504_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B7: int,C5: int] :
( ( A
= ( times_times_int @ B7 @ C5 ) )
& ( dvd_dvd_int @ B7 @ B )
& ( dvd_dvd_int @ C5 @ C ) ) ) ).
% division_decomp
thf(fact_4505_dvd__add,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ C )
=> ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4506_dvd__add,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4507_dvd__add,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ C )
=> ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4508_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4509_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4510_dvd__add__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ C )
=> ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4511_dvd__add__left__iff,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4512_dvd__add__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4513_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4514_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4515_dvd__add__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4516_dvd__add__right__iff,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4517_dvd__add__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( dvd_dvd_rat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4518_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4519_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4520_dvd__diff__commute,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_4521_dvd__diff__commute,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
= ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_4522_div__div__div__same,axiom,
! [D: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ D @ B )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_4523_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_4524_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_4525_dvd__div__eq__cancel,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( divide6298287555418463151nteger @ A @ C )
= ( divide6298287555418463151nteger @ B @ C ) )
=> ( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4526_dvd__div__eq__cancel,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
=> ( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4527_dvd__div__eq__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
=> ( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4528_dvd__div__eq__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
=> ( ( dvd_dvd_rat @ C @ A )
=> ( ( dvd_dvd_rat @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4529_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4530_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4531_dvd__div__eq__iff,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( ( divide6298287555418463151nteger @ A @ C )
= ( divide6298287555418463151nteger @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4532_dvd__div__eq__iff,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4533_dvd__div__eq__iff,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4534_dvd__div__eq__iff,axiom,
! [C: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ C @ A )
=> ( ( dvd_dvd_rat @ C @ B )
=> ( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4535_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4536_dvd__div__eq__iff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4537_dvd__power__same,axiom,
! [X: code_integer,Y2: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ X @ Y2 )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_4538_dvd__power__same,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_4539_dvd__power__same,axiom,
! [X: real,Y2: real,N: nat] :
( ( dvd_dvd_real @ X @ Y2 )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_4540_dvd__power__same,axiom,
! [X: int,Y2: int,N: nat] :
( ( dvd_dvd_int @ X @ Y2 )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_4541_dvd__power__same,axiom,
! [X: complex,Y2: complex,N: nat] :
( ( dvd_dvd_complex @ X @ Y2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_4542_mod__mod__cancel,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_4543_mod__mod__cancel,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
= ( modulo_modulo_int @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_4544_mod__mod__cancel,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_4545_dvd__mod,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_4546_dvd__mod,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_4547_dvd__mod,axiom,
! [K: code_integer,M: code_integer,N: code_integer] :
( ( dvd_dvd_Code_integer @ K @ M )
=> ( ( dvd_dvd_Code_integer @ K @ N )
=> ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_4548_signed__take__bit__mult,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% signed_take_bit_mult
thf(fact_4549_signed__take__bit__add,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).
% signed_take_bit_add
thf(fact_4550_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_4551_signed__take__bit__diff,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% signed_take_bit_diff
thf(fact_4552_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_4553_subset__divisors__dvd,axiom,
! [A: complex,B: complex] :
( ( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
@ ( collect_complex
@ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
= ( dvd_dvd_complex @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4554_subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4555_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4556_subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le7084787975880047091nteger
@ ( collect_Code_integer
@ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
@ ( collect_Code_integer
@ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4557_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4558_even__signed__take__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_signed_take_bit_iff
thf(fact_4559_even__signed__take__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_signed_take_bit_iff
thf(fact_4560_pinf_I9_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ Z3 @ X5 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4561_pinf_I9_J,axiom,
! [D: real,S2: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4562_pinf_I9_J,axiom,
! [D: rat,S2: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4563_pinf_I9_J,axiom,
! [D: nat,S2: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4564_pinf_I9_J,axiom,
! [D: int,S2: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4565_pinf_I10_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4566_pinf_I10_J,axiom,
! [D: real,S2: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4567_pinf_I10_J,axiom,
! [D: rat,S2: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4568_pinf_I10_J,axiom,
! [D: nat,S2: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4569_pinf_I10_J,axiom,
! [D: int,S2: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4570_minf_I9_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ X5 @ Z3 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4571_minf_I9_J,axiom,
! [D: real,S2: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4572_minf_I9_J,axiom,
! [D: rat,S2: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4573_minf_I9_J,axiom,
! [D: nat,S2: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4574_minf_I9_J,axiom,
! [D: int,S2: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4575_minf_I10_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4576_minf_I10_J,axiom,
! [D: real,S2: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4577_minf_I10_J,axiom,
! [D: rat,S2: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4578_minf_I10_J,axiom,
! [D: nat,S2: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4579_minf_I10_J,axiom,
! [D: int,S2: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4580_dvd__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4581_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4582_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4583_dvd__div__eq__0__iff,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4584_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4585_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4586_unit__mult__right__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ B @ A )
= ( times_3573771949741848930nteger @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4587_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4588_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4589_unit__mult__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ A @ B )
= ( times_3573771949741848930nteger @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4590_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4591_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4592_mult__unit__dvd__iff_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4593_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4594_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4595_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4596_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4597_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4598_mult__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4599_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4600_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4601_dvd__mult__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4602_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4603_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4604_is__unit__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
& ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% is_unit_mult_iff
thf(fact_4605_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_4606_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_4607_div__mult__div__if__dvd,axiom,
! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ D @ C )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4608_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4609_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D: int,C: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4610_dvd__mult__imp__div,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
=> ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4611_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4612_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4613_dvd__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4614_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4615_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4616_div__div__eq__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4617_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4618_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4619_div__mult__swap,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4620_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4621_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4622_dvd__div__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4623_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4624_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
= ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4625_dvd__div__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_4626_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_4627_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_4628_div__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_4629_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_4630_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_4631_unit__div__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_4632_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_4633_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_4634_div__plus__div__distrib__dvd__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_4635_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_4636_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_4637_div__plus__div__distrib__dvd__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_4638_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_4639_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_4640_div__power,axiom,
! [B: code_integer,A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
= ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% div_power
thf(fact_4641_div__power,axiom,
! [B: nat,A: nat,N: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% div_power
thf(fact_4642_div__power,axiom,
! [B: int,A: int,N: nat] :
( ( dvd_dvd_int @ B @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
= ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% div_power
thf(fact_4643_dvd__power__le,axiom,
! [X: code_integer,Y2: code_integer,N: nat,M: nat] :
( ( dvd_dvd_Code_integer @ X @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4644_dvd__power__le,axiom,
! [X: nat,Y2: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4645_dvd__power__le,axiom,
! [X: real,Y2: real,N: nat,M: nat] :
( ( dvd_dvd_real @ X @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4646_dvd__power__le,axiom,
! [X: int,Y2: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4647_dvd__power__le,axiom,
! [X: complex,Y2: complex,N: nat,M: nat] :
( ( dvd_dvd_complex @ X @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4648_power__le__dvd,axiom,
! [A: code_integer,N: nat,B: code_integer,M: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4649_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4650_power__le__dvd,axiom,
! [A: real,N: nat,B: real,M: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4651_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4652_power__le__dvd,axiom,
! [A: complex,N: nat,B: complex,M: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4653_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4654_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4655_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4656_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4657_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4658_mod__eq__dvd__iff,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
= ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% mod_eq_dvd_iff
thf(fact_4659_mod__eq__dvd__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% mod_eq_dvd_iff
thf(fact_4660_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_4661_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_4662_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_4663_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_4664_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_4665_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_4666_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_4667_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_4668_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_4669_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_4670_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_4671_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_4672_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_4673_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_4674_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_4675_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_4676_div2__even__ext__nat,axiom,
! [X: nat,Y2: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% div2_even_ext_nat
thf(fact_4677_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C2: code_integer] :
( B
!= ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_4678_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] :
( B
!= ( times_times_nat @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_4679_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] :
( B
!= ( times_times_int @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_4680_unity__coeff__ex,axiom,
! [P: code_integer > $o,L2: code_integer] :
( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X2 ) ) )
= ( ? [X2: code_integer] :
( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4681_unity__coeff__ex,axiom,
! [P: complex > $o,L2: complex] :
( ( ? [X2: complex] : ( P @ ( times_times_complex @ L2 @ X2 ) ) )
= ( ? [X2: complex] :
( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4682_unity__coeff__ex,axiom,
! [P: real > $o,L2: real] :
( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
= ( ? [X2: real] :
( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4683_unity__coeff__ex,axiom,
! [P: rat > $o,L2: rat] :
( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
= ( ? [X2: rat] :
( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4684_unity__coeff__ex,axiom,
! [P: nat > $o,L2: nat] :
( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
= ( ? [X2: nat] :
( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4685_unity__coeff__ex,axiom,
! [P: int > $o,L2: int] :
( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
= ( ? [X2: int] :
( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4686_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C @ D )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ D @ C ) )
= ( ( times_3573771949741848930nteger @ B @ C )
= ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4687_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B @ C )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4688_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B @ C )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4689_dvd__div__iff__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4690_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4691_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4692_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4693_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4694_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4695_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= C )
= ( B
= ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4696_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( ( divide_divide_nat @ B @ A )
= C )
= ( B
= ( times_times_nat @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4697_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( ( divide_divide_int @ B @ A )
= C )
= ( B
= ( times_times_int @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4698_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
thf(fact_4699_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_4700_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_4701_even__numeral,axiom,
! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_4702_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_4703_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_4704_inf__period_I3_J,axiom,
! [D: code_integer,D4: code_integer,T: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D4 )
=> ! [X5: code_integer,K4: code_integer] :
( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4705_inf__period_I3_J,axiom,
! [D: real,D4: real,T: real] :
( ( dvd_dvd_real @ D @ D4 )
=> ! [X5: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4706_inf__period_I3_J,axiom,
! [D: rat,D4: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D4 )
=> ! [X5: rat,K4: rat] :
( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4707_inf__period_I3_J,axiom,
! [D: int,D4: int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4708_inf__period_I4_J,axiom,
! [D: code_integer,D4: code_integer,T: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D4 )
=> ! [X5: code_integer,K4: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4709_inf__period_I4_J,axiom,
! [D: real,D4: real,T: real] :
( ( dvd_dvd_real @ D @ D4 )
=> ! [X5: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4710_inf__period_I4_J,axiom,
! [D: rat,D4: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D4 )
=> ! [X5: rat,K4: rat] :
( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4711_inf__period_I4_J,axiom,
! [D: int,D4: int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4712_is__unit__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4713_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4714_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4715_unit__div__mult__swap,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4716_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4717_unit__div__mult__swap,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4718_unit__div__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_4719_unit__div__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_4720_unit__div__commute,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_4721_div__mult__unit2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4722_div__mult__unit2,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4723_div__mult__unit2,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4724_unit__eq__div2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( A
= ( divide6298287555418463151nteger @ C @ B ) )
= ( ( times_3573771949741848930nteger @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4725_unit__eq__div2,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( A
= ( divide_divide_nat @ C @ B ) )
= ( ( times_times_nat @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4726_unit__eq__div2,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( A
= ( divide_divide_int @ C @ B ) )
= ( ( times_times_int @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4727_unit__eq__div1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= C )
= ( A
= ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_4728_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= C )
= ( A
= ( times_times_nat @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_4729_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= C )
= ( A
= ( times_times_int @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_4730_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_4731_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_4732_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
thf(fact_4733_is__unit__power__iff,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_4734_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_4735_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_4736_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_4737_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_4738_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_4739_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_4740_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_4741_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_4742_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( ( modulo_modulo_nat @ M @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_4743_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M5: nat] :
( X
!= ( product_Pair_nat_nat @ K2 @ M5 ) ) ).
% prod_decode_aux.cases
thf(fact_4744_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_4745_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_4746_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_4747_ln__mult,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y2 ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).
% ln_mult
thf(fact_4748_even__zero,axiom,
dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% even_zero
thf(fact_4749_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_4750_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_4751_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B2: code_integer] :
( ( B2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
= B2 )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
= A )
=> ( ( ( times_3573771949741848930nteger @ A @ B2 )
= one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ C @ A )
!= ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4752_is__unitE,axiom,
! [A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B2 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B2 )
= A )
=> ( ( ( times_times_nat @ A @ B2 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A )
!= ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4753_is__unitE,axiom,
! [A: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B2: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B2 )
=> ( ( ( divide_divide_int @ one_one_int @ B2 )
= A )
=> ( ( ( times_times_int @ A @ B2 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A )
!= ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4754_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4755_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4756_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4757_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4758_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4759_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4760_evenE,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: code_integer] :
( A
!= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_4761_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_4762_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_4763_odd__one,axiom,
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% odd_one
thf(fact_4764_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_4765_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_4766_odd__even__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_4767_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_4768_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_4769_bit__eq__rec,axiom,
( ( ^ [Y5: code_integer,Z5: code_integer] : Y5 = Z5 )
= ( ^ [A4: code_integer,B3: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
& ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ B3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_4770_bit__eq__rec,axiom,
( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
= ( ^ [A4: nat,B3: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
& ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_4771_bit__eq__rec,axiom,
( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
= ( ^ [A4: int,B3: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
& ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_4772_dvd__power__iff,axiom,
! [X: code_integer,M: nat,N: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
= ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_4773_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_4774_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_4775_dvd__power,axiom,
! [N: nat,X: code_integer] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_Code_integer ) )
=> ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% dvd_power
thf(fact_4776_dvd__power,axiom,
! [N: nat,X: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_rat ) )
=> ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% dvd_power
thf(fact_4777_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_4778_dvd__power,axiom,
! [N: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% dvd_power
thf(fact_4779_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_4780_dvd__power,axiom,
! [N: nat,X: complex] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_complex ) )
=> ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% dvd_power
thf(fact_4781_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_4782_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_4783_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_4784_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq_nat @ Q2 @ N )
=> ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_4785_power__dvd__imp__le,axiom,
! [I2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_4786_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M: nat] :
( ( ord_less_nat @ R2 @ N )
=> ( ( ord_less_eq_nat @ R2 @ M )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
=> ( ( modulo_modulo_nat @ M @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_4787_mod__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
= ( ( dvd_dvd_int @ L2 @ K )
| ( ( L2 = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K ) )
| ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% mod_int_pos_iff
thf(fact_4788_bset_I9_J,axiom,
! [D: int,D4: int,B4: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% bset(9)
thf(fact_4789_bset_I10_J,axiom,
! [D: int,D4: int,B4: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% bset(10)
thf(fact_4790_aset_I9_J,axiom,
! [D: int,D4: int,A2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(9)
thf(fact_4791_aset_I10_J,axiom,
! [D: int,D4: int,A2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(10)
thf(fact_4792_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_4793_ln__diff__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y2 ) @ Y2 ) ) ) ) ).
% ln_diff_le
thf(fact_4794_even__two__times__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_4795_even__two__times__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_4796_even__two__times__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_4797_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_4798_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_4799_even__iff__mod__2__eq__zero,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_4800_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_4801_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_4802_odd__iff__mod__2__eq__one,axiom,
! [A: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_4803_power__mono__odd,axiom,
! [N: nat,A: real,B: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_4804_power__mono__odd,axiom,
! [N: nat,A: rat,B: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_4805_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_4806_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_4807_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_4808_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_4809_even__unset__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_4810_even__unset__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_4811_even__unset__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_4812_even__set__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_4813_even__set__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_4814_even__set__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_4815_even__flip__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4816_even__flip__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4817_even__flip__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4818_even__diff__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% even_diff_iff
thf(fact_4819_oddE,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: code_integer] :
( A
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).
% oddE
thf(fact_4820_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_4821_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_4822_parity__cases,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_4823_parity__cases,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_4824_parity__cases,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger ) )
=> ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer ) ) ) ).
% parity_cases
thf(fact_4825_mod2__eq__if,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_4826_mod2__eq__if,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_4827_mod2__eq__if,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ) ).
% mod2_eq_if
thf(fact_4828_zero__le__even__power,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_4829_zero__le__even__power,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_4830_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_4831_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_4832_zero__le__odd__power,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% zero_le_odd_power
thf(fact_4833_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_4834_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_4835_zero__le__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_4836_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_4837_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_4838_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_4839_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_4840_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_4841_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B
= ( plus_plus_rat @ B @ A ) )
= ( A = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_4842_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_4843_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_4844_crossproduct__eq,axiom,
! [W: real,Y2: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y2 ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_4845_crossproduct__eq,axiom,
! [W: rat,Y2: rat,X: rat,Z: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y2 ) @ ( times_times_rat @ X @ Z ) )
= ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_4846_crossproduct__eq,axiom,
! [W: nat,Y2: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y2 ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_4847_crossproduct__eq,axiom,
! [W: int,Y2: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y2 ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_4848_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4849_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
!= ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4850_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4851_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4852_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_4853_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_4854_zero__less__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_4855_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_4856_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
=> ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_4857_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_4858_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_4859_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_4860_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_4861_power__le__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_4862_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_4863_option_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_option_nat @ X @ none_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_4864_option_Osize__gen_I1_J,axiom,
! [X: product_prod_nat_nat > nat] :
( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_4865_option_Osize__gen_I1_J,axiom,
! [X: num > nat] :
( ( size_option_num @ X @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_4866_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_4867_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_4868_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4869_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4870_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4871_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4872_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4873_even__mult__exp__div__exp__iff,axiom,
! [A: int,M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4874_ln__2__less__1,axiom,
ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% ln_2_less_1
thf(fact_4875_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_4876_vebt__buildup_Oelims,axiom,
! [X: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y2 )
=> ( ( ( X = zero_zero_nat )
=> ( Y2
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( Y2
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y2
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y2
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_4877_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( tanh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% tanh_ln_real
thf(fact_4878_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_4879_signed__take__bit__rec,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4880_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4881_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_4882_intind,axiom,
! [I2: nat,N: nat,P: int > $o,X: int] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P @ X )
=> ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_4883_intind,axiom,
! [I2: nat,N: nat,P: nat > $o,X: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P @ X )
=> ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_4884_intind,axiom,
! [I2: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P @ X )
=> ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_4885_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4886_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4887_add_Oinverse__inverse,axiom,
! [A: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4888_add_Oinverse__inverse,axiom,
! [A: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4889_add_Oinverse__inverse,axiom,
! [A: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4890_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4891_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4892_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4893_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4894_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4895_Compl__anti__mono,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B4 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_4896_Compl__subset__Compl__iff,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B4 ) )
= ( ord_less_eq_set_int @ B4 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_4897_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_4898_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_4899_abs__idempotent,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_idempotent
thf(fact_4900_abs__idempotent,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_idempotent
thf(fact_4901_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4902_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4903_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4904_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4905_compl__le__compl__iff,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y2 ) )
= ( ord_less_eq_set_int @ Y2 @ X ) ) ).
% compl_le_compl_iff
thf(fact_4906_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_4907_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_4908_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_4909_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_4910_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_4911_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4912_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4913_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4914_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A ) )
= ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4915_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A ) )
= ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4916_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_4917_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_4918_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_4919_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_4920_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
thf(fact_4921_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_4922_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_4923_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_4924_equal__neg__zero,axiom,
! [A: rat] :
( ( A
= ( uminus_uminus_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_4925_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_4926_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_4927_neg__equal__zero,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_4928_neg__equal__zero,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= A )
= ( A = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_4929_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4930_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4931_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4932_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4933_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4934_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4935_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4936_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4937_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4938_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4939_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4940_mult__minus__right,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4941_mult__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4942_mult__minus__right,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4943_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4944_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4945_minus__mult__minus,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( times_times_complex @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4946_minus__mult__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( times_3573771949741848930nteger @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4947_minus__mult__minus,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( times_times_rat @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4948_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4949_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4950_mult__minus__left,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4951_mult__minus__left,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4952_mult__minus__left,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4953_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_4954_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_4955_minus__add__distrib,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_add_distrib
thf(fact_4956_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_add_distrib
thf(fact_4957_minus__add__distrib,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_add_distrib
thf(fact_4958_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4959_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4960_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4961_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4962_minus__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4963_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4964_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4965_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4966_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4967_add__minus__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4968_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4969_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4970_minus__diff__eq,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
= ( minus_minus_complex @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4971_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4972_minus__diff__eq,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
= ( minus_minus_rat @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4973_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_4974_div__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ).
% div_minus_minus
thf(fact_4975_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( abs_abs_Code_integer @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
thf(fact_4976_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_4977_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( abs_abs_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% abs_0_eq
thf(fact_4978_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_4979_abs__eq__0,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
thf(fact_4980_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_4981_abs__eq__0,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0
thf(fact_4982_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_4983_abs__zero,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_zero
thf(fact_4984_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_4985_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_zero
thf(fact_4986_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_4987_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_numeral
thf(fact_4988_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_numeral
thf(fact_4989_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_numeral
thf(fact_4990_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_numeral
thf(fact_4991_abs__mult__self__eq,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
= ( times_3573771949741848930nteger @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_4992_abs__mult__self__eq,axiom,
! [A: real] :
( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
= ( times_times_real @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_4993_abs__mult__self__eq,axiom,
! [A: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
= ( times_times_rat @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_4994_abs__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
= ( times_times_int @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_4995_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
= ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_add_abs
thf(fact_4996_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_4997_abs__add__abs,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_add_abs
thf(fact_4998_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_4999_abs__divide,axiom,
! [A: complex,B: complex] :
( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% abs_divide
thf(fact_5000_abs__divide,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_divide
thf(fact_5001_abs__divide,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_divide
thf(fact_5002_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_5003_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_5004_abs__minus__cancel,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus_cancel
thf(fact_5005_abs__minus__cancel,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus_cancel
thf(fact_5006_mod__minus__minus,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_5007_mod__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_5008_length__replicate,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_5009_length__replicate,axiom,
! [N: nat,X: $o] :
( ( size_size_list_o @ ( replicate_o @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_5010_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_5011_length__replicate,axiom,
! [N: nat,X: int] :
( ( size_size_list_int @ ( replicate_int @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_5012_tanh__real__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% tanh_real_le_iff
thf(fact_5013_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_5014_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_le_iff_le
thf(fact_5015_neg__0__le__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_5016_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_5017_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_5018_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_le_0_iff_le
thf(fact_5019_neg__le__0__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_le_0_iff_le
thf(fact_5020_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_5021_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_5022_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_5023_less__eq__neg__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_5024_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_5025_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_5026_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_5027_neg__less__eq__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_5028_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_5029_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_5030_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_5031_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_0_iff_less
thf(fact_5032_neg__less__0__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_0_iff_less
thf(fact_5033_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_5034_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_5035_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_less_iff_less
thf(fact_5036_neg__0__less__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_5037_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_5038_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_5039_neg__less__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_pos
thf(fact_5040_neg__less__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_pos
thf(fact_5041_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_5042_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_5043_less__neg__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_5044_less__neg__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_5045_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_5046_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_5047_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_5048_ab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_5049_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_5050_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_5051_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_5052_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_5053_add_Oright__inverse,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_5054_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_5055_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_5056_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_5057_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_5058_diff__0,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
= ( uminus1351360451143612070nteger @ A ) ) ).
% diff_0
thf(fact_5059_diff__0,axiom,
! [A: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A )
= ( uminus_uminus_rat @ A ) ) ).
% diff_0
thf(fact_5060_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5061_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5062_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5063_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5064_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5065_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_5066_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_5067_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_5068_mult__minus1,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1
thf(fact_5069_mult__minus1,axiom,
! [Z: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1
thf(fact_5070_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_5071_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_5072_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_5073_mult__minus1__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1_right
thf(fact_5074_mult__minus1__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1_right
thf(fact_5075_abs__of__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5076_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5077_abs__of__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5078_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5079_abs__le__self__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% abs_le_self_iff
thf(fact_5080_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_5081_abs__le__self__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% abs_le_self_iff
thf(fact_5082_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_5083_abs__le__zero__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
thf(fact_5084_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_5085_abs__le__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
thf(fact_5086_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_5087_zero__less__abs__iff,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
= ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
thf(fact_5088_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_5089_zero__less__abs__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
thf(fact_5090_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_5091_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_5092_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_5093_uminus__add__conv__diff,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( minus_minus_complex @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_5094_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_5095_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( minus_minus_rat @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_5096_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_5097_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_5098_diff__minus__eq__add,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( plus_plus_complex @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_5099_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_5100_diff__minus__eq__add,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( plus_plus_rat @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_5101_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_5102_div__minus1__right,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ A ) ) ).
% div_minus1_right
thf(fact_5103_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_5104_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_5105_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_5106_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_neg_numeral
thf(fact_5107_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_neg_numeral
thf(fact_5108_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_neg_numeral
thf(fact_5109_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_neg_numeral
thf(fact_5110_abs__neg__one,axiom,
( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
= one_one_real ) ).
% abs_neg_one
thf(fact_5111_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_5112_abs__neg__one,axiom,
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= one_one_Code_integer ) ).
% abs_neg_one
thf(fact_5113_abs__neg__one,axiom,
( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= one_one_rat ) ).
% abs_neg_one
thf(fact_5114_minus__mod__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_5115_minus__mod__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_5116_abs__power__minus,axiom,
! [A: real,N: nat] :
( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_5117_abs__power__minus,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_5118_abs__power__minus,axiom,
! [A: code_integer,N: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_5119_abs__power__minus,axiom,
! [A: rat,N: nat] :
( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_5120_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% signed_take_bit_of_minus_1
thf(fact_5121_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_5122_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_5123_Ball__set__replicate,axiom,
! [N: nat,A: int,P: int > $o] :
( ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_5124_Ball__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_5125_Ball__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_5126_Bex__set__replicate,axiom,
! [N: nat,A: int,P: int > $o] :
( ( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_5127_Bex__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_5128_Bex__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ? [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_5129_in__set__replicate,axiom,
! [X: real,N: nat,Y2: real] :
( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5130_in__set__replicate,axiom,
! [X: complex,N: nat,Y2: complex] :
( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5131_in__set__replicate,axiom,
! [X: product_prod_nat_nat,N: nat,Y2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5132_in__set__replicate,axiom,
! [X: int,N: nat,Y2: int] :
( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5133_in__set__replicate,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5134_in__set__replicate,axiom,
! [X: vEBT_VEBT,N: nat,Y2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5135_nth__replicate,axiom,
! [I2: nat,N: nat,X: int] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5136_nth__replicate,axiom,
! [I2: nat,N: nat,X: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5137_nth__replicate,axiom,
! [I2: nat,N: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5138_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_5139_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% tanh_real_nonpos_iff
thf(fact_5140_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5141_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5142_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5143_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5144_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5145_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_5146_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_5147_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_5148_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_5149_add__neg__numeral__special_I7_J,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(7)
thf(fact_5150_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% add_neg_numeral_special(7)
thf(fact_5151_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_5152_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_5153_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_5154_add__neg__numeral__special_I8_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(8)
thf(fact_5155_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= zero_zero_rat ) ).
% add_neg_numeral_special(8)
thf(fact_5156_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_5157_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_5158_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_5159_diff__numeral__special_I12_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(12)
thf(fact_5160_diff__numeral__special_I12_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% diff_numeral_special(12)
thf(fact_5161_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5162_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5163_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5164_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5165_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5166_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5167_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5168_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5169_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5170_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5171_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_5172_zero__le__divide__abs__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( B = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_5173_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A @ zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_5174_divide__le__0__abs__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
= ( ( ord_less_eq_rat @ A @ zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
thf(fact_5175_abs__of__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5176_abs__of__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5177_abs__of__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5178_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5179_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5180_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5181_left__minus__one__mult__self,axiom,
! [N: nat,A: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5182_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5183_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5184_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_5185_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_5186_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_5187_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
= one_one_Code_integer ) ).
% minus_one_mult_self
thf(fact_5188_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
= one_one_rat ) ).
% minus_one_mult_self
thf(fact_5189_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_5190_mod__minus1__right,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_5191_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_5192_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( numera6620942414471956472nteger @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_5193_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( numeral_numeral_rat @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_5194_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_5195_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5196_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ V ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5197_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5198_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5199_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5200_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5201_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5202_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5203_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(168)
thf(fact_5204_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(168)
thf(fact_5205_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y2: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(168)
thf(fact_5206_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y2: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y2 ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(168)
thf(fact_5207_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y2: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(168)
thf(fact_5208_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5209_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5210_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5211_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5212_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5213_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5214_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5215_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5216_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5217_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5218_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5219_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5220_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5221_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5222_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5223_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5224_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5225_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5226_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5227_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5228_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5229_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5230_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5231_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5232_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5233_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y2 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_5234_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y2 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_5235_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y2 ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_5236_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y2 ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_5237_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y2 ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_5238_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_5239_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_5240_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_5241_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y2 ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_5242_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_5243_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(172)
thf(fact_5244_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(172)
thf(fact_5245_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y2: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(172)
thf(fact_5246_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y2: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y2 ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(172)
thf(fact_5247_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y2: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(172)
thf(fact_5248_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5249_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5250_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5251_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5252_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5253_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5254_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5255_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5256_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5257_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5258_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5259_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5260_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_5261_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_5262_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_5263_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_5264_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5265_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= A )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5266_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5267_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5268_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= B ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5269_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5270_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5271_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5272_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5273_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5274_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_5275_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_5276_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_5277_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_5278_power2__minus,axiom,
! [A: real] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5279_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5280_power2__minus,axiom,
! [A: complex] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5281_power2__minus,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5282_power2__minus,axiom,
! [A: rat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5283_zero__less__power__abs__iff,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
= ( ( A != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5284_zero__less__power__abs__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= ( ( A != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5285_zero__less__power__abs__iff,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
= ( ( A != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5286_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5287_abs__power2,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_5288_abs__power2,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_5289_abs__power2,axiom,
! [A: real] :
( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_5290_abs__power2,axiom,
! [A: int] :
( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_5291_power2__abs,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_5292_power2__abs,axiom,
! [A: rat] :
( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_5293_power2__abs,axiom,
! [A: real] :
( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_5294_power2__abs,axiom,
! [A: int] :
( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_5295_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5296_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5297_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5298_add__neg__numeral__special_I9_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5299_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5300_diff__numeral__special_I11_J,axiom,
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5301_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5302_diff__numeral__special_I11_J,axiom,
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5303_diff__numeral__special_I11_J,axiom,
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5304_diff__numeral__special_I11_J,axiom,
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5305_diff__numeral__special_I10_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5306_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5307_diff__numeral__special_I10_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5308_diff__numeral__special_I10_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5309_diff__numeral__special_I10_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5310_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_5311_minus__1__div__2__eq,axiom,
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% minus_1_div_2_eq
thf(fact_5312_minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% minus_1_mod_2_eq
thf(fact_5313_minus__1__mod__2__eq,axiom,
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% minus_1_mod_2_eq
thf(fact_5314_bits__minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_minus_1_mod_2_eq
thf(fact_5315_bits__minus__1__mod__2__eq,axiom,
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% bits_minus_1_mod_2_eq
thf(fact_5316_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5317_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5318_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5319_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5320_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5321_power__minus__odd,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5322_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5323_power__minus__odd,axiom,
! [N: nat,A: complex] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5324_power__minus__odd,axiom,
! [N: nat,A: code_integer] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5325_power__minus__odd,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5326_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5327_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5328_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: complex] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_power_complex @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5329_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5330_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5331_power__even__abs__numeral,axiom,
! [W: num,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5332_power__even__abs__numeral,axiom,
! [W: num,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5333_power__even__abs__numeral,axiom,
! [W: num,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5334_power__even__abs__numeral,axiom,
! [W: num,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5335_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5336_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5337_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5338_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5339_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5340_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5341_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5342_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5343_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5344_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5345_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_5346_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5347_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5348_dbl__simps_I4_J,axiom,
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5349_dbl__simps_I4_J,axiom,
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5350_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5351_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_real ) ).
% power_minus1_even
thf(fact_5352_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_5353_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_complex ) ).
% power_minus1_even
thf(fact_5354_power__minus1__even,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_Code_integer ) ).
% power_minus1_even
thf(fact_5355_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_rat ) ).
% power_minus1_even
thf(fact_5356_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= ( uminus_uminus_real @ one_one_real ) ) ) ).
% neg_one_odd_power
thf(fact_5357_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_5358_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% neg_one_odd_power
thf(fact_5359_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% neg_one_odd_power
thf(fact_5360_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% neg_one_odd_power
thf(fact_5361_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= one_one_real ) ) ).
% neg_one_even_power
thf(fact_5362_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_5363_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= one_one_complex ) ) ).
% neg_one_even_power
thf(fact_5364_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) ) ).
% neg_one_even_power
thf(fact_5365_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) ) ).
% neg_one_even_power
thf(fact_5366_signed__take__bit__0,axiom,
! [A: code_integer] :
( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5367_signed__take__bit__0,axiom,
! [A: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5368_abs__leI,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
=> ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5369_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5370_abs__leI,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5371_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5372_abs__le__D2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5373_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5374_abs__le__D2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5375_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5376_abs__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5377_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le3102999989581377725nteger @ A @ B )
& ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5378_abs__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_eq_rat @ A @ B )
& ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5379_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5380_abs__ge__minus__self,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% abs_ge_minus_self
thf(fact_5381_abs__ge__minus__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_minus_self
thf(fact_5382_abs__ge__minus__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% abs_ge_minus_self
thf(fact_5383_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_5384_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_5385_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_5386_abs__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_real @ A @ B )
& ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_5387_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_5388_abs__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le6747313008572928689nteger @ A @ B )
& ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_5389_abs__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_rat @ A @ B )
& ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_5390_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_5391_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_5392_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% equation_minus_iff
thf(fact_5393_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% equation_minus_iff
thf(fact_5394_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% equation_minus_iff
thf(fact_5395_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_5396_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_5397_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_5398_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( uminus1351360451143612070nteger @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_5399_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( uminus_uminus_rat @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_5400_abs__eq__iff_H,axiom,
! [A: real,B: real] :
( ( ( abs_abs_real @ A )
= B )
= ( ( ord_less_eq_real @ zero_zero_real @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5401_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= B )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
& ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5402_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( ( abs_abs_rat @ A )
= B )
= ( ( ord_less_eq_rat @ zero_zero_rat @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5403_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5404_eq__abs__iff_H,axiom,
! [A: real,B: real] :
( ( A
= ( abs_abs_real @ B ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_real @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5405_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( abs_abs_Code_integer @ B ) )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ( B = A )
| ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5406_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A
= ( abs_abs_rat @ B ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_rat @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5407_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5408_abs__minus__le__zero,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% abs_minus_le_zero
thf(fact_5409_abs__minus__le__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_5410_abs__minus__le__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% abs_minus_le_zero
thf(fact_5411_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_5412_abs__if__raw,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_5413_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_5414_abs__if__raw,axiom,
( abs_abs_Code_integer
= ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_5415_abs__if__raw,axiom,
( abs_abs_rat
= ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_5416_abs__if,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_5417_abs__if,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_5418_abs__if,axiom,
( abs_abs_Code_integer
= ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_5419_abs__if,axiom,
( abs_abs_rat
= ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_5420_abs__of__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_neg
thf(fact_5421_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_5422_abs__of__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_neg
thf(fact_5423_abs__of__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_neg
thf(fact_5424_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_real_def
thf(fact_5425_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_5426_abs__ge__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_self
thf(fact_5427_abs__ge__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% abs_ge_self
thf(fact_5428_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_5429_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_5430_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% abs_le_D1
thf(fact_5431_abs__le__D1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% abs_le_D1
thf(fact_5432_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_5433_abs__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_mult
thf(fact_5434_abs__mult,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_mult
thf(fact_5435_abs__mult,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_mult
thf(fact_5436_abs__mult,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_mult
thf(fact_5437_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5438_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5439_abs__minus__commute,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5440_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5441_power__abs,axiom,
! [A: code_integer,N: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
= ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% power_abs
thf(fact_5442_power__abs,axiom,
! [A: rat,N: nat] :
( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
= ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% power_abs
thf(fact_5443_power__abs,axiom,
! [A: real,N: nat] :
( ( abs_abs_real @ ( power_power_real @ A @ N ) )
= ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% power_abs
thf(fact_5444_power__abs,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ A @ N ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% power_abs
thf(fact_5445_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5446_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5447_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5448_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5449_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5450_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5451_minus__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5452_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5453_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_5454_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_minus_iff
thf(fact_5455_le__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% le_minus_iff
thf(fact_5456_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_5457_compl__mono,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X @ Y2 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y2 ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% compl_mono
thf(fact_5458_compl__le__swap1,axiom,
! [Y2: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y2 @ ( uminus1532241313380277803et_int @ X ) )
=> ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_5459_compl__le__swap2,axiom,
! [Y2: set_int,X: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y2 ) @ X )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_5460_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5461_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5462_verit__negate__coefficient_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5463_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5464_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_5465_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_5466_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% less_minus_iff
thf(fact_5467_less__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% less_minus_iff
thf(fact_5468_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5469_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5470_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5471_minus__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5472_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5473_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5474_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
!= ( numera6690914467698888265omplex @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5475_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
!= ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5476_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
!= ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5477_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5478_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5479_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6690914467698888265omplex @ M )
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5480_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6620942414471956472nteger @ M )
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5481_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_rat @ M )
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5482_minus__mult__commute,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_5483_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_5484_minus__mult__commute,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_mult_commute
thf(fact_5485_minus__mult__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_mult_commute
thf(fact_5486_minus__mult__commute,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_mult_commute
thf(fact_5487_square__eq__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5488_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5489_square__eq__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ A )
= ( times_times_complex @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5490_square__eq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ A )
= ( times_3573771949741848930nteger @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5491_square__eq__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ A )
= ( times_times_rat @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5492_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_5493_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_5494_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_5495_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_5496_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_5497_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5498_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5499_is__num__normalize_I8_J,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5500_is__num__normalize_I8_J,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5501_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5502_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5503_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5504_add_Oinverse__distrib__swap,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5505_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5506_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5507_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5508_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5509_group__cancel_Oneg1,axiom,
! [A2: complex,K: complex,A: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( uminus1482373934393186551omplex @ A2 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5510_group__cancel_Oneg1,axiom,
! [A2: code_integer,K: code_integer,A: code_integer] :
( ( A2
= ( plus_p5714425477246183910nteger @ K @ A ) )
=> ( ( uminus1351360451143612070nteger @ A2 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5511_group__cancel_Oneg1,axiom,
! [A2: rat,K: rat,A: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( uminus_uminus_rat @ A2 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5512_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_5513_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_5514_minus__diff__commute,axiom,
! [B: complex,A: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_5515_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
= ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_5516_minus__diff__commute,axiom,
! [B: rat,A: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_5517_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_5518_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_5519_minus__diff__minus,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_5520_minus__diff__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_5521_minus__diff__minus,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_5522_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_5523_div__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% div_minus_right
thf(fact_5524_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_5525_minus__divide__right,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_divide_right
thf(fact_5526_minus__divide__right,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_divide_right
thf(fact_5527_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_5528_minus__divide__divide,axiom,
! [A: complex,B: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ).
% minus_divide_divide
thf(fact_5529_minus__divide__divide,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ).
% minus_divide_divide
thf(fact_5530_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_5531_minus__divide__left,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_5532_minus__divide__left,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_5533_mod__minus__eq,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_5534_mod__minus__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_5535_mod__minus__cong,axiom,
! [A: int,B: int,A5: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A5 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_5536_mod__minus__cong,axiom,
! [A: code_integer,B: code_integer,A5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= ( modulo364778990260209775nteger @ A5 @ B ) )
=> ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_5537_mod__minus__right,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_5538_mod__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_5539_abs__ge__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_zero
thf(fact_5540_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_5541_abs__ge__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% abs_ge_zero
thf(fact_5542_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_5543_abs__of__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5544_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5545_abs__of__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5546_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5547_abs__not__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_5548_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_5549_abs__not__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_5550_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_5551_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_5552_abs__triangle__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_5553_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_5554_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_5555_abs__mult__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
=> ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_5556_abs__mult__less,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
=> ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_5557_abs__mult__less,axiom,
! [A: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
=> ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
=> ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_5558_abs__mult__less,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_5559_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_5560_abs__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_5561_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_5562_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_5563_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_5564_abs__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_5565_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_5566_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_5567_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5568_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5569_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5570_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5571_nonzero__abs__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_5572_nonzero__abs__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_5573_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5574_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5575_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5576_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5577_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5578_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5579_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5580_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5581_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5582_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5583_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5584_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5585_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5586_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5587_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5588_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5589_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5590_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5591_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5592_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5593_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5594_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_5595_le__minus__one__simps_I2_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% le_minus_one_simps(2)
thf(fact_5596_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_5597_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_5598_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_5599_le__minus__one__simps_I4_J,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(4)
thf(fact_5600_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_5601_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_5602_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_5603_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_5604_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_5605_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_5606_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_5607_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5608_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5609_neg__eq__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5610_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5611_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5612_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5613_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5614_eq__neg__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5615_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5616_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5617_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5618_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5619_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5620_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5621_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5622_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5623_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5624_ab__group__add__class_Oab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5625_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5626_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5627_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5628_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5629_add__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5630_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5631_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5632_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_5633_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_5634_less__minus__one__simps_I2_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% less_minus_one_simps(2)
thf(fact_5635_less__minus__one__simps_I2_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% less_minus_one_simps(2)
thf(fact_5636_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_5637_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_5638_less__minus__one__simps_I4_J,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(4)
thf(fact_5639_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(4)
thf(fact_5640_numeral__times__minus__swap,axiom,
! [W: num,X: real] :
( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
= ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5641_numeral__times__minus__swap,axiom,
! [W: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5642_numeral__times__minus__swap,axiom,
! [W: num,X: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
= ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5643_numeral__times__minus__swap,axiom,
! [W: num,X: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
= ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5644_numeral__times__minus__swap,axiom,
! [W: num,X: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
= ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5645_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5646_nonzero__minus__divide__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5647_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5648_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5649_nonzero__minus__divide__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5650_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5651_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5652_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5653_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5654_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5655_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5656_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_5657_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_5658_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ N )
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% numeral_neq_neg_one
thf(fact_5659_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ N )
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% numeral_neq_neg_one
thf(fact_5660_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_rat @ N )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% numeral_neq_neg_one
thf(fact_5661_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_5662_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_5663_square__eq__1__iff,axiom,
! [X: complex] :
( ( ( times_times_complex @ X @ X )
= one_one_complex )
= ( ( X = one_one_complex )
| ( X
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% square_eq_1_iff
thf(fact_5664_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( ( times_3573771949741848930nteger @ X @ X )
= one_one_Code_integer )
= ( ( X = one_one_Code_integer )
| ( X
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% square_eq_1_iff
thf(fact_5665_square__eq__1__iff,axiom,
! [X: rat] :
( ( ( times_times_rat @ X @ X )
= one_one_rat )
= ( ( X = one_one_rat )
| ( X
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% square_eq_1_iff
thf(fact_5666_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5667_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5668_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5669_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5670_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5671_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5672_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5673_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5674_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5675_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5676_group__cancel_Osub2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B4 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5677_group__cancel_Osub2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5678_group__cancel_Osub2,axiom,
! [B4: complex,K: complex,B: complex,A: complex] :
( ( B4
= ( plus_plus_complex @ K @ B ) )
=> ( ( minus_minus_complex @ A @ B4 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5679_group__cancel_Osub2,axiom,
! [B4: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B4
= ( plus_p5714425477246183910nteger @ K @ B ) )
=> ( ( minus_8373710615458151222nteger @ A @ B4 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5680_group__cancel_Osub2,axiom,
! [B4: rat,K: rat,B: rat,A: rat] :
( ( B4
= ( plus_plus_rat @ K @ B ) )
=> ( ( minus_minus_rat @ A @ B4 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5681_replicate__eqI,axiom,
! [Xs2: list_real,N: nat,X: real] :
( ( ( size_size_list_real @ Xs2 )
= N )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_real @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5682_replicate__eqI,axiom,
! [Xs2: list_complex,N: nat,X: complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= N )
=> ( ! [Y3: complex] :
( ( member_complex @ Y3 @ ( set_complex2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_complex @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5683_replicate__eqI,axiom,
! [Xs2: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= N )
=> ( ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5684_replicate__eqI,axiom,
! [Xs2: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= N )
=> ( ! [Y3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5685_replicate__eqI,axiom,
! [Xs2: list_o,N: nat,X: $o] :
( ( ( size_size_list_o @ Xs2 )
= N )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_o @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5686_replicate__eqI,axiom,
! [Xs2: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5687_replicate__eqI,axiom,
! [Xs2: list_int,N: nat,X: int] :
( ( ( size_size_list_int @ Xs2 )
= N )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_int @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5688_replicate__length__same,axiom,
! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5689_replicate__length__same,axiom,
! [Xs2: list_o,X: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5690_replicate__length__same,axiom,
! [Xs2: list_nat,X: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5691_replicate__length__same,axiom,
! [Xs2: list_int,X: int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5692_dvd__neg__div,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5693_dvd__neg__div,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5694_dvd__neg__div,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5695_dvd__neg__div,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5696_dvd__neg__div,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5697_dvd__div__neg,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5698_dvd__div__neg,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5699_dvd__div__neg,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5700_dvd__div__neg,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5701_dvd__div__neg,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5702_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_5703_subset__Compl__self__eq,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% subset_Compl_self_eq
thf(fact_5704_subset__Compl__self__eq,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_5705_sin__bound__lemma,axiom,
! [X: real,Y2: real,U: real,V: real] :
( ( X = Y2 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y2 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_5706_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_5707_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_5708_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_5709_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_5710_dense__eq0__I,axiom,
! [X: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_5711_dense__eq0__I,axiom,
! [X: rat] :
( ! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
thf(fact_5712_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
| ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
& ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
| ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
=> ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5713_abs__eq__mult,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( ord_less_eq_real @ A @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B )
| ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5714_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( ord_less_eq_rat @ A @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B )
| ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5715_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5716_abs__mult__pos,axiom,
! [X: code_integer,Y2: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y2 ) @ X )
= ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y2 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5717_abs__mult__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ ( abs_abs_real @ Y2 ) @ X )
= ( abs_abs_real @ ( times_times_real @ Y2 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5718_abs__mult__pos,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y2 ) @ X )
= ( abs_abs_rat @ ( times_times_rat @ Y2 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5719_abs__mult__pos,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y2 ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y2 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5720_abs__div__pos,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y2 )
= ( abs_abs_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% abs_div_pos
thf(fact_5721_abs__div__pos,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y2 )
= ( abs_abs_rat @ ( divide_divide_rat @ X @ Y2 ) ) ) ) ).
% abs_div_pos
thf(fact_5722_zero__le__power__abs,axiom,
! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5723_zero__le__power__abs,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5724_zero__le__power__abs,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5725_zero__le__power__abs,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5726_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_5727_abs__triangle__ineq4,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_5728_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_5729_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_5730_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5731_abs__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5732_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5733_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5734_abs__diff__le__iff,axiom,
! [X: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
= ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
& ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5735_abs__diff__le__iff,axiom,
! [X: real,A: real,R2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
& ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5736_abs__diff__le__iff,axiom,
! [X: rat,A: rat,R2: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
& ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5737_abs__diff__le__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5738_abs__diff__less__iff,axiom,
! [X: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
= ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
& ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5739_abs__diff__less__iff,axiom,
! [X: real,A: real,R2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
= ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5740_abs__diff__less__iff,axiom,
! [X: rat,A: rat,R2: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
= ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5741_abs__diff__less__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5742_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5743_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5744_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5745_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5746_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_5747_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_5748_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_5749_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_5750_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_5751_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_5752_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_5753_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_5754_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5755_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5756_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5757_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5758_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_5759_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_5760_le__minus__one__simps_I1_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% le_minus_one_simps(1)
thf(fact_5761_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_5762_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_5763_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_5764_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(3)
thf(fact_5765_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_5766_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_5767_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_5768_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_5769_less__minus__one__simps_I1_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% less_minus_one_simps(1)
thf(fact_5770_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_5771_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_5772_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_5773_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(3)
thf(fact_5774_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5775_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5776_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5777_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5778_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_5779_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_5780_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_le_neg_one
thf(fact_5781_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_5782_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_5783_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% neg_numeral_le_neg_one
thf(fact_5784_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% neg_numeral_le_neg_one
thf(fact_5785_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_5786_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_le_numeral
thf(fact_5787_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_le_numeral
thf(fact_5788_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_le_numeral
thf(fact_5789_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_5790_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_5791_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_5792_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_le_one
thf(fact_5793_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_5794_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_5795_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_5796_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_less_one
thf(fact_5797_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_less_one
thf(fact_5798_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_less_numeral
thf(fact_5799_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_5800_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_less_numeral
thf(fact_5801_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_less_numeral
thf(fact_5802_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_5803_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_5804_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_less_neg_one
thf(fact_5805_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_less_neg_one
thf(fact_5806_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5807_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5808_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5809_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5810_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5811_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5812_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5813_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5814_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5815_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5816_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= ( uminus_uminus_rat @ B ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5817_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5818_minus__divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
= A )
= ( ( ( C != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B )
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5819_minus__divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
= A )
= ( ( ( C != zero_zero_rat )
=> ( ( uminus_uminus_rat @ B )
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5820_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5821_nonzero__neg__divide__eq__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( B != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= C )
= ( ( uminus1482373934393186551omplex @ A )
= ( times_times_complex @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5822_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( B != zero_zero_rat )
=> ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= C )
= ( ( uminus_uminus_rat @ A )
= ( times_times_rat @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5823_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5824_nonzero__neg__divide__eq__eq2,axiom,
! [B: complex,C: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( C
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ( times_times_complex @ C @ B )
= ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5825_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
= ( ( times_times_rat @ C @ B )
= ( uminus_uminus_rat @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5826_mult__1s__ring__1_I2_J,axiom,
! [B: real] :
( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5827_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5828_mult__1s__ring__1_I2_J,axiom,
! [B: complex] :
( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
= ( uminus1482373934393186551omplex @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5829_mult__1s__ring__1_I2_J,axiom,
! [B: code_integer] :
( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
= ( uminus1351360451143612070nteger @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5830_mult__1s__ring__1_I2_J,axiom,
! [B: rat] :
( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
= ( uminus_uminus_rat @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5831_mult__1s__ring__1_I1_J,axiom,
! [B: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5832_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5833_mult__1s__ring__1_I1_J,axiom,
! [B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5834_mult__1s__ring__1_I1_J,axiom,
! [B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5835_mult__1s__ring__1_I1_J,axiom,
! [B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
= ( uminus_uminus_rat @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5836_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5837_divide__eq__minus__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B != zero_zero_complex )
& ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5838_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ( B != zero_zero_rat )
& ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5839_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_5840_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_5841_uminus__numeral__One,axiom,
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% uminus_numeral_One
thf(fact_5842_uminus__numeral__One,axiom,
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% uminus_numeral_One
thf(fact_5843_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_5844_power__minus,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% power_minus
thf(fact_5845_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_5846_power__minus,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_minus
thf(fact_5847_power__minus,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_minus
thf(fact_5848_power__minus,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_minus
thf(fact_5849_power__minus__Bit0,axiom,
! [X: real,K: num] :
( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5850_power__minus__Bit0,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5851_power__minus__Bit0,axiom,
! [X: complex,K: num] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5852_power__minus__Bit0,axiom,
! [X: code_integer,K: num] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5853_power__minus__Bit0,axiom,
! [X: rat,K: num] :
( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5854_real__0__less__add__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y2 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_5855_real__add__less__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
= ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_5856_real__0__le__add__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y2 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_5857_real__add__le__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_5858_abs__add__one__gt__zero,axiom,
! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5859_abs__add__one__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5860_abs__add__one__gt__zero,axiom,
! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5861_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5862_power__even__abs,axiom,
! [N: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_even_abs
thf(fact_5863_power__even__abs,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) ) ).
% power_even_abs
thf(fact_5864_power__even__abs,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) ) ).
% power_even_abs
thf(fact_5865_power__even__abs,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% power_even_abs
thf(fact_5866_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5867_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5868_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5869_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5870_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5871_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5872_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5873_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5874_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5875_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5876_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5877_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5878_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5879_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C: complex,W: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5880_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5881_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5882_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: complex,C: complex] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5883_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5884_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5885_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5886_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5887_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5888_minus__divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y2: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y2 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5889_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y2 )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5890_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5891_add__divide__eq__if__simps_I6_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5892_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5893_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5894_add__divide__eq__if__simps_I5_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5895_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5896_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y2 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5897_minus__divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y2: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y2 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5898_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y2 )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5899_even__minus,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_5900_even__minus,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_5901_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
=> ( ( ord_less_eq_real @ A @ Y4 )
& ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_5902_power2__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus_real @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_5903_power2__eq__iff,axiom,
! [X: int,Y2: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus_int @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_5904_power2__eq__iff,axiom,
! [X: complex,Y2: complex] :
( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus1482373934393186551omplex @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_5905_power2__eq__iff,axiom,
! [X: code_integer,Y2: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus1351360451143612070nteger @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_5906_power2__eq__iff,axiom,
! [X: rat,Y2: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus_rat @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_5907_uminus__power__if,axiom,
! [N: nat,A: real] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5908_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5909_uminus__power__if,axiom,
! [N: nat,A: complex] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_power_complex @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5910_uminus__power__if,axiom,
! [N: nat,A: code_integer] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5911_uminus__power__if,axiom,
! [N: nat,A: rat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5912_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_5913_verit__less__mono__div__int2,axiom,
! [A2: int,B4: int,N: int] :
( ( ord_less_eq_int @ A2 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_5914_abs__le__square__iff,axiom,
! [X: code_integer,Y2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y2 ) )
= ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5915_abs__le__square__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y2 ) )
= ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5916_abs__le__square__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y2 ) )
= ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5917_abs__le__square__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y2 ) )
= ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5918_power__mono__even,axiom,
! [N: nat,A: code_integer,B: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5919_power__mono__even,axiom,
! [N: nat,A: real,B: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5920_power__mono__even,axiom,
! [N: nat,A: rat,B: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5921_power__mono__even,axiom,
! [N: nat,A: int,B: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5922_abs__square__eq__1,axiom,
! [X: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( abs_abs_Code_integer @ X )
= one_one_Code_integer ) ) ).
% abs_square_eq_1
thf(fact_5923_abs__square__eq__1,axiom,
! [X: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( abs_abs_rat @ X )
= one_one_rat ) ) ).
% abs_square_eq_1
thf(fact_5924_abs__square__eq__1,axiom,
! [X: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( abs_abs_real @ X )
= one_one_real ) ) ).
% abs_square_eq_1
thf(fact_5925_abs__square__eq__1,axiom,
! [X: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% abs_square_eq_1
thf(fact_5926_pos__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5927_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5928_pos__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5929_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5930_neg__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5931_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5932_neg__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5933_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5934_minus__divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5935_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5936_le__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5937_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5938_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5939_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5940_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5941_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5942_power2__eq__1__iff,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( A = one_one_real )
| ( A
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5943_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5944_power2__eq__1__iff,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex )
= ( ( A = one_one_complex )
| ( A
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5945_power2__eq__1__iff,axiom,
! [A: code_integer] :
( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( A = one_one_Code_integer )
| ( A
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5946_power2__eq__1__iff,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( A = one_one_rat )
| ( A
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5947_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= one_one_real ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% minus_one_power_iff
thf(fact_5948_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_5949_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= one_one_complex ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% minus_one_power_iff
thf(fact_5950_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% minus_one_power_iff
thf(fact_5951_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% minus_one_power_iff
thf(fact_5952_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5953_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5954_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5955_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5956_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5957_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_5958_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_5959_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_5960_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_5961_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_5962_minus__mod__int__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L2 )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
= ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% minus_mod_int_eq
thf(fact_5963_power2__le__iff__abs__le,axiom,
! [Y2: code_integer,X: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y2 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5964_power2__le__iff__abs__le,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y2 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5965_power2__le__iff__abs__le,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y2 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5966_power2__le__iff__abs__le,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y2 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5967_abs__sqrt__wlog,axiom,
! [P: code_integer > code_integer > $o,X: code_integer] :
( ! [X3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
=> ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5968_abs__sqrt__wlog,axiom,
! [P: real > real > $o,X: real] :
( ! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5969_abs__sqrt__wlog,axiom,
! [P: rat > rat > $o,X: rat] :
( ! [X3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5970_abs__sqrt__wlog,axiom,
! [P: int > int > $o,X: int] :
( ! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5971_abs__square__le__1,axiom,
! [X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% abs_square_le_1
thf(fact_5972_abs__square__le__1,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
= ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% abs_square_le_1
thf(fact_5973_abs__square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% abs_square_le_1
thf(fact_5974_abs__square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% abs_square_le_1
thf(fact_5975_abs__square__less__1,axiom,
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% abs_square_less_1
thf(fact_5976_abs__square__less__1,axiom,
! [X: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
= ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% abs_square_less_1
thf(fact_5977_abs__square__less__1,axiom,
! [X: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% abs_square_less_1
thf(fact_5978_abs__square__less__1,axiom,
! [X: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% abs_square_less_1
thf(fact_5979_zminus1__lemma,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( B != zero_zero_int )
=> ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5980_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5981_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5982_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5983_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5984_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5985_square__le__1,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% square_le_1
thf(fact_5986_square__le__1,axiom,
! [X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
=> ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% square_le_1
thf(fact_5987_square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% square_le_1
thf(fact_5988_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_5989_minus__power__mult__self,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5990_minus__power__mult__self,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5991_minus__power__mult__self,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5992_minus__power__mult__self,axiom,
! [A: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5993_minus__power__mult__self,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5994_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_5995_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_5996_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_5997_div__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_5998_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% power_minus1_odd
thf(fact_5999_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_6000_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power_minus1_odd
thf(fact_6001_power__minus1__odd,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% power_minus1_odd
thf(fact_6002_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% power_minus1_odd
thf(fact_6003_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_6004_eq__diff__eq_H,axiom,
! [X: real,Y2: real,Z: real] :
( ( X
= ( minus_minus_real @ Y2 @ Z ) )
= ( Y2
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_6005_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_6006_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6007_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_6008_arctan__double,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_6009_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_6010_vebt__buildup_Opelims,axiom,
! [X: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y2 )
=> ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y2
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y2
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_6011_flip__bit__0,axiom,
! [A: code_integer] :
( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
= ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6012_flip__bit__0,axiom,
! [A: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6013_flip__bit__0,axiom,
! [A: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6014_set__decode__0,axiom,
! [X: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_6015_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_6016_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( set_or1266510415728281911st_int @ M @ N ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_6017_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_6018_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_6019_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_6020_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_6021_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_6022_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_6023_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_6024_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_6025_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_6026_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_6027_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_6028_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_6029_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_6030_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_6031_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_6032_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_6033_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_6034_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_6035_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_6036_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_6037_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_6038_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6039_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6040_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6041_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6042_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6043_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
= ( zero_n2687167440665602831ol_nat
@ ( N
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_6044_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_6045_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_6046_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% zero_le_arctan_iff
thf(fact_6047_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% arctan_le_zero_iff
thf(fact_6048_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_6049_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_6050_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_6051_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_6052_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_6053_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_6054_sum__abs,axiom,
! [F: int > int,A2: set_int] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_6055_sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_6056_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_6057_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_6058_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_6059_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_6060_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_6061_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_6062_sum__abs__ge__zero,axiom,
! [F: int > int,A2: set_int] :
( ord_less_eq_int @ zero_zero_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_6063_sum__abs__ge__zero,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_6064_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_6065_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_6066_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_6067_odd__of__bool__self,axiom,
! [P4: $o] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
= P4 ) ).
% odd_of_bool_self
thf(fact_6068_odd__of__bool__self,axiom,
! [P4: $o] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
= P4 ) ).
% odd_of_bool_self
thf(fact_6069_odd__of__bool__self,axiom,
! [P4: $o] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
= P4 ) ).
% odd_of_bool_self
thf(fact_6070_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_6071_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_6072_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% of_bool_half_eq_0
thf(fact_6073_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_6074_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_6075_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_6076_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_6077_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6078_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6079_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6080_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6081_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6082_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6083_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_6084_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6085_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6086_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6087_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_6088_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n3304061248610475627l_real
@ ( P
& Q ) )
= ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% of_bool_conj
thf(fact_6089_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2052037380579107095ol_rat
@ ( P
& Q ) )
= ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% of_bool_conj
thf(fact_6090_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P
& Q ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% of_bool_conj
thf(fact_6091_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P
& Q ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% of_bool_conj
thf(fact_6092_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n356916108424825756nteger
@ ( P
& Q ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% of_bool_conj
thf(fact_6093_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_6094_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_6095_arctan__monotone_H,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y2 ) ) ) ).
% arctan_monotone'
thf(fact_6096_arctan__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% arctan_le_iff
thf(fact_6097_sum__mono,axiom,
! [K5: set_nat,F: nat > rat,G: nat > rat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6098_sum__mono,axiom,
! [K5: set_real,F: real > rat,G: real > rat] :
( ! [I4: real] :
( ( member_real @ I4 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6099_sum__mono,axiom,
! [K5: set_int,F: int > rat,G: int > rat] :
( ! [I4: int] :
( ( member_int @ I4 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6100_sum__mono,axiom,
! [K5: set_complex,F: complex > rat,G: complex > rat] :
( ! [I4: complex] :
( ( member_complex @ I4 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6101_sum__mono,axiom,
! [K5: set_real,F: real > nat,G: real > nat] :
( ! [I4: real] :
( ( member_real @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6102_sum__mono,axiom,
! [K5: set_int,F: int > nat,G: int > nat] :
( ! [I4: int] :
( ( member_int @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6103_sum__mono,axiom,
! [K5: set_complex,F: complex > nat,G: complex > nat] :
( ! [I4: complex] :
( ( member_complex @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6104_sum__mono,axiom,
! [K5: set_nat,F: nat > int,G: nat > int] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6105_sum__mono,axiom,
! [K5: set_real,F: real > int,G: real > int] :
( ! [I4: real] :
( ( member_real @ I4 @ K5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6106_sum__mono,axiom,
! [K5: set_complex,F: complex > int,G: complex > int] :
( ! [I4: complex] :
( ( member_complex @ I4 @ K5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_6107_sum__product,axiom,
! [F: int > int,A2: set_int,G: int > int,B4: set_int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B4 ) )
= ( groups4538972089207619220nt_int
@ ^ [I3: int] :
( groups4538972089207619220nt_int
@ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_6108_sum__product,axiom,
! [F: complex > complex,A2: set_complex,G: complex > complex,B4: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B4 ) )
= ( groups7754918857620584856omplex
@ ^ [I3: complex] :
( groups7754918857620584856omplex
@ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_6109_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B4: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B4 ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_6110_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: nat > real,B4: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B4 ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] :
( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_6111_sum__distrib__right,axiom,
! [F: int > int,A2: set_int,R2: int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
= ( groups4538972089207619220nt_int
@ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_6112_sum__distrib__right,axiom,
! [F: complex > complex,A2: set_complex,R2: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_6113_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R2: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_6114_sum__distrib__right,axiom,
! [F: nat > real,A2: set_nat,R2: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_6115_sum__distrib__left,axiom,
! [R2: int,F: int > int,A2: set_int] :
( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [N2: int] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_6116_sum__distrib__left,axiom,
! [R2: complex,F: complex > complex,A2: set_complex] :
( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_6117_sum__distrib__left,axiom,
! [R2: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_6118_sum__distrib__left,axiom,
! [R2: real,F: nat > real,A2: set_nat] :
( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_6119_sum_Odistrib,axiom,
! [G: int > int,H2: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_6120_sum_Odistrib,axiom,
! [G: complex > complex,H2: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_6121_sum_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_6122_sum_Odistrib,axiom,
! [G: nat > real,H2: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_6123_sum__divide__distrib,axiom,
! [F: complex > complex,A2: set_complex,R2: complex] :
( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_6124_sum__divide__distrib,axiom,
! [F: nat > real,A2: set_nat,R2: real] :
( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_6125_mod__sum__eq,axiom,
! [F: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_6126_mod__sum__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_6127_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6128_sum__nonneg,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6129_sum__nonneg,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6130_sum__nonneg,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6131_sum__nonneg,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6132_sum__nonneg,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6133_sum__nonneg,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6134_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6135_sum__nonneg,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6136_sum__nonneg,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_6137_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_6138_sum__nonpos,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_6139_sum__nonpos,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_6140_sum__nonpos,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6141_sum__nonpos,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6142_sum__nonpos,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6143_sum__nonpos,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6144_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_6145_sum__nonpos,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_6146_sum__nonpos,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_6147_sum__mono__inv,axiom,
! [F: real > rat,I5: set_real,G: real > rat,I2: real] :
( ( ( groups1300246762558778688al_rat @ F @ I5 )
= ( groups1300246762558778688al_rat @ G @ I5 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_real @ I2 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6148_sum__mono__inv,axiom,
! [F: nat > rat,I5: set_nat,G: nat > rat,I2: nat] :
( ( ( groups2906978787729119204at_rat @ F @ I5 )
= ( groups2906978787729119204at_rat @ G @ I5 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6149_sum__mono__inv,axiom,
! [F: int > rat,I5: set_int,G: int > rat,I2: int] :
( ( ( groups3906332499630173760nt_rat @ F @ I5 )
= ( groups3906332499630173760nt_rat @ G @ I5 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I2 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6150_sum__mono__inv,axiom,
! [F: complex > rat,I5: set_complex,G: complex > rat,I2: complex] :
( ( ( groups5058264527183730370ex_rat @ F @ I5 )
= ( groups5058264527183730370ex_rat @ G @ I5 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6151_sum__mono__inv,axiom,
! [F: real > nat,I5: set_real,G: real > nat,I2: real] :
( ( ( groups1935376822645274424al_nat @ F @ I5 )
= ( groups1935376822645274424al_nat @ G @ I5 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_real @ I2 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6152_sum__mono__inv,axiom,
! [F: int > nat,I5: set_int,G: int > nat,I2: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I5 )
= ( groups4541462559716669496nt_nat @ G @ I5 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I2 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6153_sum__mono__inv,axiom,
! [F: complex > nat,I5: set_complex,G: complex > nat,I2: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I5 )
= ( groups5693394587270226106ex_nat @ G @ I5 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6154_sum__mono__inv,axiom,
! [F: real > int,I5: set_real,G: real > int,I2: real] :
( ( ( groups1932886352136224148al_int @ F @ I5 )
= ( groups1932886352136224148al_int @ G @ I5 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_real @ I2 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6155_sum__mono__inv,axiom,
! [F: nat > int,I5: set_nat,G: nat > int,I2: nat] :
( ( ( groups3539618377306564664at_int @ F @ I5 )
= ( groups3539618377306564664at_int @ G @ I5 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6156_sum__mono__inv,axiom,
! [F: complex > int,I5: set_complex,G: complex > int,I2: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I5 )
= ( groups5690904116761175830ex_int @ G @ I5 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6157_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_6158_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_6159_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_6160_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_6161_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_6162_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% of_bool_less_eq_one
thf(fact_6163_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% of_bool_less_eq_one
thf(fact_6164_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_6165_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_6166_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% of_bool_less_eq_one
thf(fact_6167_num_Oexhaust,axiom,
! [Y2: num] :
( ( Y2 != one )
=> ( ! [X23: num] :
( Y2
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y2
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_6168_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
=> ( ! [M5: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M5: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
=> ( ! [M5: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M5: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_6169_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_6170_sum__le__included,axiom,
! [S2: set_int,T: set_int,G: int > real,I2: int > int,F: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6171_sum__le__included,axiom,
! [S2: set_int,T: set_complex,G: complex > real,I2: complex > int,F: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6172_sum__le__included,axiom,
! [S2: set_complex,T: set_int,G: int > real,I2: int > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6173_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6174_sum__le__included,axiom,
! [S2: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6175_sum__le__included,axiom,
! [S2: set_nat,T: set_int,G: int > rat,I2: int > nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6176_sum__le__included,axiom,
! [S2: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6177_sum__le__included,axiom,
! [S2: set_int,T: set_nat,G: nat > rat,I2: nat > int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6178_sum__le__included,axiom,
! [S2: set_int,T: set_int,G: int > rat,I2: int > int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6179_sum__le__included,axiom,
! [S2: set_int,T: set_complex,G: complex > rat,I2: complex > int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_6180_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > real] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A2 )
= zero_zero_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6181_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: int > real] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A2 )
= zero_zero_real )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6182_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A2 )
= zero_zero_real )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6183_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6184_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6185_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6186_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6187_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6188_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6189_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_6190_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6191_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6192_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6193_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6194_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6195_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6196_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6197_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6198_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6199_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_6200_sum_Orelated,axiom,
! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6201_sum_Orelated,axiom,
! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6202_sum_Orelated,axiom,
! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6203_sum_Orelated,axiom,
! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6204_sum_Orelated,axiom,
! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6205_sum_Orelated,axiom,
! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6206_sum_Orelated,axiom,
! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6207_sum_Orelated,axiom,
! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6208_sum_Orelated,axiom,
! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6209_sum_Orelated,axiom,
! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X15: int,Y15: int,X23: int,Y23: int] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_6210_sum__strict__mono,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6211_sum__strict__mono,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6212_sum__strict__mono,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6213_sum__strict__mono,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6214_sum__strict__mono,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6215_sum__strict__mono,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6216_sum__strict__mono,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6217_sum__strict__mono,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6218_sum__strict__mono,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6219_sum__strict__mono,axiom,
! [A2: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6220_numeral__Bit1,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_Bit1
thf(fact_6221_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_6222_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_6223_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_6224_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_6225_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_6226_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_6227_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_6228_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_6229_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_6230_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_6231_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_6232_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_6233_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_6234_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_6235_power__minus__Bit1,axiom,
! [X: real,K: num] :
( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6236_power__minus__Bit1,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6237_power__minus__Bit1,axiom,
! [X: complex,K: num] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6238_power__minus__Bit1,axiom,
! [X: code_integer,K: num] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6239_power__minus__Bit1,axiom,
! [X: rat,K: num] :
( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6240_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > real,B4: real,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S2 )
= B4 )
=> ( ( member_real @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6241_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > real,B4: real,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S2 )
= B4 )
=> ( ( member_int @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6242_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > real,B4: real,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S2 )
= B4 )
=> ( ( member_complex @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6243_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > rat,B4: rat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S2 )
= B4 )
=> ( ( member_real @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6244_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > rat,B4: rat,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S2 )
= B4 )
=> ( ( member_nat @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6245_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > rat,B4: rat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
= B4 )
=> ( ( member_int @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6246_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > rat,B4: rat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
= B4 )
=> ( ( member_complex @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6247_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > nat,B4: nat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S2 )
= B4 )
=> ( ( member_real @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6248_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > nat,B4: nat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
= B4 )
=> ( ( member_int @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6249_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > nat,B4: nat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= B4 )
=> ( ( member_complex @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_6250_sum__nonneg__0,axiom,
! [S2: set_real,F: real > real,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_real @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6251_sum__nonneg__0,axiom,
! [S2: set_int,F: int > real,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_int @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6252_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > real,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_complex @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6253_sum__nonneg__0,axiom,
! [S2: set_real,F: real > rat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_real @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6254_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > rat,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_nat @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6255_sum__nonneg__0,axiom,
! [S2: set_int,F: int > rat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_int @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6256_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > rat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_complex @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6257_sum__nonneg__0,axiom,
! [S2: set_real,F: real > nat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_real @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6258_sum__nonneg__0,axiom,
! [S2: set_int,F: int > nat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_int @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6259_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > nat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_complex @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_6260_dvd__imp__le__int,axiom,
! [I2: int,D: int] :
( ( I2 != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I2 ) ) ) ) ).
% dvd_imp_le_int
thf(fact_6261_abs__mod__less,axiom,
! [L2: int,K: int] :
( ( L2 != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% abs_mod_less
thf(fact_6262_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_code(3)
thf(fact_6263_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_6264_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_6265_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_6266_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_6267_power__numeral__odd,axiom,
! [Z: complex,W: num] :
( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_6268_power__numeral__odd,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_6269_power__numeral__odd,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_6270_power__numeral__odd,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_6271_power__numeral__odd,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_6272_sum__pos2,axiom,
! [I5: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6273_sum__pos2,axiom,
! [I5: set_int,I2: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6274_sum__pos2,axiom,
! [I5: set_complex,I2: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6275_sum__pos2,axiom,
! [I5: set_real,I2: real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6276_sum__pos2,axiom,
! [I5: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6277_sum__pos2,axiom,
! [I5: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6278_sum__pos2,axiom,
! [I5: set_complex,I2: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6279_sum__pos2,axiom,
! [I5: set_real,I2: real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6280_sum__pos2,axiom,
! [I5: set_int,I2: int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6281_sum__pos2,axiom,
! [I5: set_complex,I2: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_6282_sum__pos,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6283_sum__pos,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6284_sum__pos,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6285_sum__pos,axiom,
! [I5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6286_sum__pos,axiom,
! [I5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6287_sum__pos,axiom,
! [I5: set_int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6288_sum__pos,axiom,
! [I5: set_real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6289_sum__pos,axiom,
! [I5: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6290_sum__pos,axiom,
! [I5: set_int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6291_sum__pos,axiom,
! [I5: set_real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_6292_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ T3 )
= ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6293_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ T3 )
= ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6294_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ T3 )
= ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6295_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ T3 )
= ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6296_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T3 )
= ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6297_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ T3 )
= ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6298_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T3 )
= ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6299_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ T3 )
= ( groups1932886352136224148al_int @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6300_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ T3 )
= ( groups5690904116761175830ex_int @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6301_sum_Omono__neutral__cong__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ T3 )
= ( groups2073611262835488442omplex @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_6302_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_complex ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S3 )
= ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6303_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S3 )
= ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6304_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_real ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S3 )
= ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6305_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_rat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ S3 )
= ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6306_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_rat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S3 )
= ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6307_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S3 )
= ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6308_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S3 )
= ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6309_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_int ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ S3 )
= ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6310_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_int ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ S3 )
= ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6311_sum_Omono__neutral__cong__left,axiom,
! [T3: set_nat,S3: set_nat,H2: nat > complex,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= zero_zero_complex ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ S3 )
= ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_6312_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ T3 )
= ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6313_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T3 )
= ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6314_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T3 )
= ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6315_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ T3 )
= ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6316_sum_Omono__neutral__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ T3 )
= ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6317_sum_Omono__neutral__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ T3 )
= ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6318_sum_Omono__neutral__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > int] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ T3 )
= ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6319_sum_Omono__neutral__right,axiom,
! [T3: set_int,S3: set_int,G: int > complex] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ T3 )
= ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6320_sum_Omono__neutral__right,axiom,
! [T3: set_int,S3: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ T3 )
= ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6321_sum_Omono__neutral__right,axiom,
! [T3: set_int,S3: set_int,G: int > rat] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ T3 )
= ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_6322_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ S3 )
= ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6323_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S3 )
= ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6324_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S3 )
= ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6325_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ S3 )
= ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6326_sum_Omono__neutral__left,axiom,
! [T3: set_nat,S3: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ S3 )
= ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6327_sum_Omono__neutral__left,axiom,
! [T3: set_nat,S3: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ S3 )
= ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6328_sum_Omono__neutral__left,axiom,
! [T3: set_nat,S3: set_nat,G: nat > int] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ S3 )
= ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6329_sum_Omono__neutral__left,axiom,
! [T3: set_int,S3: set_int,G: int > complex] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ S3 )
= ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6330_sum_Omono__neutral__left,axiom,
! [T3: set_int,S3: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ S3 )
= ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6331_sum_Omono__neutral__left,axiom,
! [T3: set_int,S3: set_int,G: int > rat] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ S3 )
= ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_6332_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_complex ) )
=> ( ( ( groups5754745047067104278omplex @ G @ C4 )
= ( groups5754745047067104278omplex @ H2 @ C4 ) )
=> ( ( groups5754745047067104278omplex @ G @ A2 )
= ( groups5754745047067104278omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6333_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ C4 )
= ( groups8097168146408367636l_real @ H2 @ C4 ) )
=> ( ( groups8097168146408367636l_real @ G @ A2 )
= ( groups8097168146408367636l_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6334_sum_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ C4 )
= ( groups5808333547571424918x_real @ H2 @ C4 ) )
=> ( ( groups5808333547571424918x_real @ G @ A2 )
= ( groups5808333547571424918x_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6335_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ C4 )
= ( groups1300246762558778688al_rat @ H2 @ C4 ) )
=> ( ( groups1300246762558778688al_rat @ G @ A2 )
= ( groups1300246762558778688al_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6336_sum_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
= ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
=> ( ( groups5058264527183730370ex_rat @ G @ A2 )
= ( groups5058264527183730370ex_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6337_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ C4 )
= ( groups1935376822645274424al_nat @ H2 @ C4 ) )
=> ( ( groups1935376822645274424al_nat @ G @ A2 )
= ( groups1935376822645274424al_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6338_sum_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
= ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
=> ( ( groups5693394587270226106ex_nat @ G @ A2 )
= ( groups5693394587270226106ex_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6339_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ C4 )
= ( groups1932886352136224148al_int @ H2 @ C4 ) )
=> ( ( groups1932886352136224148al_int @ G @ A2 )
= ( groups1932886352136224148al_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6340_sum_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ C4 )
= ( groups5690904116761175830ex_int @ H2 @ C4 ) )
=> ( ( groups5690904116761175830ex_int @ G @ A2 )
= ( groups5690904116761175830ex_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6341_sum_Osame__carrierI,axiom,
! [C4: set_nat,A2: set_nat,B4: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_complex ) )
=> ( ( ( groups2073611262835488442omplex @ G @ C4 )
= ( groups2073611262835488442omplex @ H2 @ C4 ) )
=> ( ( groups2073611262835488442omplex @ G @ A2 )
= ( groups2073611262835488442omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_6342_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_complex ) )
=> ( ( ( groups5754745047067104278omplex @ G @ A2 )
= ( groups5754745047067104278omplex @ H2 @ B4 ) )
= ( ( groups5754745047067104278omplex @ G @ C4 )
= ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6343_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ A2 )
= ( groups8097168146408367636l_real @ H2 @ B4 ) )
= ( ( groups8097168146408367636l_real @ G @ C4 )
= ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6344_sum_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ A2 )
= ( groups5808333547571424918x_real @ H2 @ B4 ) )
= ( ( groups5808333547571424918x_real @ G @ C4 )
= ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6345_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ A2 )
= ( groups1300246762558778688al_rat @ H2 @ B4 ) )
= ( ( groups1300246762558778688al_rat @ G @ C4 )
= ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6346_sum_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
= ( groups5058264527183730370ex_rat @ H2 @ B4 ) )
= ( ( groups5058264527183730370ex_rat @ G @ C4 )
= ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6347_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ A2 )
= ( groups1935376822645274424al_nat @ H2 @ B4 ) )
= ( ( groups1935376822645274424al_nat @ G @ C4 )
= ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6348_sum_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
= ( groups5693394587270226106ex_nat @ H2 @ B4 ) )
= ( ( groups5693394587270226106ex_nat @ G @ C4 )
= ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6349_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ A2 )
= ( groups1932886352136224148al_int @ H2 @ B4 ) )
= ( ( groups1932886352136224148al_int @ G @ C4 )
= ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6350_sum_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ A2 )
= ( groups5690904116761175830ex_int @ H2 @ B4 ) )
= ( ( groups5690904116761175830ex_int @ G @ C4 )
= ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6351_sum_Osame__carrier,axiom,
! [C4: set_nat,A2: set_nat,B4: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A2 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= zero_zero_complex ) )
=> ( ( ( groups2073611262835488442omplex @ G @ A2 )
= ( groups2073611262835488442omplex @ H2 @ B4 ) )
= ( ( groups2073611262835488442omplex @ G @ C4 )
= ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_6352_sum_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5808333547571424918x_real @ G @ A2 )
= ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6353_sum_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5058264527183730370ex_rat @ G @ A2 )
= ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6354_sum_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5693394587270226106ex_nat @ G @ A2 )
= ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6355_sum_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5690904116761175830ex_int @ G @ A2 )
= ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6356_sum_Osubset__diff,axiom,
! [B4: set_nat,A2: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G @ A2 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6357_sum_Osubset__diff,axiom,
! [B4: set_nat,A2: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6358_sum_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups8778361861064173332t_real @ G @ A2 )
= ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups8778361861064173332t_real @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6359_sum_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G @ A2 )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups3906332499630173760nt_rat @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6360_sum_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G @ A2 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups4541462559716669496nt_nat @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6361_sum_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups4538972089207619220nt_int @ G @ B4 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_6362_sum__diff,axiom,
! [A2: set_complex,B4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
= ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6363_sum__diff,axiom,
! [A2: set_complex,B4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
= ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6364_sum__diff,axiom,
! [A2: set_complex,B4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
= ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6365_sum__diff,axiom,
! [A2: set_nat,B4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B4 ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6366_sum__diff,axiom,
! [A2: set_nat,B4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B4 ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6367_sum__diff,axiom,
! [A2: set_int,B4: set_int,F: int > real] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B4 ) )
= ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6368_sum__diff,axiom,
! [A2: set_int,B4: set_int,F: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B4 ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6369_sum__diff,axiom,
! [A2: set_int,B4: set_int,F: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B4 ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6370_sum__diff,axiom,
! [A2: set_complex,B4: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6371_sum__diff,axiom,
! [A2: set_nat,B4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B4 ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B4 ) ) ) ) ) ).
% sum_diff
thf(fact_6372_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_6373_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_6374_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_6375_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_6376_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_6377_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_6378_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_6379_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_6380_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(3)
thf(fact_6381_power3__eq__cube,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6382_power3__eq__cube,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6383_power3__eq__cube,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6384_power3__eq__cube,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6385_power3__eq__cube,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6386_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_6387_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_6388_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_6389_sum__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > real] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6390_sum__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6391_sum__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > rat] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6392_sum__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6393_sum__mono2,axiom,
! [B4: set_nat,A2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6394_sum__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > nat] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6395_sum__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6396_sum__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > int] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6397_sum__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6398_sum__mono2,axiom,
! [B4: set_nat,A2: set_nat,F: nat > int] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_6399_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_6400_of__bool__odd__eq__mod__2,axiom,
! [A: nat] :
( ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_6401_of__bool__odd__eq__mod__2,axiom,
! [A: int] :
( ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_6402_of__bool__odd__eq__mod__2,axiom,
! [A: code_integer] :
( ( zero_n356916108424825756nteger
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_6403_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6404_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6405_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6406_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6407_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6408_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6409_even__abs__add__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% even_abs_add_iff
thf(fact_6410_even__add__abs__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% even_add_abs_iff
thf(fact_6411_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_6412_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_6413_sum__strict__mono2,axiom,
! [B4: set_real,A2: set_real,B: real,F: real > real] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6414_sum__strict__mono2,axiom,
! [B4: set_complex,A2: set_complex,B: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6415_sum__strict__mono2,axiom,
! [B4: set_real,A2: set_real,B: real,F: real > rat] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6416_sum__strict__mono2,axiom,
! [B4: set_complex,A2: set_complex,B: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6417_sum__strict__mono2,axiom,
! [B4: set_nat,A2: set_nat,B: nat,F: nat > rat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B4 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6418_sum__strict__mono2,axiom,
! [B4: set_real,A2: set_real,B: real,F: real > nat] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6419_sum__strict__mono2,axiom,
! [B4: set_complex,A2: set_complex,B: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6420_sum__strict__mono2,axiom,
! [B4: set_real,A2: set_real,B: real,F: real > int] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6421_sum__strict__mono2,axiom,
! [B4: set_complex,A2: set_complex,B: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6422_sum__strict__mono2,axiom,
! [B4: set_nat,A2: set_nat,B: nat,F: nat > int] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B4 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_6423_bits__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [A3: nat] :
( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ A3 ) )
=> ( ! [A3: nat,B2: $o] :
( ( P @ A3 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6424_bits__induct,axiom,
! [P: int > $o,A: int] :
( ! [A3: int] :
( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ A3 ) )
=> ( ! [A3: int,B2: $o] :
( ( P @ A3 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6425_bits__induct,axiom,
! [P: code_integer > $o,A: code_integer] :
( ! [A3: code_integer] :
( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ A3 ) )
=> ( ! [A3: code_integer,B2: $o] :
( ( P @ A3 )
=> ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6426_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
=> ( ( ( groups7501900531339628137nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6427_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
=> ( ( ( groups7713935264441627589nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7713935264441627589nteger
@ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6428_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
=> ( ( ( groups7873554091576472773nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7873554091576472773nteger
@ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6429_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
=> ( ( ( groups6621422865394947399nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups6621422865394947399nteger
@ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6430_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ X @ I5 )
= one_one_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6431_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
=> ( ( ( groups8778361861064173332t_real @ X @ I5 )
= one_one_real )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6432_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ X @ I5 )
= one_one_real )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6433_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6434_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups1300246762558778688al_rat
@ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6435_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups3906332499630173760nt_rat
@ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6436_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ( ord_less_eq_nat @ M @ I4 )
& ( ord_less_nat @ I4 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ M @ I4 )
& ( ord_less_eq_nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_6437_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= zero_zero_nat )
| ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= one_one_nat )
| ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
| ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_6438_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_6439_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_6440_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6441_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6442_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6443_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_6444_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_6445_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6446_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6447_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger
@ ( zero_n356916108424825756nteger
@ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
!= zero_z3403309356797280102nteger )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6448_arctan__add,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y2 ) )
= ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y2 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_6449_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_6450_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect_nat
@ ^ [N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_6451_signed__take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_6452_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6453_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6454_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6455_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6456_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6457_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_6458_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_6459_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_6460_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_6461_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_6462_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_6463_signed__take__bit__numeral__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_6464_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8557863876264182079omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6465_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6466_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6467_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6468_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6511756317524482435omplex @ one_one_complex )
= one_one_complex ) ).
% dbl_dec_simps(3)
thf(fact_6469_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_6470_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
= one_one_rat ) ).
% dbl_dec_simps(3)
thf(fact_6471_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_6472_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_6473_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_6474_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_6475_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_6476_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_6477_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_6478_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_6479_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_6480_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_6481_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_6482_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_6483_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_inc_simps(4)
thf(fact_6484_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6485_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6486_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
= ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6487_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6488_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_6489_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_6490_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_6491_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_6492_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_6493_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_6494_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_6495_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_6496_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_6497_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_6498_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_6499_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_6500_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_6501_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_dec_simps(2)
thf(fact_6502_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6503_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6504_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6505_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6506_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6507_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6508_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6509_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6510_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6511_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6512_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6513_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6514_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6515_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6516_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6517_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_6518_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_6519_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_6520_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique5052692396658037445od_int @ M @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(2)
thf(fact_6521_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique5055182867167087721od_nat @ M @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(2)
thf(fact_6522_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique3479559517661332726nteger @ M @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_6523_sum__zero__power,axiom,
! [A2: set_nat,C: nat > complex] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A2 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_6524_sum__zero__power,axiom,
! [A2: set_nat,C: nat > rat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A2 )
= zero_zero_rat ) ) ) ).
% sum_zero_power
thf(fact_6525_sum__zero__power,axiom,
! [A2: set_nat,C: nat > real] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A2 )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_6526_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6527_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6528_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6529_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6530_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6531_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6532_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > complex,D: nat > complex] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A2 )
= ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A2 )
= zero_zero_complex ) ) ) ).
% sum_zero_power'
thf(fact_6533_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > rat,D: nat > rat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
@ A2 )
= ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
@ A2 )
= zero_zero_rat ) ) ) ).
% sum_zero_power'
thf(fact_6534_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > real,D: nat > real] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A2 )
= ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A2 )
= zero_zero_real ) ) ) ).
% sum_zero_power'
thf(fact_6535_signed__take__bit__numeral__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_6536_signed__take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_6537_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_6538_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_6539_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_6540_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6541_sum__subtractf__nat,axiom,
! [A2: set_int,G: int > nat,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6542_sum__subtractf__nat,axiom,
! [A2: set_complex,G: complex > nat,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6543_sum__subtractf__nat,axiom,
! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups977919841031483927at_nat
@ ^ [X2: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6544_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6545_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_6546_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_6547_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_6548_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M: nat,K: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_6549_sum__eq__Suc0__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: int] :
( ( member_int @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6550_sum__eq__Suc0__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: complex] :
( ( member_complex @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6551_sum__eq__Suc0__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: nat] :
( ( member_nat @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6552_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_6553_sum__power__add,axiom,
! [X: complex,M: nat,I5: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6554_sum__power__add,axiom,
! [X: rat,M: nat,I5: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6555_sum__power__add,axiom,
! [X: int,M: nat,I5: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6556_sum__power__add,axiom,
! [X: real,M: nat,I5: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6557_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_6558_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N: nat,M: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_6559_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_6560_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_6561_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_6562_sum__diff__nat,axiom,
! [B4: set_complex,A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6563_sum__diff__nat,axiom,
! [B4: set_int,A2: set_int,F: int > nat] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B4 ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B4 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6564_sum__diff__nat,axiom,
! [B4: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B4 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6565_sum__shift__lb__Suc0__0,axiom,
! [F: nat > complex,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6566_sum__shift__lb__Suc0__0,axiom,
! [F: nat > rat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_rat )
=> ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6567_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6568_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6569_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6570_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6571_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6572_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6573_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6574_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6575_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6576_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6577_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6578_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6579_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6580_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6581_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6582_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_6583_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_6584_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_6585_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_6586_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ M )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6587_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ M )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6588_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6589_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ M )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6590_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_6591_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_6592_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_6593_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6594_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6595_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6596_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6597_divmod__int__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_6598_divmod__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_6599_divmod__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_6600_divmod__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_6601_divmod_H__nat__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_6602_dbl__dec__def,axiom,
( neg_nu6511756317524482435omplex
= ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% dbl_dec_def
thf(fact_6603_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_6604_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_6605_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_6606_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > complex] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_complex ) ) ) ).
% sum_natinterval_diff
thf(fact_6607_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_6608_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_6609_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_6610_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_6611_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_6612_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_6613_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_6614_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_6615_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6616_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6617_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6618_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6619_sum_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6620_sum_Oin__pairs,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6621_sum_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6622_sum_Oin__pairs,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6623_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_6624_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_6625_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_6626_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_6627_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M6: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6628_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M6: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6629_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M6: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6630_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_6631_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_6632_infinite__int__iff__unbounded__le,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ! [M6: int] :
? [N2: int] :
( ( ord_less_eq_int @ M6 @ ( abs_abs_int @ N2 ) )
& ( member_int @ N2 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_6633_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_6634_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_6635_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R2: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_6636_infinite__nat__iff__unbounded,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M6: nat] :
? [N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
& ( member_nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_6637_unbounded__k__infinite,axiom,
! [K: nat,S3: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N7: nat] :
( ( ord_less_nat @ M5 @ N7 )
& ( member_nat @ N7 @ S3 ) ) )
=> ~ ( finite_finite_nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_6638_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M6: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
& ( member_nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_6639_sum__gp,axiom,
! [N: nat,M: nat,X: complex] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6640_sum__gp,axiom,
! [N: nat,M: nat,X: rat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6641_sum__gp,axiom,
! [N: nat,M: nat,X: real] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6642_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% divmod_BitM_2_eq
thf(fact_6643_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6644_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6645_of__int__code__if,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] :
( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
@ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_real
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6646_of__int__code__if,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] :
( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
@ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_int
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6647_of__int__code__if,axiom,
( ring_17405671764205052669omplex
= ( ^ [K3: int] :
( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
@ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_complex
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6648_of__int__code__if,axiom,
( ring_18347121197199848620nteger
= ( ^ [K3: int] :
( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6649_of__int__code__if,axiom,
( ring_1_of_int_rat
= ( ^ [K3: int] :
( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
@ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_rat
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6650_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
@ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6651_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
@ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6652_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6653_sum__gp__offset,axiom,
! [X: complex,M: nat,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6654_sum__gp__offset,axiom,
! [X: rat,M: nat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6655_sum__gp__offset,axiom,
! [X: real,M: nat,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6656_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_6657_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_6658_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_6659_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri681578069525770553at_rat @ M )
= ( semiri681578069525770553at_rat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_6660_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_6661_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( semiri4939895301339042750nteger @ N ) ) ).
% abs_of_nat
thf(fact_6662_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_6663_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_6664_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% abs_of_nat
thf(fact_6665_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_6666_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6667_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6668_case__prod__conv,axiom,
! [F: int > int > product_prod_int_int,A: int,B: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6669_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6670_case__prod__conv,axiom,
! [F: int > int > int,A: int,B: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6671_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_6672_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_6673_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_6674_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_6675_of__nat__0,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% of_nat_0
thf(fact_6676_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6677_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6678_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6679_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6680_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6681_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6682_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6683_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6684_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6685_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri681578069525770553at_rat @ M )
= zero_zero_rat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6686_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6687_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6688_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6689_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6690_of__nat__numeral,axiom,
! [N: num] :
( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
= ( numera6690914467698888265omplex @ N ) ) ).
% of_nat_numeral
thf(fact_6691_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_6692_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_6693_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_6694_of__nat__numeral,axiom,
! [N: num] :
( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% of_nat_numeral
thf(fact_6695_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6696_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6697_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6698_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6699_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_6700_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_6701_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_6702_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_add
thf(fact_6703_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_6704_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_6705_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_6706_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_mult
thf(fact_6707_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_6708_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_6709_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_6710_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_6711_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_6712_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6713_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6714_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6715_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6716_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6717_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6718_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6719_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6720_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6721_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri681578069525770553at_rat @ N )
= one_one_rat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6722_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_6723_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_6724_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_6725_of__int__numeral,axiom,
! [K: num] :
( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
= ( numera6690914467698888265omplex @ K ) ) ).
% of_int_numeral
thf(fact_6726_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_real @ K ) ) ).
% of_int_numeral
thf(fact_6727_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_rat @ K ) ) ).
% of_int_numeral
thf(fact_6728_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_6729_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_17405671764205052669omplex @ Z )
= ( numera6690914467698888265omplex @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6730_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_real @ Z )
= ( numeral_numeral_real @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6731_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_rat @ Z )
= ( numeral_numeral_rat @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6732_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_int @ Z )
= ( numeral_numeral_int @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6733_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_6734_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_6735_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_6736_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6737_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6738_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6739_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6740_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri681578069525770553at_rat @ X )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6741_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6742_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6743_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6744_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6745_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
= ( semiri681578069525770553at_rat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6746_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6747_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6748_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6749_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6750_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6751_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_6752_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_mult
thf(fact_6753_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_6754_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_6755_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_6756_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_add
thf(fact_6757_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_6758_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
= ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% of_int_power
thf(fact_6759_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% of_int_power
thf(fact_6760_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% of_int_power
thf(fact_6761_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% of_int_power
thf(fact_6762_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
= ( ring_1_of_int_rat @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6763_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6764_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6765_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
= ( ring_17405671764205052669omplex @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6766_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_rat @ X )
= ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6767_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6768_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6769_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_17405671764205052669omplex @ X )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6770_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% of_nat_of_bool
thf(fact_6771_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% of_nat_of_bool
thf(fact_6772_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% of_nat_of_bool
thf(fact_6773_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_nat_of_bool
thf(fact_6774_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% of_nat_of_bool
thf(fact_6775_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_6776_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_6777_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
= ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_6778_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_6779_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6780_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6781_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6782_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6783_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_6784_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_6785_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_6786_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_6787_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% of_nat_Suc
thf(fact_6788_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_6789_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_6790_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_6791_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_6792_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6793_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6794_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6795_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6796_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6797_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6798_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6799_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6800_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6801_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6802_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6803_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6804_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6805_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6806_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6807_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6808_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6809_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6810_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6811_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6812_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6813_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6814_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6815_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6816_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( semiri8010041392384452111omplex @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6817_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6818_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6819_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6820_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( semiri681578069525770553at_rat @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6821_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri8010041392384452111omplex @ Y2 )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6822_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y2 )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6823_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y2 )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6824_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y2 )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6825_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri681578069525770553at_rat @ Y2 )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6826_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6827_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6828_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6829_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6830_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6831_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6832_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6833_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6834_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6835_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6836_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6837_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6838_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6839_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6840_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6841_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6842_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6843_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6844_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6845_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6846_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6847_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6848_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6849_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6850_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6851_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6852_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6853_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6854_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6855_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6856_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6857_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6858_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6859_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6860_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6861_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6862_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6863_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6864_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y2 )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y2
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6865_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y2 )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y2
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6866_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y2 )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y2
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6867_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y2 )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y2
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6868_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( ring_17405671764205052669omplex @ Y2 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6869_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( ring_1_of_int_real @ Y2 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6870_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( ring_1_of_int_rat @ Y2 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6871_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( ring_1_of_int_int @ Y2 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6872_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6873_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6874_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6875_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6876_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6877_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6878_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6879_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6880_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6881_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6882_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6883_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6884_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6885_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6886_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6887_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6888_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6889_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6890_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6891_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6892_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6893_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6894_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6895_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6896_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6897_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6898_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6899_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6900_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6901_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6902_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6903_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6904_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6905_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6906_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6907_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6908_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6909_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6910_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6911_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6912_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6913_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
= ( ring_1_of_int_real @ Y2 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y2 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6914_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= ( ring_1_of_int_int @ Y2 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y2 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6915_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
= ( ring_17405671764205052669omplex @ Y2 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y2 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6916_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
= ( ring_18347121197199848620nteger @ Y2 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y2 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6917_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
= ( ring_1_of_int_rat @ Y2 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y2 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6918_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y2 )
= ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( Y2
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6919_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y2 )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( Y2
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6920_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y2 )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
= ( Y2
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6921_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_18347121197199848620nteger @ Y2 )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( Y2
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6922_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y2 )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( Y2
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6923_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_6924_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_6925_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_6926_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6927_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6928_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6929_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6930_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6931_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6932_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6933_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6934_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6935_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6936_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6937_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6938_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6939_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6940_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6941_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6942_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6943_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6944_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6945_mult__of__int__commute,axiom,
! [X: int,Y2: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y2 )
= ( times_times_real @ Y2 @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6946_mult__of__int__commute,axiom,
! [X: int,Y2: rat] :
( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y2 )
= ( times_times_rat @ Y2 @ ( ring_1_of_int_rat @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6947_mult__of__int__commute,axiom,
! [X: int,Y2: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y2 )
= ( times_times_int @ Y2 @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6948_mult__of__nat__commute,axiom,
! [X: nat,Y2: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y2 )
= ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6949_mult__of__nat__commute,axiom,
! [X: nat,Y2: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y2 )
= ( times_times_real @ Y2 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6950_mult__of__nat__commute,axiom,
! [X: nat,Y2: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y2 )
= ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6951_mult__of__nat__commute,axiom,
! [X: nat,Y2: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y2 )
= ( times_times_rat @ Y2 @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6952_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6953_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6954_old_Oprod_Ocase,axiom,
! [F: int > int > product_prod_int_int,X1: int,X22: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6955_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X22: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6956_old_Oprod_Ocase,axiom,
! [F: int > int > int,X1: int,X22: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6957_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_6958_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6959_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6960_case__prodE2,axiom,
! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6961_case__prodE2,axiom,
! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6962_case__prodE2,axiom,
! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6963_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6964_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6965_case__prod__eta,axiom,
! [F: product_prod_int_int > product_prod_int_int] :
( ( produc4245557441103728435nt_int
@ ^ [X2: int,Y: int] : ( F @ ( product_Pair_int_int @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6966_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X2: int,Y: int] : ( F @ ( product_Pair_int_int @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6967_case__prod__eta,axiom,
! [F: product_prod_int_int > int] :
( ( produc8211389475949308722nt_int
@ ^ [X2: int,Y: int] : ( F @ ( product_Pair_int_int @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6968_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X3: nat,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
=> ( ( produc27273713700761075at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6969_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
=> ( ( produc8739625826339149834_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6970_cond__case__prod__eta,axiom,
! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc4245557441103728435nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6971_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6972_cond__case__prod__eta,axiom,
! [F: int > int > int,G: product_prod_int_int > int] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc8211389475949308722nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6973_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6974_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6975_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6976_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6977_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_6978_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_6979_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_6980_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_6981_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_6982_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_6983_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_6984_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_6985_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_6986_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_6987_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_6988_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6989_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6990_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6991_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6992_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6993_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6994_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6995_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6996_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_6997_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% of_nat_mono
thf(fact_6998_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_6999_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_7000_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_7001_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_7002_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_7003_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_7004_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_7005_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_7006_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_7007_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_7008_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_7009_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_7010_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_7011_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_7012_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_7013_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mod
thf(fact_7014_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mod
thf(fact_7015_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mod
thf(fact_7016_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_7017_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_7018_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_7019_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_7020_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z2: int] :
? [N2: nat] :
( Z2
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_7021_zdiv__int,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% zdiv_int
thf(fact_7022_of__nat__max,axiom,
! [X: nat,Y2: nat] :
( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y2 ) )
= ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y2 ) ) ) ).
% of_nat_max
thf(fact_7023_of__nat__max,axiom,
! [X: nat,Y2: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y2 ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) ).
% of_nat_max
thf(fact_7024_of__nat__max,axiom,
! [X: nat,Y2: nat] :
( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y2 ) )
= ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y2 ) ) ) ).
% of_nat_max
thf(fact_7025_of__nat__max,axiom,
! [X: nat,Y2: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y2 ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y2 ) ) ) ).
% of_nat_max
thf(fact_7026_of__nat__max,axiom,
! [X: nat,Y2: nat] :
( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y2 ) )
= ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y2 ) ) ) ).
% of_nat_max
thf(fact_7027_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_7028_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_7029_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_7030_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_7031_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_7032_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_7033_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_7034_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_7035_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_7036_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_7037_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_7038_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_7039_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_7040_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_7041_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_7042_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z2: int] :
? [N2: nat] :
( Z2
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_7043_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_7044_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_7045_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_7046_real__of__int__div,axiom,
! [D: int,N: int] :
( ( dvd_dvd_int @ D @ N )
=> ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
= ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div
thf(fact_7047_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_7048_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_7049_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_7050_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_7051_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_nonneg
thf(fact_7052_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_7053_of__int__leD,axiom,
! [N: int,X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_leD
thf(fact_7054_of__int__leD,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% of_int_leD
thf(fact_7055_of__int__leD,axiom,
! [N: int,X: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% of_int_leD
thf(fact_7056_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_7057_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_7058_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_pos
thf(fact_7059_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_7060_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_lessD
thf(fact_7061_of__int__lessD,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% of_int_lessD
thf(fact_7062_of__int__lessD,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% of_int_lessD
thf(fact_7063_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_7064_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7065_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7066_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7067_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_7068_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_7069_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_7070_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_7071_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_7072_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_7073_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_7074_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_7075_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_7076_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% int_less_real_le
thf(fact_7077_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_7078_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_7079_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_7080_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% nat_less_real_le
thf(fact_7081_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_7082_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_7083_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_7084_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_7085_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_7086_real__of__int__div__aux,axiom,
! [X: int,D: int] :
( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div_aux
thf(fact_7087_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] :
( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_7088_numeral__BitM,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bitM @ N ) )
= ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% numeral_BitM
thf(fact_7089_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bitM @ N ) )
= ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% numeral_BitM
thf(fact_7090_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bitM @ N ) )
= ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% numeral_BitM
thf(fact_7091_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_7092_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_7093_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_7094_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_7095_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_7096_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_7097_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_7098_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_7099_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_7100_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_7101_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_7102_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_7103_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_7104_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_7105_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_7106_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_7107_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_7108_even__of__int__iff,axiom,
! [K: int] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% even_of_int_iff
thf(fact_7109_even__of__int__iff,axiom,
! [K: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% even_of_int_iff
thf(fact_7110_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_7111_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_7112_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L: num] :
( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_7113_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L: num] :
( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_7114_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_7115_double__arith__series,axiom,
! [A: complex,D: complex,N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7116_double__arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7117_double__arith__series,axiom,
! [A: rat,D: rat,N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7118_double__arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7119_double__arith__series,axiom,
! [A: real,D: real,N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7120_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum
thf(fact_7121_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_7122_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_7123_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_7124_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum
thf(fact_7125_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L: num] :
( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_7126_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L: num] :
( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_7127_divmod__step__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L: num] :
( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_7128_arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7129_arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7130_gauss__sum,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7131_gauss__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7132_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7133_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7134_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7135_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7136_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7137_nat__approx__posE,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N3: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).
% nat_approx_posE
thf(fact_7138_nat__approx__posE,axiom,
! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ~ ! [N3: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).
% nat_approx_posE
thf(fact_7139_floor__exists1,axiom,
! [X: real] :
? [X3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
=> ( Y4 = X3 ) ) ) ).
% floor_exists1
thf(fact_7140_floor__exists1,axiom,
! [X: rat] :
? [X3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
=> ( Y4 = X3 ) ) ) ).
% floor_exists1
thf(fact_7141_floor__exists,axiom,
! [X: real] :
? [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_7142_floor__exists,axiom,
! [X: rat] :
? [Z3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_7143_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( topolo6980174941875973593q_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_7144_lemma__termdiff3,axiom,
! [H2: real,Z: real,K5: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_7145_lemma__termdiff3,axiom,
! [H2: complex,Z: complex,K5: real,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_7146_ex__less__of__nat__mult,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7147_ex__less__of__nat__mult,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] : ( ord_less_rat @ Y2 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7148_case__prodI2,axiom,
! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_7149_case__prodI2,axiom,
! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_7150_case__prodI2,axiom,
! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P4
= ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_7151_case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_7152_case__prodI2,axiom,
! [P4: product_prod_int_int,C: int > int > $o] :
( ! [A3: int,B2: int] :
( ( P4
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_7153_case__prodI,axiom,
! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F @ A @ B )
=> ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_7154_case__prodI,axiom,
! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F @ A @ B )
=> ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_7155_case__prodI,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_7156_case__prodI,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_7157_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_7158_mem__case__prodI2,axiom,
! [P4: product_prod_int_int,Z: nat,C: int > int > set_nat] :
( ! [A3: int,B2: int] :
( ( P4
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7159_mem__case__prodI2,axiom,
! [P4: product_prod_int_int,Z: real,C: int > int > set_real] :
( ! [A3: int,B2: int] :
( ( P4
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7160_mem__case__prodI2,axiom,
! [P4: product_prod_int_int,Z: int,C: int > int > set_int] :
( ! [A3: int,B2: int] :
( ( P4
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7161_mem__case__prodI2,axiom,
! [P4: product_prod_int_int,Z: complex,C: int > int > set_complex] :
( ! [A3: int,B2: int] :
( ( P4
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_complex @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7162_mem__case__prodI2,axiom,
! [P4: product_prod_int_int,Z: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
( ! [A3: int,B2: int] :
( ( P4
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member8440522571783428010at_nat @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc1656060378719767003at_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7163_mem__case__prodI2,axiom,
! [P4: produc8763457246119570046nteger,Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7164_mem__case__prodI2,axiom,
! [P4: produc8763457246119570046nteger,Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7165_mem__case__prodI2,axiom,
! [P4: produc8763457246119570046nteger,Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7166_mem__case__prodI2,axiom,
! [P4: produc8763457246119570046nteger,Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_complex @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7167_mem__case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_7168_mem__case__prodI,axiom,
! [Z: nat,C: int > int > set_nat,A: int,B: int] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7169_mem__case__prodI,axiom,
! [Z: real,C: int > int > set_real,A: int,B: int] :
( ( member_real @ Z @ ( C @ A @ B ) )
=> ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7170_mem__case__prodI,axiom,
! [Z: int,C: int > int > set_int,A: int,B: int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7171_mem__case__prodI,axiom,
! [Z: complex,C: int > int > set_complex,A: int,B: int] :
( ( member_complex @ Z @ ( C @ A @ B ) )
=> ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7172_mem__case__prodI,axiom,
! [Z: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7173_mem__case__prodI,axiom,
! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7174_mem__case__prodI,axiom,
! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_real @ Z @ ( C @ A @ B ) )
=> ( member_real @ Z @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7175_mem__case__prodI,axiom,
! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7176_mem__case__prodI,axiom,
! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_complex @ Z @ ( C @ A @ B ) )
=> ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7177_mem__case__prodI,axiom,
! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_7178_case__prodI2_H,axiom,
! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B2 )
= P4 )
=> ( C @ A3 @ B2 @ X ) )
=> ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).
% case_prodI2'
thf(fact_7179_mem__case__prodE,axiom,
! [Z: nat,C: int > int > set_nat,P4: product_prod_int_int] :
( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7180_mem__case__prodE,axiom,
! [Z: real,C: int > int > set_real,P4: product_prod_int_int] :
( ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7181_mem__case__prodE,axiom,
! [Z: int,C: int > int > set_int,P4: product_prod_int_int] :
( ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7182_mem__case__prodE,axiom,
! [Z: complex,C: int > int > set_complex,P4: product_prod_int_int] :
( ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_complex @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7183_mem__case__prodE,axiom,
! [Z: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P4: product_prod_int_int] :
( ( member8440522571783428010at_nat @ Z @ ( produc1656060378719767003at_nat @ C @ P4 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7184_mem__case__prodE,axiom,
! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
( ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7185_mem__case__prodE,axiom,
! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
( ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7186_mem__case__prodE,axiom,
! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
( ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7187_mem__case__prodE,axiom,
! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P4: produc8763457246119570046nteger] :
( ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_complex @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7188_mem__case__prodE,axiom,
! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
=> ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7189_case__prodE,axiom,
! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
( ( produc127349428274296955eger_o @ C @ P4 )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P4
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7190_case__prodE,axiom,
! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
( ( produc6253627499356882019eger_o @ C @ P4 )
=> ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7191_case__prodE,axiom,
! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
( ( produc1573362020775583542_int_o @ C @ P4 )
=> ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P4
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7192_case__prodE,axiom,
! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
( ( produc2558449545302689196_int_o @ C @ P4 )
=> ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7193_case__prodE,axiom,
! [C: int > int > $o,P4: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C @ P4 )
=> ~ ! [X3: int,Y3: int] :
( ( P4
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7194_case__prodD,axiom,
! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7195_case__prodD,axiom,
! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7196_case__prodD,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7197_case__prodD,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7198_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7199_case__prodE_H,axiom,
! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
=> ~ ! [X3: nat,Y3: nat] :
( ( P4
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_7200_case__prodD_H,axiom,
! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
=> ( R @ A @ B @ C ) ) ).
% case_prodD'
thf(fact_7201_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_7202_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% complex_mod_triangle_ineq2
thf(fact_7203_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( produc8211389475949308722nt_int
@ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_7204_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_7205_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_7206_real__arch__simple,axiom,
! [X: rat] :
? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% real_arch_simple
thf(fact_7207_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_7208_reals__Archimedean2,axiom,
! [X: rat] :
? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% reals_Archimedean2
thf(fact_7209_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_7210_ex__le__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_le_of_int
thf(fact_7211_ex__le__of__int,axiom,
! [X: rat] :
? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% ex_le_of_int
thf(fact_7212_ex__less__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_less_of_int
thf(fact_7213_ex__less__of__int,axiom,
! [X: rat] :
? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% ex_less_of_int
thf(fact_7214_ex__of__int__less,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_7215_ex__of__int__less,axiom,
! [X: rat] :
? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_7216_norm__divide__numeral,axiom,
! [A: real,W: num] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_divide_numeral
thf(fact_7217_norm__divide__numeral,axiom,
! [A: complex,W: num] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_divide_numeral
thf(fact_7218_norm__mult__numeral1,axiom,
! [W: num,A: real] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_mult_numeral1
thf(fact_7219_norm__mult__numeral1,axiom,
! [W: num,A: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% norm_mult_numeral1
thf(fact_7220_norm__mult__numeral2,axiom,
! [A: real,W: num] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
= ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_mult_numeral2
thf(fact_7221_norm__mult__numeral2,axiom,
! [A: complex,W: num] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
= ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_mult_numeral2
thf(fact_7222_norm__neg__numeral,axiom,
! [W: num] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_neg_numeral
thf(fact_7223_norm__neg__numeral,axiom,
! [W: num] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_neg_numeral
thf(fact_7224_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_7225_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_7226_norm__numeral,axiom,
! [W: num] :
( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_numeral
thf(fact_7227_norm__numeral,axiom,
! [W: num] :
( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_numeral
thf(fact_7228_Collect__case__prod__mono,axiom,
! [A2: int > int > $o,B4: int > int > $o] :
( ( ord_le6741204236512500942_int_o @ A2 @ B4 )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_7229_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_7230_norm__mult,axiom,
! [X: real,Y2: real] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y2 ) )
= ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).
% norm_mult
thf(fact_7231_norm__mult,axiom,
! [X: complex,Y2: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y2 ) )
= ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).
% norm_mult
thf(fact_7232_sum__norm__le,axiom,
! [S3: set_real,F: real > complex,G: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S3 ) ) @ ( groups8097168146408367636l_real @ G @ S3 ) ) ) ).
% sum_norm_le
thf(fact_7233_sum__norm__le,axiom,
! [S3: set_int,F: int > complex,G: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S3 ) ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ).
% sum_norm_le
thf(fact_7234_sum__norm__le,axiom,
! [S3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > complex,G: product_prod_nat_nat > real] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6381953495645901045omplex @ F @ S3 ) ) @ ( groups4567486121110086003t_real @ G @ S3 ) ) ) ).
% sum_norm_le
thf(fact_7235_sum__norm__le,axiom,
! [S3: set_nat,F: nat > complex,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).
% sum_norm_le
thf(fact_7236_sum__norm__le,axiom,
! [S3: set_complex,F: complex > complex,G: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S3 ) ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ).
% sum_norm_le
thf(fact_7237_sum__norm__le,axiom,
! [S3: set_nat,F: nat > real,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).
% sum_norm_le
thf(fact_7238_norm__divide,axiom,
! [A: real,B: real] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_divide
thf(fact_7239_norm__divide,axiom,
! [A: complex,B: complex] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_divide
thf(fact_7240_norm__power,axiom,
! [X: real,N: nat] :
( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
= ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% norm_power
thf(fact_7241_norm__power,axiom,
! [X: complex,N: nat] :
( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% norm_power
thf(fact_7242_norm__sum,axiom,
! [F: nat > complex,A2: set_nat] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
@ A2 ) ) ).
% norm_sum
thf(fact_7243_norm__sum,axiom,
! [F: complex > complex,A2: set_complex] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
@ A2 ) ) ).
% norm_sum
thf(fact_7244_norm__sum,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
@ A2 ) ) ).
% norm_sum
thf(fact_7245_norm__uminus__minus,axiom,
! [X: real,Y2: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y2 ) )
= ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) ) ).
% norm_uminus_minus
thf(fact_7246_norm__uminus__minus,axiom,
! [X: complex,Y2: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y2 ) )
= ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y2 ) ) ) ).
% norm_uminus_minus
thf(fact_7247_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_7248_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_7249_power__eq__imp__eq__norm,axiom,
! [W: real,N: nat,Z: real] :
( ( ( power_power_real @ W @ N )
= ( power_power_real @ Z @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W )
= ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_7250_power__eq__imp__eq__norm,axiom,
! [W: complex,N: nat,Z: complex] :
( ( ( power_power_complex @ W @ N )
= ( power_power_complex @ Z @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W )
= ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_7251_norm__mult__less,axiom,
! [X: real,R2: real,Y2: real,S2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y2 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% norm_mult_less
thf(fact_7252_norm__mult__less,axiom,
! [X: complex,R2: real,Y2: complex,S2: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y2 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% norm_mult_less
thf(fact_7253_norm__mult__ineq,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y2 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).
% norm_mult_ineq
thf(fact_7254_norm__mult__ineq,axiom,
! [X: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y2 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).
% norm_mult_ineq
thf(fact_7255_norm__add__less,axiom,
! [X: real,R2: real,Y2: real,S2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_add_less
thf(fact_7256_norm__add__less,axiom,
! [X: complex,R2: real,Y2: complex,S2: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y2 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_add_less
thf(fact_7257_norm__triangle__lt,axiom,
! [X: real,Y2: real,E: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ E ) ) ).
% norm_triangle_lt
thf(fact_7258_norm__triangle__lt,axiom,
! [X: complex,Y2: complex,E: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y2 ) ) @ E ) ) ).
% norm_triangle_lt
thf(fact_7259_norm__power__ineq,axiom,
! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% norm_power_ineq
thf(fact_7260_norm__power__ineq,axiom,
! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% norm_power_ineq
thf(fact_7261_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_7262_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_7263_norm__triangle__le,axiom,
! [X: real,Y2: real,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_7264_norm__triangle__le,axiom,
! [X: complex,Y2: complex,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y2 ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_7265_norm__triangle__ineq,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).
% norm_triangle_ineq
thf(fact_7266_norm__triangle__ineq,axiom,
! [X: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).
% norm_triangle_ineq
thf(fact_7267_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_7268_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S2: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_7269_norm__diff__triangle__less,axiom,
! [X: real,Y2: real,E1: real,Z: real,E22: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y2 ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y2 @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_7270_norm__diff__triangle__less,axiom,
! [X: complex,Y2: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y2 ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y2 @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_7271_norm__triangle__sub,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y2 ) ) ) ) ).
% norm_triangle_sub
thf(fact_7272_norm__triangle__sub,axiom,
! [X: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y2 ) ) ) ) ).
% norm_triangle_sub
thf(fact_7273_norm__triangle__ineq4,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_triangle_ineq4
thf(fact_7274_norm__triangle__ineq4,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_triangle_ineq4
thf(fact_7275_norm__diff__triangle__le,axiom,
! [X: real,Y2: real,E1: real,Z: real,E22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y2 ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y2 @ Z ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_7276_norm__diff__triangle__le,axiom,
! [X: complex,Y2: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y2 ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y2 @ Z ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_7277_norm__triangle__le__diff,axiom,
! [X: real,Y2: real,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y2 ) ) @ E ) ) ).
% norm_triangle_le_diff
thf(fact_7278_norm__triangle__le__diff,axiom,
! [X: complex,Y2: complex,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y2 ) ) @ E ) ) ).
% norm_triangle_le_diff
thf(fact_7279_norm__diff__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_7280_norm__diff__ineq,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_7281_norm__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_7282_norm__triangle__ineq2,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_7283_power__eq__1__iff,axiom,
! [W: real,N: nat] :
( ( ( power_power_real @ W @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_7284_power__eq__1__iff,axiom,
! [W: complex,N: nat] :
( ( ( power_power_complex @ W @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_7285_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_7286_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_7287_norm__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_7288_norm__triangle__ineq3,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_7289_square__norm__one,axiom,
! [X: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
=> ( ( real_V7735802525324610683m_real @ X )
= one_one_real ) ) ).
% square_norm_one
thf(fact_7290_square__norm__one,axiom,
! [X: complex] :
( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex )
=> ( ( real_V1022390504157884413omplex @ X )
= one_one_real ) ) ).
% square_norm_one
thf(fact_7291_norm__power__diff,axiom,
! [Z: real,W: real,M: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_7292_norm__power__diff,axiom,
! [Z: complex,W: complex,M: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_7293_ln__series,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ( ln_ln_real @ X )
= ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_7294_of__nat__code__if,axiom,
( semiri8010041392384452111omplex
= ( ^ [N2: nat] :
( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
@ ( produc1917071388513777916omplex
@ ^ [M6: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ one_one_complex ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7295_of__nat__code__if,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
@ ( produc6840382203811409530at_int
@ ^ [M6: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ one_one_int ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7296_of__nat__code__if,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
@ ( produc1703576794950452218t_real
@ ^ [M6: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ one_one_real ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7297_of__nat__code__if,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( produc6842872674320459806at_nat
@ ^ [M6: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ one_one_nat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7298_of__nat__code__if,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
@ ( produc6207742614233964070at_rat
@ ^ [M6: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ one_one_rat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7299_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M6: nat,N2: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N2 = zero_zero_nat )
| ( ord_less_nat @ M6 @ N2 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
@ ( produc2626176000494625587at_nat
@ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_7300_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_7301_round__unique,axiom,
! [X: real,Y2: int] :
( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y2 ) )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y2 ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= Y2 ) ) ) ).
% round_unique
thf(fact_7302_round__unique,axiom,
! [X: rat,Y2: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y2 ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y2 ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= Y2 ) ) ) ).
% round_unique
thf(fact_7303_lemma__termdiff2,axiom,
! [H2: complex,Z: complex,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_complex @ H2
@ ( groups2073611262835488442omplex
@ ^ [P5: nat] :
( groups2073611262835488442omplex
@ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_7304_lemma__termdiff2,axiom,
! [H2: rat,Z: rat,N: nat] :
( ( H2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_rat @ H2
@ ( groups2906978787729119204at_rat
@ ^ [P5: nat] :
( groups2906978787729119204at_rat
@ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_7305_lemma__termdiff2,axiom,
! [H2: real,Z: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_real @ H2
@ ( groups6591440286371151544t_real
@ ^ [P5: nat] :
( groups6591440286371151544t_real
@ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_7306_lessThan__iff,axiom,
! [I2: rat,K: rat] :
( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
= ( ord_less_rat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7307_lessThan__iff,axiom,
! [I2: num,K: num] :
( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7308_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7309_lessThan__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7310_lessThan__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7311_lessThan__subset__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y2 ) )
= ( ord_less_eq_rat @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_7312_lessThan__subset__iff,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y2 ) )
= ( ord_less_eq_num @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_7313_lessThan__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_7314_lessThan__subset__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_7315_lessThan__subset__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_7316_round__numeral,axiom,
! [N: num] :
( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% round_numeral
thf(fact_7317_round__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% round_numeral
thf(fact_7318_sum_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7319_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7320_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7321_sum_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7322_round__neg__numeral,axiom,
! [N: num] :
( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_7323_round__neg__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_7324_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_7325_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_7326_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X2: rat] : ( ord_less_rat @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7327_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X2: num] : ( ord_less_num @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7328_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7329_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7330_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7331_lessThan__strict__subset__iff,axiom,
! [M: rat,N: rat] :
( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
= ( ord_less_rat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7332_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7333_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7334_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7335_lessThan__strict__subset__iff,axiom,
! [M: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7336_finite__nat__bounded,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_7337_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S4: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_7338_round__mono,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y2 ) ) ) ).
% round_mono
thf(fact_7339_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_7340_sum_Onat__diff__reindex,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_7341_sum__diff__distrib,axiom,
! [Q: int > nat,P: int > nat,N: int] :
( ! [X3: int] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan_int @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_7342_sum__diff__distrib,axiom,
! [Q: real > nat,P: real > nat,N: real] :
( ! [X3: real] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_7343_sum__diff__distrib,axiom,
! [Q: nat > nat,P: nat > nat,N: nat] :
( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_7344_sum_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7345_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7346_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7347_sum_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7348_sumr__diff__mult__const2,axiom,
! [F: nat > int,N: nat,R2: int] :
( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R2 ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_7349_sumr__diff__mult__const2,axiom,
! [F: nat > rat,N: nat,R2: rat] :
( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R2 ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_7350_sumr__diff__mult__const2,axiom,
! [F: nat > real,N: nat,R2: real] :
( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R2 ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_7351_sum__lessThan__telescope,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_7352_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_7353_sum__lessThan__telescope,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_7354_sum__lessThan__telescope_H,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_7355_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_7356_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_7357_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_7358_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_7359_power__diff__1__eq,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
= ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7360_power__diff__1__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
= ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7361_power__diff__1__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7362_power__diff__1__eq,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
= ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7363_one__diff__power__eq,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7364_one__diff__power__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7365_one__diff__power__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7366_one__diff__power__eq,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7367_geometric__sum,axiom,
! [X: complex,N: nat] :
( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% geometric_sum
thf(fact_7368_geometric__sum,axiom,
! [X: rat,N: nat] :
( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% geometric_sum
thf(fact_7369_geometric__sum,axiom,
! [X: real,N: nat] :
( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% geometric_sum
thf(fact_7370_round__diff__minimal,axiom,
! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_7371_round__diff__minimal,axiom,
! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_7372_sum__gp__strict,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_7373_sum__gp__strict,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_7374_sum__gp__strict,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_7375_lemma__termdiff1,axiom,
! [Z: complex,H2: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7376_lemma__termdiff1,axiom,
! [Z: rat,H2: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7377_lemma__termdiff1,axiom,
! [Z: int,H2: int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups3539618377306564664at_int
@ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7378_lemma__termdiff1,axiom,
! [Z: real,H2: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7379_power__diff__sumr2,axiom,
! [X: complex,N: nat,Y2: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y2 @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y2 )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7380_power__diff__sumr2,axiom,
! [X: rat,N: nat,Y2: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y2 @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y2 )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7381_power__diff__sumr2,axiom,
! [X: int,N: nat,Y2: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y2 )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7382_power__diff__sumr2,axiom,
! [X: real,N: nat,Y2: real] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y2 )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7383_diff__power__eq__sum,axiom,
! [X: complex,N: nat,Y2: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y2 @ ( suc @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y2 )
@ ( groups2073611262835488442omplex
@ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7384_diff__power__eq__sum,axiom,
! [X: rat,N: nat,Y2: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y2 @ ( suc @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y2 )
@ ( groups2906978787729119204at_rat
@ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7385_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y2: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y2 @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y2 )
@ ( groups3539618377306564664at_int
@ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y2 @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7386_diff__power__eq__sum,axiom,
! [X: real,N: nat,Y2: real] :
( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y2 @ ( suc @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y2 )
@ ( groups6591440286371151544t_real
@ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7387_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > rat,K5: rat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7388_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > int,K5: int,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K5 )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7389_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > nat,K5: nat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7390_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > real,K5: real,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ K5 )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7391_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M6: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N2 ) @ ( modulo_modulo_nat @ M6 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_7392_one__diff__power__eq_H,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7393_one__diff__power__eq_H,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7394_one__diff__power__eq_H,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7395_one__diff__power__eq_H,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7396_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_7397_of__int__round__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_7398_of__int__round__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_7399_of__int__round__ge,axiom,
! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% of_int_round_ge
thf(fact_7400_of__int__round__ge,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_ge
thf(fact_7401_of__int__round__gt,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% of_int_round_gt
thf(fact_7402_of__int__round__gt,axiom,
! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_gt
thf(fact_7403_of__int__round__abs__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_7404_of__int__round__abs__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_7405_round__unique_H,axiom,
! [X: real,N: int] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= N ) ) ).
% round_unique'
thf(fact_7406_round__unique_H,axiom,
! [X: rat,N: int] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= N ) ) ).
% round_unique'
thf(fact_7407_suminf__geometric,axiom,
! [C: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
=> ( ( suminf_real @ ( power_power_real @ C ) )
= ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% suminf_geometric
thf(fact_7408_suminf__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( ( suminf_complex @ ( power_power_complex @ C ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% suminf_geometric
thf(fact_7409_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_7410_sum__bounds__lt__plus1,axiom,
! [F: nat > real,Mm: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_7411_pi__series,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suminf_real
@ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% pi_series
thf(fact_7412_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_7413_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_7414_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X2: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
@ ^ [X2: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ S3 ) )
= ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_7415_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X2: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ R )
@ ^ [X2: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ S3 ) )
= ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_7416_pred__subset__eq2,axiom,
! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
( ( ord_le3636971675376928563eger_o
@ ^ [X2: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X2 @ Y ) @ R )
@ ^ [X2: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X2 @ Y ) @ S3 ) )
= ( ord_le3216752416896350996nteger @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_7417_pred__subset__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
( ( ord_le4340812435750786203eger_o
@ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y ) @ R )
@ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y ) @ S3 ) )
= ( ord_le653643898420964396nteger @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_7418_pred__subset__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
( ( ord_le5643404153117327598_int_o
@ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y ) @ R )
@ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y ) @ S3 ) )
= ( ord_le8725513860283290265nt_int @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_7419_pred__subset__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
( ( ord_le2124322318746777828_int_o
@ ^ [X2: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y ) @ R )
@ ^ [X2: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y ) @ S3 ) )
= ( ord_le135402666524580259nt_int @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_7420_summable__iff__shift,axiom,
! [F: nat > real,K: nat] :
( ( summable_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
= ( summable_real @ F ) ) ).
% summable_iff_shift
thf(fact_7421_summable__cmult__iff,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_cmult_iff
thf(fact_7422_summable__cmult__iff,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_cmult_iff
thf(fact_7423_summable__divide__iff,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_divide_iff
thf(fact_7424_summable__divide__iff,axiom,
! [F: nat > real,C: real] :
( ( summable_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_7425_summable__geometric__iff,axiom,
! [C: real] :
( ( summable_real @ ( power_power_real @ C ) )
= ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% summable_geometric_iff
thf(fact_7426_summable__geometric__iff,axiom,
! [C: complex] :
( ( summable_complex @ ( power_power_complex @ C ) )
= ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% summable_geometric_iff
thf(fact_7427_summable__comparison__test_H,axiom,
! [G: nat > real,N5: nat,F: nat > real] :
( ( summable_real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test'
thf(fact_7428_summable__comparison__test_H,axiom,
! [G: nat > real,N5: nat,F: nat > complex] :
( ( summable_real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test'
thf(fact_7429_summable__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test
thf(fact_7430_summable__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test
thf(fact_7431_summable__mult2,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).
% summable_mult2
thf(fact_7432_summable__mult,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).
% summable_mult
thf(fact_7433_summable__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).
% summable_divide
thf(fact_7434_summable__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).
% summable_divide
thf(fact_7435_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_7436_summable__ignore__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_7437_summable__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% summable_add
thf(fact_7438_summable__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( summable_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% summable_add
thf(fact_7439_summable__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( summable_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% summable_add
thf(fact_7440_suminf__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% suminf_le
thf(fact_7441_suminf__le,axiom,
! [F: nat > nat,G: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% suminf_le
thf(fact_7442_suminf__le,axiom,
! [F: nat > int,G: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% suminf_le
thf(fact_7443_summable__mult__D,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
=> ( ( C != zero_zero_complex )
=> ( summable_complex @ F ) ) ) ).
% summable_mult_D
thf(fact_7444_summable__mult__D,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
=> ( ( C != zero_zero_real )
=> ( summable_real @ F ) ) ) ).
% summable_mult_D
thf(fact_7445_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_7446_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_7447_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_7448_suminf__mult,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
= ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% suminf_mult
thf(fact_7449_suminf__mult2,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( times_times_real @ ( suminf_real @ F ) @ C )
= ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).
% suminf_mult2
thf(fact_7450_suminf__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
= ( suminf_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% suminf_add
thf(fact_7451_suminf__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
= ( suminf_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% suminf_add
thf(fact_7452_suminf__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
= ( suminf_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% suminf_add
thf(fact_7453_suminf__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
= ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_7454_suminf__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
= ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_7455_powser__insidea,axiom,
! [F: nat > real,X: real,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_7456_powser__insidea,axiom,
! [F: nat > complex,X: complex,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_7457_suminf__nonneg,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_7458_suminf__nonneg,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_7459_suminf__nonneg,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_7460_suminf__eq__zero__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ( suminf_real @ F )
= zero_zero_real )
= ( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_real ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_7461_suminf__eq__zero__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ( suminf_nat @ F )
= zero_zero_nat )
= ( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_nat ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_7462_suminf__eq__zero__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ( suminf_int @ F )
= zero_zero_int )
= ( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_int ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_7463_suminf__pos,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_pos
thf(fact_7464_suminf__pos,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_pos
thf(fact_7465_suminf__pos,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_pos
thf(fact_7466_summable__zero__power_H,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% summable_zero_power'
thf(fact_7467_summable__zero__power_H,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% summable_zero_power'
thf(fact_7468_summable__zero__power_H,axiom,
! [F: nat > int] :
( summable_int
@ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% summable_zero_power'
thf(fact_7469_summable__0__powser,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% summable_0_powser
thf(fact_7470_summable__0__powser,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% summable_0_powser
thf(fact_7471_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% powser_split_head(3)
thf(fact_7472_powser__split__head_I3_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% powser_split_head(3)
thf(fact_7473_summable__powser__split__head,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
= ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% summable_powser_split_head
thf(fact_7474_summable__powser__split__head,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
= ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% summable_powser_split_head
thf(fact_7475_summable__powser__ignore__initial__segment,axiom,
! [F: nat > complex,M: nat,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z @ N2 ) ) )
= ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_7476_summable__powser__ignore__initial__segment,axiom,
! [F: nat > real,M: nat,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z @ N2 ) ) )
= ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_7477_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
% pi_ge_zero
thf(fact_7478_summable__norm__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_7479_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_7480_summable__rabs,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% summable_rabs
thf(fact_7481_suminf__pos2,axiom,
! [F: nat > real,I2: nat] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_7482_suminf__pos2,axiom,
! [F: nat > nat,I2: nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_7483_suminf__pos2,axiom,
! [F: nat > int,I2: nat] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_7484_suminf__pos__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
= ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_7485_suminf__pos__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
= ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_7486_suminf__pos__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
= ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_7487_suminf__le__const,axiom,
! [F: nat > int,X: int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_7488_suminf__le__const,axiom,
! [F: nat > nat,X: nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_7489_suminf__le__const,axiom,
! [F: nat > real,X: real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_7490_summableI__nonneg__bounded,axiom,
! [F: nat > int,X: int] :
( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( summable_int @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_7491_summableI__nonneg__bounded,axiom,
! [F: nat > nat,X: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( summable_nat @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_7492_summableI__nonneg__bounded,axiom,
! [F: nat > real,X: real] :
( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( summable_real @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_7493_summable__geometric,axiom,
! [C: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
=> ( summable_real @ ( power_power_real @ C ) ) ) ).
% summable_geometric
thf(fact_7494_summable__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% summable_geometric
thf(fact_7495_complete__algebra__summable__geometric,axiom,
! [X: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
=> ( summable_real @ ( power_power_real @ X ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_7496_complete__algebra__summable__geometric,axiom,
! [X: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_7497_suminf__split__head,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_7498_summable__norm,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% summable_norm
thf(fact_7499_summable__norm,axiom,
! [F: nat > complex] :
( ( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
@ ( suminf_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% summable_norm
thf(fact_7500_sum__le__suminf,axiom,
! [F: nat > int,I5: set_nat] :
( ( summable_int @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_7501_sum__le__suminf,axiom,
! [F: nat > nat,I5: set_nat] :
( ( summable_nat @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_7502_sum__le__suminf,axiom,
! [F: nat > real,I5: set_nat] :
( ( summable_real @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_7503_suminf__split__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ( suminf_real @ F )
= ( plus_plus_real
@ ( suminf_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
@ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_7504_suminf__minus__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_7505_powser__inside,axiom,
! [F: nat > real,X: real,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% powser_inside
thf(fact_7506_powser__inside,axiom,
! [F: nat > complex,X: complex,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% powser_inside
thf(fact_7507_sum__less__suminf,axiom,
! [F: nat > int,N: nat] :
( ( summable_int @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N @ M5 )
=> ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_7508_sum__less__suminf,axiom,
! [F: nat > nat,N: nat] :
( ( summable_nat @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N @ M5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_7509_sum__less__suminf,axiom,
! [F: nat > real,N: nat] :
( ( summable_real @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N @ M5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_7510_pi__less__4,axiom,
ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% pi_less_4
thf(fact_7511_pi__ge__two,axiom,
ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% pi_ge_two
thf(fact_7512_pi__half__neq__two,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_neq_two
thf(fact_7513_powser__split__head_I1_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
=> ( ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
= ( plus_plus_complex @ ( F @ zero_zero_nat )
@ ( times_times_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
@ Z ) ) ) ) ).
% powser_split_head(1)
thf(fact_7514_powser__split__head_I1_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
=> ( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
= ( plus_plus_real @ ( F @ zero_zero_nat )
@ ( times_times_real
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
@ Z ) ) ) ) ).
% powser_split_head(1)
thf(fact_7515_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
@ Z )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_7516_powser__split__head_I2_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
@ Z )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_7517_summable__partial__sum__bound,axiom,
! [F: nat > complex,E: real] :
( ( summable_complex @ F )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N9: nat] :
~ ! [M2: nat] :
( ( ord_less_eq_nat @ N9 @ M2 )
=> ! [N7: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_7518_summable__partial__sum__bound,axiom,
! [F: nat > real,E: real] :
( ( summable_real @ F )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N9: nat] :
~ ! [M2: nat] :
( ( ord_less_eq_nat @ N9 @ M2 )
=> ! [N7: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_7519_suminf__exist__split,axiom,
! [R2: real,F: nat > real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_real @ F )
=> ? [N9: nat] :
! [N7: nat] :
( ( ord_less_eq_nat @ N9 @ N7 )
=> ( ord_less_real
@ ( real_V7735802525324610683m_real
@ ( suminf_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N7 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_7520_suminf__exist__split,axiom,
! [R2: real,F: nat > complex] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_complex @ F )
=> ? [N9: nat] :
! [N7: nat] :
( ( ord_less_eq_nat @ N9 @ N7 )
=> ( ord_less_real
@ ( real_V1022390504157884413omplex
@ ( suminf_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N7 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_7521_summable__power__series,axiom,
! [F: nat > real,Z: real] :
( ! [I4: nat] : ( ord_less_eq_real @ ( F @ I4 ) @ one_one_real )
=> ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( summable_real
@ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_7522_Abel__lemma,axiom,
! [R2: real,R0: real,A: nat > complex,M7: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_real @ R2 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_7523_pred__equals__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X2: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R ) )
= ( ^ [X2: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_7524_pred__equals__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ( ( ^ [X2: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ R ) )
= ( ^ [X2: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_7525_pred__equals__eq2,axiom,
! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
( ( ( ^ [X2: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X2 @ Y ) @ R ) )
= ( ^ [X2: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X2 @ Y ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_7526_pred__equals__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
( ( ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y ) @ R ) )
= ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_7527_pred__equals__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
( ( ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y ) @ R ) )
= ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_7528_pred__equals__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
( ( ( ^ [X2: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y ) @ R ) )
= ( ^ [X2: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_7529_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X2: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_7530_bot__empty__eq2,axiom,
( bot_bot_int_int_o
= ( ^ [X2: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_7531_bot__empty__eq2,axiom,
( bot_bo5358457235160185703eger_o
= ( ^ [X2: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X2 @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).
% bot_empty_eq2
thf(fact_7532_bot__empty__eq2,axiom,
( bot_bo3000040243691356879eger_o
= ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).
% bot_empty_eq2
thf(fact_7533_bot__empty__eq2,axiom,
( bot_bo8662317086119403298_int_o
= ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_7534_bot__empty__eq2,axiom,
( bot_bo1403522918969695512_int_o
= ( ^ [X2: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_7535_summable__ratio__test,axiom,
! [C: real,N5: nat,F: nat > real] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_ratio_test
thf(fact_7536_summable__ratio__test,axiom,
! [C: real,N5: nat,F: nat > complex] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_ratio_test
thf(fact_7537_sum__less__suminf2,axiom,
! [F: nat > int,N: nat,I2: nat] :
( ( summable_int @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N @ M5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
=> ( ( ord_less_eq_nat @ N @ I2 )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_7538_sum__less__suminf2,axiom,
! [F: nat > nat,N: nat,I2: nat] :
( ( summable_nat @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N @ M5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
=> ( ( ord_less_eq_nat @ N @ I2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_7539_sum__less__suminf2,axiom,
! [F: nat > real,N: nat,I2: nat] :
( ( summable_real @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N @ M5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
=> ( ( ord_less_eq_nat @ N @ I2 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_7540_pi__half__neq__zero,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% pi_half_neq_zero
thf(fact_7541_pi__half__less__two,axiom,
ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% pi_half_less_two
thf(fact_7542_pi__half__le__two,axiom,
ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% pi_half_le_two
thf(fact_7543_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_gt_zero
thf(fact_7544_pi__half__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_ge_zero
thf(fact_7545_m2pi__less__pi,axiom,
ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% m2pi_less_pi
thf(fact_7546_arctan__ubound,axiom,
! [Y2: real] : ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arctan_ubound
thf(fact_7547_arctan__one,axiom,
( ( arctan @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% arctan_one
thf(fact_7548_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_7549_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% subrelI
thf(fact_7550_subrelI,axiom,
! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
( ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ R2 )
=> ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le3216752416896350996nteger @ R2 @ S2 ) ) ).
% subrelI
thf(fact_7551_subrelI,axiom,
! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R2 )
=> ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le653643898420964396nteger @ R2 @ S2 ) ) ).
% subrelI
thf(fact_7552_subrelI,axiom,
! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R2 )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le8725513860283290265nt_int @ R2 @ S2 ) ) ).
% subrelI
thf(fact_7553_subrelI,axiom,
! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R2 )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le135402666524580259nt_int @ R2 @ S2 ) ) ).
% subrelI
thf(fact_7554_minus__pi__half__less__zero,axiom,
ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% minus_pi_half_less_zero
thf(fact_7555_arctan__lbound,axiom,
! [Y2: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) ) ).
% arctan_lbound
thf(fact_7556_arctan__bounded,axiom,
! [Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) )
& ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arctan_bounded
thf(fact_7557_pred__subset__eq,axiom,
! [R: set_nat,S3: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S3 ) )
= ( ord_less_eq_set_nat @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_7558_pred__subset__eq,axiom,
! [R: set_real,S3: set_real] :
( ( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ R )
@ ^ [X2: real] : ( member_real @ X2 @ S3 ) )
= ( ord_less_eq_set_real @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_7559_pred__subset__eq,axiom,
! [R: set_complex,S3: set_complex] :
( ( ord_le4573692005234683329plex_o
@ ^ [X2: complex] : ( member_complex @ X2 @ R )
@ ^ [X2: complex] : ( member_complex @ X2 @ S3 ) )
= ( ord_le211207098394363844omplex @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_7560_pred__subset__eq,axiom,
! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le704812498762024988_nat_o
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ R )
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ S3 ) )
= ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_7561_pred__subset__eq,axiom,
! [R: set_int,S3: set_int] :
( ( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ R )
@ ^ [X2: int] : ( member_int @ X2 @ S3 ) )
= ( ord_less_eq_set_int @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_7562_sum__pos__lt__pair,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_7563_machin__Euler,axiom,
( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% machin_Euler
thf(fact_7564_machin,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% machin
thf(fact_7565_sin__cos__npi,axiom,
! [N: nat] :
( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% sin_cos_npi
thf(fact_7566_cos__pi__eq__zero,axiom,
! [M: nat] :
( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_eq_zero
thf(fact_7567_accp__subset,axiom,
! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
=> ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).
% accp_subset
thf(fact_7568_accp__subset,axiom,
! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
=> ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_7569_accp__subset,axiom,
! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
=> ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_7570_accp__subset,axiom,
! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
=> ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_7571_accp__subset,axiom,
! [R1: nat > nat > $o,R22: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
=> ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_7572_geometric__deriv__sums,axiom,
! [Z: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
@ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_7573_geometric__deriv__sums,axiom,
! [Z: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
=> ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
@ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_7574_monoseq__def,axiom,
( topolo6980174941875973593q_real
= ( ^ [X6: nat > real] :
( ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) )
| ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ M6 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7575_monoseq__def,axiom,
( topolo3100542954746470799et_int
= ( ^ [X6: nat > set_int] :
( ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_set_int @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) )
| ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_set_int @ ( X6 @ N2 ) @ ( X6 @ M6 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7576_monoseq__def,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X6: nat > rat] :
( ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_rat @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) )
| ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ M6 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7577_monoseq__def,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X6: nat > num] :
( ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_num @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) )
| ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ M6 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7578_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_nat @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) )
| ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ M6 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7579_monoseq__def,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X6: nat > int] :
( ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_int @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) )
| ! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ M6 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7580_monoI2,axiom,
! [X8: nat > real] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% monoI2
thf(fact_7581_monoI2,axiom,
! [X8: nat > set_int] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_set_int @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% monoI2
thf(fact_7582_monoI2,axiom,
! [X8: nat > rat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% monoI2
thf(fact_7583_monoI2,axiom,
! [X8: nat > num] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% monoI2
thf(fact_7584_monoI2,axiom,
! [X8: nat > nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% monoI2
thf(fact_7585_monoI2,axiom,
! [X8: nat > int] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% monoI2
thf(fact_7586_cos__periodic__pi,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_7587_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_7588_sin__periodic__pi,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_7589_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_7590_sin__cos__squared__add3,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
= one_one_complex ) ).
% sin_cos_squared_add3
thf(fact_7591_sin__cos__squared__add3,axiom,
! [X: real] :
( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
= one_one_real ) ).
% sin_cos_squared_add3
thf(fact_7592_sin__npi2,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi2
thf(fact_7593_sin__npi,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% sin_npi
thf(fact_7594_sin__npi__int,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi_int
thf(fact_7595_powser__sums__zero__iff,axiom,
! [A: nat > complex,X: complex] :
( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_7596_powser__sums__zero__iff,axiom,
! [A: nat > real,X: real] :
( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_7597_cos__pi__half,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_half
thf(fact_7598_sin__two__pi,axiom,
( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= zero_zero_real ) ).
% sin_two_pi
thf(fact_7599_sin__pi__half,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_pi_half
thf(fact_7600_cos__two__pi,axiom,
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_real ) ).
% cos_two_pi
thf(fact_7601_cos__periodic,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( cos_real @ X ) ) ).
% cos_periodic
thf(fact_7602_sin__periodic,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( sin_real @ X ) ) ).
% sin_periodic
thf(fact_7603_cos__2pi__minus,axiom,
! [X: real] :
( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
= ( cos_real @ X ) ) ).
% cos_2pi_minus
thf(fact_7604_cos__npi,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% cos_npi
thf(fact_7605_cos__npi2,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% cos_npi2
thf(fact_7606_sin__cos__squared__add,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_cos_squared_add
thf(fact_7607_sin__cos__squared__add,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_cos_squared_add
thf(fact_7608_sin__cos__squared__add2,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_cos_squared_add2
thf(fact_7609_sin__cos__squared__add2,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_cos_squared_add2
thf(fact_7610_sin__2npi,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
= zero_zero_real ) ).
% sin_2npi
thf(fact_7611_cos__2npi,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
= one_one_real ) ).
% cos_2npi
thf(fact_7612_sin__2pi__minus,axiom,
! [X: real] :
( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_2pi_minus
thf(fact_7613_sin__int__2pin,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_int_2pin
thf(fact_7614_cos__int__2pin,axiom,
! [N: int] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= one_one_real ) ).
% cos_int_2pin
thf(fact_7615_cos__3over2__pi,axiom,
( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
= zero_zero_real ) ).
% cos_3over2_pi
thf(fact_7616_sin__3over2__pi,axiom,
( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% sin_3over2_pi
thf(fact_7617_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= one_one_real ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% cos_npi_int
thf(fact_7618_sums__le,axiom,
! [F: nat > real,G: nat > real,S2: real,T: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_real @ F @ S2 )
=> ( ( sums_real @ G @ T )
=> ( ord_less_eq_real @ S2 @ T ) ) ) ) ).
% sums_le
thf(fact_7619_sums__le,axiom,
! [F: nat > nat,G: nat > nat,S2: nat,T: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_nat @ F @ S2 )
=> ( ( sums_nat @ G @ T )
=> ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).
% sums_le
thf(fact_7620_sums__le,axiom,
! [F: nat > int,G: nat > int,S2: int,T: int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_int @ F @ S2 )
=> ( ( sums_int @ G @ T )
=> ( ord_less_eq_int @ S2 @ T ) ) ) ) ).
% sums_le
thf(fact_7621_sin__add,axiom,
! [X: real,Y2: real] :
( ( sin_real @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y2 ) ) ) ) ).
% sin_add
thf(fact_7622_polar__Ex,axiom,
! [X: real,Y2: real] :
? [R3: real,A3: real] :
( ( X
= ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
& ( Y2
= ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).
% polar_Ex
thf(fact_7623_sin__diff,axiom,
! [X: real,Y2: real] :
( ( sin_real @ ( minus_minus_real @ X @ Y2 ) )
= ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y2 ) ) ) ) ).
% sin_diff
thf(fact_7624_sums__mult,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
@ ( times_times_real @ C @ A ) ) ) ).
% sums_mult
thf(fact_7625_sums__mult2,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
@ ( times_times_real @ A @ C ) ) ) ).
% sums_mult2
thf(fact_7626_sums__add,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_real @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7627_sums__add,axiom,
! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
( ( sums_nat @ F @ A )
=> ( ( sums_nat @ G @ B )
=> ( sums_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_nat @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7628_sums__add,axiom,
! [F: nat > int,A: int,G: nat > int,B: int] :
( ( sums_int @ F @ A )
=> ( ( sums_int @ G @ B )
=> ( sums_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_int @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7629_sums__divide,axiom,
! [F: nat > complex,A: complex,C: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
@ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% sums_divide
thf(fact_7630_sums__divide,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
@ ( divide_divide_real @ A @ C ) ) ) ).
% sums_divide
thf(fact_7631_cos__add,axiom,
! [X: real,Y2: real] :
( ( cos_real @ ( plus_plus_real @ X @ Y2 ) )
= ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y2 ) ) ) ) ).
% cos_add
thf(fact_7632_cos__diff,axiom,
! [X: real,Y2: real] :
( ( cos_real @ ( minus_minus_real @ X @ Y2 ) )
= ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y2 ) ) ) ) ).
% cos_diff
thf(fact_7633_sin__double,axiom,
! [X: complex] :
( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% sin_double
thf(fact_7634_sin__double,axiom,
! [X: real] :
( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% sin_double
thf(fact_7635_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
& ( ord_less_eq_real @ Y3 @ pi )
& ( ( sin_real @ Y3 )
= ( sin_real @ X ) )
& ( ( cos_real @ Y3 )
= ( cos_real @ X ) ) ) ).
% sincos_principal_value
thf(fact_7636_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_7637_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% sin_le_one
thf(fact_7638_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% cos_le_one
thf(fact_7639_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_7640_sums__mult__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
@ ( times_times_complex @ C @ D ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_7641_sums__mult__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
@ ( times_times_real @ C @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_7642_sums__mult2__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
@ ( times_times_complex @ D @ C ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_7643_sums__mult2__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
@ ( times_times_real @ D @ C ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_7644_sin__cos__le1,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) ) ) @ one_one_real ) ).
% sin_cos_le1
thf(fact_7645_cos__squared__eq,axiom,
! [X: complex] :
( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_squared_eq
thf(fact_7646_cos__squared__eq,axiom,
! [X: real] :
( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_squared_eq
thf(fact_7647_sin__squared__eq,axiom,
! [X: complex] :
( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sin_squared_eq
thf(fact_7648_sin__squared__eq,axiom,
! [X: real] :
( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sin_squared_eq
thf(fact_7649_sums__mult__D,axiom,
! [C: complex,F: nat > complex,A: complex] :
( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
@ A )
=> ( ( C != zero_zero_complex )
=> ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_7650_sums__mult__D,axiom,
! [C: real,F: nat > real,A: real] :
( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
@ A )
=> ( ( C != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_7651_sums__Suc__imp,axiom,
! [F: nat > complex,S2: complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S2 )
=> ( sums_complex @ F @ S2 ) ) ) ).
% sums_Suc_imp
thf(fact_7652_sums__Suc__imp,axiom,
! [F: nat > real,S2: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S2 )
=> ( sums_real @ F @ S2 ) ) ) ).
% sums_Suc_imp
thf(fact_7653_sums__Suc__iff,axiom,
! [F: nat > real,S2: real] :
( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S2 )
= ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc_iff
thf(fact_7654_sums__Suc,axiom,
! [F: nat > real,L2: real] :
( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L2 )
=> ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7655_sums__Suc,axiom,
! [F: nat > nat,L2: nat] :
( ( sums_nat
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L2 )
=> ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7656_sums__Suc,axiom,
! [F: nat > int,L2: int] :
( ( sums_int
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L2 )
=> ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7657_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_7658_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_7659_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > complex,S2: complex] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( F @ I4 )
= zero_zero_complex ) )
=> ( ( sums_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S2 )
= ( sums_complex @ F @ S2 ) ) ) ).
% sums_zero_iff_shift
thf(fact_7660_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > real,S2: real] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( F @ I4 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S2 )
= ( sums_real @ F @ S2 ) ) ) ).
% sums_zero_iff_shift
thf(fact_7661_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% sin_ge_minus_one
thf(fact_7662_cos__inj__pi,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ pi )
=> ( ( ( cos_real @ X )
= ( cos_real @ Y2 ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_7663_cos__mono__le__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ pi )
=> ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_7664_cos__monotone__0__pi__le,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_7665_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% cos_ge_minus_one
thf(fact_7666_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% abs_sin_le_one
thf(fact_7667_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% abs_cos_le_one
thf(fact_7668_sin__times__sin,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7669_sin__times__sin,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7670_sin__times__cos,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7671_sin__times__cos,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7672_cos__times__sin,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7673_cos__times__sin,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7674_sin__plus__sin,axiom,
! [W: complex,Z: complex] :
( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7675_sin__plus__sin,axiom,
! [W: real,Z: real] :
( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7676_sin__diff__sin,axiom,
! [W: complex,Z: complex] :
( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7677_sin__diff__sin,axiom,
! [W: real,Z: real] :
( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7678_cos__diff__cos,axiom,
! [W: complex,Z: complex] :
( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7679_cos__diff__cos,axiom,
! [W: real,Z: real] :
( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7680_cos__double,axiom,
! [X: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_double
thf(fact_7681_cos__double,axiom,
! [X: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_double
thf(fact_7682_cos__double__sin,axiom,
! [W: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_7683_cos__double__sin,axiom,
! [W: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_7684_powser__sums__if,axiom,
! [M: nat,Z: complex] :
( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
@ ( power_power_complex @ Z @ M ) ) ).
% powser_sums_if
thf(fact_7685_powser__sums__if,axiom,
! [M: nat,Z: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
@ ( power_power_real @ Z @ M ) ) ).
% powser_sums_if
thf(fact_7686_powser__sums__if,axiom,
! [M: nat,Z: int] :
( sums_int
@ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
@ ( power_power_int @ Z @ M ) ) ).
% powser_sums_if
thf(fact_7687_powser__sums__zero,axiom,
! [A: nat > complex] :
( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7688_powser__sums__zero,axiom,
! [A: nat > real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7689_cos__two__neq__zero,axiom,
( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% cos_two_neq_zero
thf(fact_7690_cos__mono__less__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ pi )
=> ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_7691_cos__monotone__0__pi,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_7692_sums__iff__shift,axiom,
! [F: nat > real,N: nat,S2: real] :
( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S2 )
= ( sums_real @ F @ ( plus_plus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_iff_shift
thf(fact_7693_sums__iff__shift_H,axiom,
! [F: nat > real,N: nat,S2: real] :
( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ ( minus_minus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
= ( sums_real @ F @ S2 ) ) ).
% sums_iff_shift'
thf(fact_7694_sums__split__initial__segment,axiom,
! [F: nat > real,S2: real,N: nat] :
( ( sums_real @ F @ S2 )
=> ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ ( minus_minus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_7695_cos__monotone__minus__pi__0_H,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( cos_real @ Y2 ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_7696_sums__If__finite__set_H,axiom,
! [G: nat > real,S3: real,A2: set_nat,S5: real,F: nat > real] :
( ( sums_real @ G @ S3 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( S5
= ( plus_plus_real @ S3
@ ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ A2 ) ) )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
@ S5 ) ) ) ) ).
% sums_If_finite_set'
thf(fact_7697_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [I3: int] :
( X
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_7698_sincos__total__pi,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T4: real] :
( ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ pi )
& ( X
= ( cos_real @ T4 ) )
& ( Y2
= ( sin_real @ T4 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_7699_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_7700_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_7701_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_7702_cos__two__less__zero,axiom,
ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_less_zero
thf(fact_7703_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X3 )
= zero_zero_real )
& ! [Y4: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
& ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ Y4 )
= zero_zero_real ) )
=> ( Y4 = X3 ) ) ) ).
% cos_is_zero
thf(fact_7704_cos__two__le__zero,axiom,
ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_le_zero
thf(fact_7705_cos__monotone__minus__pi__0,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cos_real @ Y2 ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_7706_cos__total,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ pi )
& ( ( cos_real @ X3 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
& ( ord_less_eq_real @ Y4 @ pi )
& ( ( cos_real @ Y4 )
= Y2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_7707_sincos__total__pi__half,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T4: real] :
( ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( X
= ( cos_real @ T4 ) )
& ( Y2
= ( sin_real @ T4 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_7708_sincos__total__2pi__le,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T4: real] :
( ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
& ( X
= ( cos_real @ T4 ) )
& ( Y2
= ( sin_real @ T4 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_7709_sincos__total__2pi,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ~ ! [T4: real] :
( ( ord_less_eq_real @ zero_zero_real @ T4 )
=> ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X
= ( cos_real @ T4 ) )
=> ( Y2
!= ( sin_real @ T4 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_7710_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_7711_cos__times__cos,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7712_cos__times__cos,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7713_cos__plus__cos,axiom,
! [W: complex,Z: complex] :
( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7714_cos__plus__cos,axiom,
! [W: real,Z: real] :
( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7715_geometric__sums,axiom,
! [C: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
=> ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% geometric_sums
thf(fact_7716_geometric__sums,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% geometric_sums
thf(fact_7717_power__half__series,axiom,
( sums_real
@ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
@ one_one_real ) ).
% power_half_series
thf(fact_7718_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_7719_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less_real @ pi @ X )
=> ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_lt_zero
thf(fact_7720_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% cos_double_less_one
thf(fact_7721_sin__30,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_30
thf(fact_7722_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_7723_sin__inj__pi,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( sin_real @ X )
= ( sin_real @ Y2 ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_7724_sin__mono__le__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_7725_sin__monotone__2pi__le,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sin_real @ Y2 ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_7726_cos__60,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_60
thf(fact_7727_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X2: int] :
( X
= ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_7728_cos__double__cos,axiom,
! [W: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% cos_double_cos
thf(fact_7729_cos__double__cos,axiom,
! [W: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% cos_double_cos
thf(fact_7730_cos__treble__cos,axiom,
! [X: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% cos_treble_cos
thf(fact_7731_cos__treble__cos,axiom,
! [X: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% cos_treble_cos
thf(fact_7732_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_7733_sums__if,axiom,
! [G: nat > real,X: real,F: nat > real,Y2: real] :
( ( sums_real @ G @ X )
=> ( ( sums_real @ F @ Y2 )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ X @ Y2 ) ) ) ) ).
% sums_if
thf(fact_7734_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ pi @ X )
=> ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_le_zero
thf(fact_7735_sin__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_less_zero
thf(fact_7736_sin__mono__less__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_7737_sin__monotone__2pi,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sin_real @ Y2 ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_7738_sin__total,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ? [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X3 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
& ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ Y4 )
= Y2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_7739_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_7740_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_7741_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X2: nat] :
( X
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
| ? [X2: nat] :
( X
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_7742_accp__subset__induct,axiom,
! [D4: product_prod_num_num > $o,R: product_prod_num_num > product_prod_num_num > $o,X: product_prod_num_num,P: product_prod_num_num > $o] :
( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R ) )
=> ( ! [X3: product_prod_num_num,Z3: product_prod_num_num] :
( ( D4 @ X3 )
=> ( ( R @ Z3 @ X3 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: product_prod_num_num] :
( ( D4 @ X3 )
=> ( ! [Z4: product_prod_num_num] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7743_accp__subset__induct,axiom,
! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
=> ( ! [X3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( D4 @ X3 )
=> ( ( R @ Z3 @ X3 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: product_prod_nat_nat] :
( ( D4 @ X3 )
=> ( ! [Z4: product_prod_nat_nat] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7744_accp__subset__induct,axiom,
! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
=> ( ! [X3: product_prod_int_int,Z3: product_prod_int_int] :
( ( D4 @ X3 )
=> ( ( R @ Z3 @ X3 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: product_prod_int_int] :
( ( D4 @ X3 )
=> ( ! [Z4: product_prod_int_int] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7745_accp__subset__induct,axiom,
! [D4: list_nat > $o,R: list_nat > list_nat > $o,X: list_nat,P: list_nat > $o] :
( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
=> ( ! [X3: list_nat,Z3: list_nat] :
( ( D4 @ X3 )
=> ( ( R @ Z3 @ X3 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: list_nat] :
( ( D4 @ X3 )
=> ( ! [Z4: list_nat] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7746_accp__subset__induct,axiom,
! [D4: nat > $o,R: nat > nat > $o,X: nat,P: nat > $o] :
( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
=> ( ! [X3: nat,Z3: nat] :
( ( D4 @ X3 )
=> ( ( R @ Z3 @ X3 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: nat] :
( ( D4 @ X3 )
=> ( ! [Z4: nat] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7747_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_7748_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [I3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
& ( X
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_7749_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= zero_zero_real )
= ( ? [I3: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
& ( X
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_7750_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ? [N3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_7751_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_7752_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( cos_real @ X )
= zero_zero_real )
=> ? [N3: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_7753_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos_real @ X )
= zero_zero_real )
= ( ? [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_7754_mono__SucI1,axiom,
! [X8: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% mono_SucI1
thf(fact_7755_mono__SucI1,axiom,
! [X8: nat > set_int] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% mono_SucI1
thf(fact_7756_mono__SucI1,axiom,
! [X8: nat > rat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% mono_SucI1
thf(fact_7757_mono__SucI1,axiom,
! [X8: nat > num] :
( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% mono_SucI1
thf(fact_7758_mono__SucI1,axiom,
! [X8: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% mono_SucI1
thf(fact_7759_mono__SucI1,axiom,
! [X8: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% mono_SucI1
thf(fact_7760_mono__SucI2,axiom,
! [X8: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% mono_SucI2
thf(fact_7761_mono__SucI2,axiom,
! [X8: nat > set_int] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% mono_SucI2
thf(fact_7762_mono__SucI2,axiom,
! [X8: nat > rat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% mono_SucI2
thf(fact_7763_mono__SucI2,axiom,
! [X8: nat > num] :
( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% mono_SucI2
thf(fact_7764_mono__SucI2,axiom,
! [X8: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% mono_SucI2
thf(fact_7765_mono__SucI2,axiom,
! [X8: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% mono_SucI2
thf(fact_7766_monoseq__Suc,axiom,
( topolo6980174941875973593q_real
= ( ^ [X6: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7767_monoseq__Suc,axiom,
( topolo3100542954746470799et_int
= ( ^ [X6: nat > set_int] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_set_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7768_monoseq__Suc,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X6: nat > rat] :
( ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7769_monoseq__Suc,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X6: nat > num] :
( ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7770_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7771_monoseq__Suc,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X6: nat > int] :
( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7772_monoI1,axiom,
! [X8: nat > real] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% monoI1
thf(fact_7773_monoI1,axiom,
! [X8: nat > set_int] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_set_int @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% monoI1
thf(fact_7774_monoI1,axiom,
! [X8: nat > rat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% monoI1
thf(fact_7775_monoI1,axiom,
! [X8: nat > num] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_num @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% monoI1
thf(fact_7776_monoI1,axiom,
! [X8: nat > nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% monoI1
thf(fact_7777_monoI1,axiom,
! [X8: nat > int] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_int @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% monoI1
thf(fact_7778_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y2: option_nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y2 )
=> ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( B2
=> ( Y2
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y2 = none_nat ) ) ) ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ( Y2 = none_nat )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( some_nat @ Ma2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_7779_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y2: option_nat] :
( ( ( vEBT_vebt_mint @ X )
= Y2 )
=> ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( A3
=> ( Y2
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( B2
=> ( Y2
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( Y2 = none_nat ) ) ) ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ( Y2 = none_nat )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( some_nat @ Mi2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_7780_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ? [T4: real] :
( ( ord_less_real @ X @ T4 )
& ( ord_less_real @ T4 @ zero_zero_real )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_7781_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ T4 )
& ( ord_less_real @ T4 @ X )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_7782_fact__Suc__0,axiom,
( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
= one_one_complex ) ).
% fact_Suc_0
thf(fact_7783_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_7784_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_7785_fact__Suc__0,axiom,
( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% fact_Suc_0
thf(fact_7786_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_7787_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_7788_fact__Suc,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_Suc
thf(fact_7789_fact__Suc,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_Suc
thf(fact_7790_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_7791_fact__2,axiom,
( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7792_fact__2,axiom,
( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7793_fact__2,axiom,
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7794_fact__2,axiom,
( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7795_fact__2,axiom,
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7796_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_zero
thf(fact_7797_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_zero
thf(fact_7798_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_zero
thf(fact_7799_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_zero
thf(fact_7800_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_gt_zero
thf(fact_7801_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_7802_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_gt_zero
thf(fact_7803_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_7804_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% fact_not_neg
thf(fact_7805_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_7806_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% fact_not_neg
thf(fact_7807_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_7808_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_1
thf(fact_7809_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_7810_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_1
thf(fact_7811_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_7812_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_mono
thf(fact_7813_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_mono
thf(fact_7814_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_mono
thf(fact_7815_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono
thf(fact_7816_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% fact_dvd
thf(fact_7817_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% fact_dvd
thf(fact_7818_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% fact_dvd
thf(fact_7819_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% fact_dvd
thf(fact_7820_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7821_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7822_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7823_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7824_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
= zero_zero_int ) ) ).
% fact_mod
thf(fact_7825_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
= zero_z3403309356797280102nteger ) ) ).
% fact_mod
thf(fact_7826_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
= zero_zero_nat ) ) ).
% fact_mod
thf(fact_7827_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7828_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7829_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7830_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7831_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7832_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7833_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7834_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7835_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7836_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% choose_dvd
thf(fact_7837_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% choose_dvd
thf(fact_7838_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% choose_dvd
thf(fact_7839_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% choose_dvd
thf(fact_7840_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% choose_dvd
thf(fact_7841_fact__numeral,axiom,
! [K: num] :
( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7842_fact__numeral,axiom,
! [K: num] :
( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
= ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7843_fact__numeral,axiom,
! [K: num] :
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
= ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7844_fact__numeral,axiom,
! [K: num] :
( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
= ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7845_fact__numeral,axiom,
! [K: num] :
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
= ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7846_square__fact__le__2__fact,axiom,
! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% square_fact_le_2_fact
thf(fact_7847_fact__num__eq__if,axiom,
( semiri5044797733671781792omplex
= ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7848_fact__num__eq__if,axiom,
( semiri1406184849735516958ct_int
= ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7849_fact__num__eq__if,axiom,
( semiri773545260158071498ct_rat
= ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7850_fact__num__eq__if,axiom,
( semiri2265585572941072030t_real
= ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7851_fact__num__eq__if,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7852_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7853_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri773545260158071498ct_rat @ N )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7854_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri2265585572941072030t_real @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7855_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7856_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > complex > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% Maclaurin_zero
thf(fact_7857_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > real > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% Maclaurin_zero
thf(fact_7858_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > rat > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% Maclaurin_zero
thf(fact_7859_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > nat > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% Maclaurin_zero
thf(fact_7860_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > int > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% Maclaurin_zero
thf(fact_7861_Maclaurin__lemma,axiom,
! [H2: real,F: real > real,J: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ? [B8: real] :
( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_7862_cos__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( cos_real @ X ) ) ).
% cos_paired
thf(fact_7863_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% cos_coeff_def
thf(fact_7864_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( sin_real @ X ) ) ).
% sin_paired
thf(fact_7865_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_7866_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ T4 )
& ( ord_less_real @ T4 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_7867_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_7868_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_7869_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_7870_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_7871_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_self
thf(fact_7872_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_7873_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_7874_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_7875_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_7876_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_7877_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_7878_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_7879_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_7880_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F: nat > nat > nat,A: nat,B: nat,Acc: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
=> ( ( ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
= Acc ) )
& ( ~ ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
= ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_7881_fold__atLeastAtMost__nat_Opelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y2: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
= Y2 )
=> ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y2
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) )
=> ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_7882_VEBT__internal_Ooption__shift_Opelims,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y2: option4927543243414619207at_nat] :
( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
= Y2 )
=> ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ Xa2 @ Xb2 ) ) )
=> ( ( ( Xa2 = none_P5556105721700978146at_nat )
=> ( ( Y2 = none_P5556105721700978146at_nat )
=> ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb2 ) ) ) ) )
=> ( ! [V2: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ V2 ) )
=> ( ( Xb2 = none_P5556105721700978146at_nat )
=> ( ( Y2 = none_P5556105721700978146at_nat )
=> ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
=> ~ ! [A3: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ! [B2: product_prod_nat_nat] :
( ( Xb2
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( ( Y2
= ( some_P7363390416028606310at_nat @ ( X @ A3 @ B2 ) ) )
=> ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_7883_VEBT__internal_Ooption__shift_Opelims,axiom,
! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y2: option_num] :
( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
= Y2 )
=> ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ Xa2 @ Xb2 ) ) )
=> ( ( ( Xa2 = none_num )
=> ( ( Y2 = none_num )
=> ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ none_num @ Xb2 ) ) ) ) )
=> ( ! [V2: num] :
( ( Xa2
= ( some_num @ V2 ) )
=> ( ( Xb2 = none_num )
=> ( ( Y2 = none_num )
=> ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
=> ~ ! [A3: num] :
( ( Xa2
= ( some_num @ A3 ) )
=> ! [B2: num] :
( ( Xb2
= ( some_num @ B2 ) )
=> ( ( Y2
= ( some_num @ ( X @ A3 @ B2 ) ) )
=> ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_7884_VEBT__internal_Ooption__shift_Opelims,axiom,
! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y2: option_nat] :
( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
= Y2 )
=> ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ Xa2 @ Xb2 ) ) )
=> ( ( ( Xa2 = none_nat )
=> ( ( Y2 = none_nat )
=> ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ none_nat @ Xb2 ) ) ) ) )
=> ( ! [V2: nat] :
( ( Xa2
= ( some_nat @ V2 ) )
=> ( ( Xb2 = none_nat )
=> ( ( Y2 = none_nat )
=> ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
=> ~ ! [A3: nat] :
( ( Xa2
= ( some_nat @ A3 ) )
=> ! [B2: nat] :
( ( Xb2
= ( some_nat @ B2 ) )
=> ( ( Y2
= ( some_nat @ ( X @ A3 @ B2 ) ) )
=> ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_7885_diffs__equiv,axiom,
! [C: nat > complex,X: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
=> ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% diffs_equiv
thf(fact_7886_diffs__equiv,axiom,
! [C: nat > real,X: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% diffs_equiv
thf(fact_7887_tan__double,axiom,
! [X: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_7888_tan__double,axiom,
! [X: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
!= zero_zero_real )
=> ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_7889_tan__periodic__pi,axiom,
! [X: real] :
( ( tan_real @ ( plus_plus_real @ X @ pi ) )
= ( tan_real @ X ) ) ).
% tan_periodic_pi
thf(fact_7890_tan__npi,axiom,
! [N: nat] :
( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% tan_npi
thf(fact_7891_tan__periodic__n,axiom,
! [X: real,N: num] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_n
thf(fact_7892_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_nat
thf(fact_7893_tan__periodic__int,axiom,
! [X: real,I2: int] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_int
thf(fact_7894_tan__periodic,axiom,
! [X: real] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic
thf(fact_7895_tan__def,axiom,
( tan_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X2 ) @ ( cos_complex @ X2 ) ) ) ) ).
% tan_def
thf(fact_7896_tan__def,axiom,
( tan_real
= ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ X2 ) @ ( cos_real @ X2 ) ) ) ) ).
% tan_def
thf(fact_7897_diffs__def,axiom,
( diffs_int
= ( ^ [C3: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% diffs_def
thf(fact_7898_diffs__def,axiom,
( diffs_real
= ( ^ [C3: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% diffs_def
thf(fact_7899_diffs__def,axiom,
( diffs_rat
= ( ^ [C3: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% diffs_def
thf(fact_7900_termdiff__converges__all,axiom,
! [C: nat > complex,X: complex] :
( ! [X3: complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% termdiff_converges_all
thf(fact_7901_termdiff__converges__all,axiom,
! [C: nat > real,X: real] :
( ! [X3: real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% termdiff_converges_all
thf(fact_7902_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y2: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
= Y2 )
=> ( ( ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y2
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_7903_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F3: nat > nat > nat,A4: nat,B3: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A4 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F3 @ A4 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_7904_tan__45,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= one_one_real ) ).
% tan_45
thf(fact_7905_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_7906_lemma__tan__total,axiom,
! [Y2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ord_less_real @ Y2 @ ( tan_real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_7907_tan__total,axiom,
! [Y2: real] :
? [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X3 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
& ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ Y4 )
= Y2 ) )
=> ( Y4 = X3 ) ) ) ).
% tan_total
thf(fact_7908_tan__monotone,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( tan_real @ Y2 ) @ ( tan_real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_7909_tan__monotone_H,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y2 @ X )
= ( ord_less_real @ ( tan_real @ Y2 ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_7910_tan__mono__lt__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_7911_lemma__tan__total1,axiom,
! [Y2: real] :
? [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X3 )
= Y2 ) ) ).
% lemma_tan_total1
thf(fact_7912_tan__minus__45,axiom,
( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% tan_minus_45
thf(fact_7913_tan__inverse,axiom,
! [Y2: real] :
( ( divide_divide_real @ one_one_real @ ( tan_real @ Y2 ) )
= ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 ) ) ) ).
% tan_inverse
thf(fact_7914_sum__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A4: nat] : ( plus_plus_complex @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_complex ) ) ).
% sum_atLeastAtMost_code
thf(fact_7915_sum__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7916_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_7917_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7918_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_7919_add__tan__eq,axiom,
! [X: complex,Y2: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y2 )
!= zero_zero_complex )
=> ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y2 ) )
= ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y2 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y2 ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_7920_add__tan__eq,axiom,
! [X: real,Y2: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y2 )
!= zero_zero_real )
=> ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) )
= ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_7921_termdiff__converges,axiom,
! [X: real,K5: real,C: nat > real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
=> ( ! [X3: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% termdiff_converges
thf(fact_7922_termdiff__converges,axiom,
! [X: complex,K5: real,C: nat > complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
=> ( ! [X3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% termdiff_converges
thf(fact_7923_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_7924_tan__total__pos,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X3 )
= Y2 ) ) ) ).
% tan_total_pos
thf(fact_7925_tan__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% tan_less_zero
thf(fact_7926_tan__mono__le__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_7927_tan__mono__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) ) ) ) ) ).
% tan_mono_le
thf(fact_7928_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% tan_bound_pi2
thf(fact_7929_arctan__unique,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( tan_real @ X )
= Y2 )
=> ( ( arctan @ Y2 )
= X ) ) ) ) ).
% arctan_unique
thf(fact_7930_arctan__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( arctan @ ( tan_real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_7931_arctan,axiom,
! [Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) )
& ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ ( arctan @ Y2 ) )
= Y2 ) ) ).
% arctan
thf(fact_7932_tan__add,axiom,
! [X: complex,Y2: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y2 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y2 ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( plus_plus_complex @ X @ Y2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y2 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y2 ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_7933_tan__add,axiom,
! [X: real,Y2: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y2 )
!= zero_zero_real )
=> ( ( ( cos_real @ ( plus_plus_real @ X @ Y2 ) )
!= zero_zero_real )
=> ( ( tan_real @ ( plus_plus_real @ X @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_7934_tan__diff,axiom,
! [X: complex,Y2: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y2 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y2 ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( minus_minus_complex @ X @ Y2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y2 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y2 ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_7935_tan__diff,axiom,
! [X: real,Y2: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y2 )
!= zero_zero_real )
=> ( ( ( cos_real @ ( minus_minus_real @ X @ Y2 ) )
!= zero_zero_real )
=> ( ( tan_real @ ( minus_minus_real @ X @ Y2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_7936_lemma__tan__add1,axiom,
! [X: complex,Y2: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y2 )
!= zero_zero_complex )
=> ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y2 ) ) )
= ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y2 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y2 ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_7937_lemma__tan__add1,axiom,
! [X: real,Y2: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y2 )
!= zero_zero_real )
=> ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y2 ) ) )
= ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_7938_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ? [Z3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
& ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
& ( ( tan_real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_7939_fact__code,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7940_fact__code,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7941_fact__code,axiom,
( semiri2265585572941072030t_real
= ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7942_fact__code,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7943_tan__half,axiom,
( tan_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).
% tan_half
thf(fact_7944_tan__half,axiom,
( tan_real
= ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).
% tan_half
thf(fact_7945_in__measure,axiom,
! [X: nat,Y2: nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( measure_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) ) ).
% in_measure
thf(fact_7946_in__measure,axiom,
! [X: int,Y2: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) ) ).
% in_measure
thf(fact_7947_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V1022390504157884413omplex @ Z )
= one_one_real )
=> ~ ! [T4: real] :
( ( ord_less_eq_real @ zero_zero_real @ T4 )
=> ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_7948_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
& ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_7949_in__finite__psubset,axiom,
! [A2: set_nat,B4: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B4 ) @ finite_psubset_nat )
= ( ( ord_less_set_nat @ A2 @ B4 )
& ( finite_finite_nat @ B4 ) ) ) ).
% in_finite_psubset
thf(fact_7950_in__finite__psubset,axiom,
! [A2: set_int,B4: set_int] :
( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A2 @ B4 ) @ finite_psubset_int )
= ( ( ord_less_set_int @ A2 @ B4 )
& ( finite_finite_int @ B4 ) ) ) ).
% in_finite_psubset
thf(fact_7951_in__finite__psubset,axiom,
! [A2: set_complex,B4: set_complex] :
( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A2 @ B4 ) @ finite8643634255014194347omplex )
= ( ( ord_less_set_complex @ A2 @ B4 )
& ( finite3207457112153483333omplex @ B4 ) ) ) ).
% in_finite_psubset
thf(fact_7952_sin__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( sin_real @ X )
= ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_7953_real__sqrt__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y2 ) )
= ( X = Y2 ) ) ).
% real_sqrt_eq_iff
thf(fact_7954_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_7955_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
% real_sqrt_zero
thf(fact_7956_real__sqrt__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ).
% real_sqrt_less_iff
thf(fact_7957_real__sqrt__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% real_sqrt_le_iff
thf(fact_7958_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% real_sqrt_eq_1_iff
thf(fact_7959_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
% real_sqrt_one
thf(fact_7960_exp__le__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% exp_le_cancel_iff
thf(fact_7961_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% real_sqrt_lt_0_iff
thf(fact_7962_real__sqrt__gt__0__iff,axiom,
! [Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y2 ) )
= ( ord_less_real @ zero_zero_real @ Y2 ) ) ).
% real_sqrt_gt_0_iff
thf(fact_7963_real__sqrt__ge__0__iff,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y2 ) )
= ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ).
% real_sqrt_ge_0_iff
thf(fact_7964_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% real_sqrt_le_0_iff
thf(fact_7965_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ).
% real_sqrt_lt_1_iff
thf(fact_7966_real__sqrt__gt__1__iff,axiom,
! [Y2: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y2 ) )
= ( ord_less_real @ one_one_real @ Y2 ) ) ).
% real_sqrt_gt_1_iff
thf(fact_7967_real__sqrt__ge__1__iff,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y2 ) )
= ( ord_less_eq_real @ one_one_real @ Y2 ) ) ).
% real_sqrt_ge_1_iff
thf(fact_7968_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% real_sqrt_le_1_iff
thf(fact_7969_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times_real @ X @ X ) )
= ( abs_abs_real @ X ) ) ).
% real_sqrt_abs2
thf(fact_7970_real__sqrt__mult__self,axiom,
! [A: real] :
( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
= ( abs_abs_real @ A ) ) ).
% real_sqrt_mult_self
thf(fact_7971_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% real_sqrt_four
thf(fact_7972_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% exp_le_one_iff
thf(fact_7973_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% one_le_exp_iff
thf(fact_7974_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( abs_abs_real @ X ) ) ).
% real_sqrt_abs
thf(fact_7975_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_7976_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_7977_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y2: real,Xa2: real,Ya: real] :
( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7978_real__sqrt__divide,axiom,
! [X: real,Y2: real] :
( ( sqrt @ ( divide_divide_real @ X @ Y2 ) )
= ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_divide
thf(fact_7979_real__sqrt__mult,axiom,
! [X: real,Y2: real] :
( ( sqrt @ ( times_times_real @ X @ Y2 ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_mult
thf(fact_7980_real__sqrt__power,axiom,
! [X: real,K: nat] :
( ( sqrt @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% real_sqrt_power
thf(fact_7981_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_7982_norm__exp,axiom,
! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% norm_exp
thf(fact_7983_norm__exp,axiom,
! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% norm_exp
thf(fact_7984_real__sqrt__le__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_le_mono
thf(fact_7985_real__sqrt__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_less_mono
thf(fact_7986_exp__times__arg__commute,axiom,
! [A2: complex] :
( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
= ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% exp_times_arg_commute
thf(fact_7987_exp__times__arg__commute,axiom,
! [A2: real] :
( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
= ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% exp_times_arg_commute
thf(fact_7988_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_7989_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sqrt @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_7990_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_7991_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_le_zero
thf(fact_7992_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_ge_zero
thf(fact_7993_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_7994_exp__add__commuting,axiom,
! [X: complex,Y2: complex] :
( ( ( times_times_complex @ X @ Y2 )
= ( times_times_complex @ Y2 @ X ) )
=> ( ( exp_complex @ ( plus_plus_complex @ X @ Y2 ) )
= ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y2 ) ) ) ) ).
% exp_add_commuting
thf(fact_7995_exp__add__commuting,axiom,
! [X: real,Y2: real] :
( ( ( times_times_real @ X @ Y2 )
= ( times_times_real @ Y2 @ X ) )
=> ( ( exp_real @ ( plus_plus_real @ X @ Y2 ) )
= ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y2 ) ) ) ) ).
% exp_add_commuting
thf(fact_7996_mult__exp__exp,axiom,
! [X: complex,Y2: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y2 ) )
= ( exp_complex @ ( plus_plus_complex @ X @ Y2 ) ) ) ).
% mult_exp_exp
thf(fact_7997_mult__exp__exp,axiom,
! [X: real,Y2: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y2 ) )
= ( exp_real @ ( plus_plus_real @ X @ Y2 ) ) ) ).
% mult_exp_exp
thf(fact_7998_exp__diff,axiom,
! [X: complex,Y2: complex] :
( ( exp_complex @ ( minus_minus_complex @ X @ Y2 ) )
= ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y2 ) ) ) ).
% exp_diff
thf(fact_7999_exp__diff,axiom,
! [X: real,Y2: real] :
( ( exp_real @ ( minus_minus_real @ X @ Y2 ) )
= ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y2 ) ) ) ).
% exp_diff
thf(fact_8000_Complex__eq__numeral,axiom,
! [A: real,B: real,W: num] :
( ( ( complex2 @ A @ B )
= ( numera6690914467698888265omplex @ W ) )
= ( ( A
= ( numeral_numeral_real @ W ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_numeral
thf(fact_8001_complex__add,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
= ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% complex_add
thf(fact_8002_complex__norm,axiom,
! [X: real,Y2: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y2 ) )
= ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_norm
thf(fact_8003_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( divide_divide_real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_8004_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y2 ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_8005_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_8006_le__real__sqrt__sumsq,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_8007_exp__minus__inverse,axiom,
! [X: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
= one_one_real ) ).
% exp_minus_inverse
thf(fact_8008_exp__minus__inverse,axiom,
! [X: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
= one_one_complex ) ).
% exp_minus_inverse
thf(fact_8009_exp__of__nat__mult,axiom,
! [N: nat,X: complex] :
( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_8010_exp__of__nat__mult,axiom,
! [N: nat,X: real] :
( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_8011_exp__of__nat2__mult,axiom,
! [X: complex,N: nat] :
( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_8012_exp__of__nat2__mult,axiom,
! [X: real,N: nat] :
( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_8013_Complex__eq__neg__numeral,axiom,
! [A: real,B: real,W: num] :
( ( ( complex2 @ A @ B )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( ( A
= ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_8014_complex__mult,axiom,
! [A: real,B: real,C: real,D: real] :
( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
= ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% complex_mult
thf(fact_8015_sqrt2__less__2,axiom,
ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% sqrt2_less_2
thf(fact_8016_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_8017_lemma__exp__total,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ one_one_real @ Y2 )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y2 @ one_one_real ) )
& ( ( exp_real @ X3 )
= Y2 ) ) ) ).
% lemma_exp_total
thf(fact_8018_ln__ge__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ ( exp_real @ Y2 ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_8019_ln__x__over__x__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y2 ) @ Y2 ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_8020_real__less__rsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ord_less_real @ X @ ( sqrt @ Y2 ) ) ) ).
% real_less_rsqrt
thf(fact_8021_real__le__rsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ord_less_eq_real @ X @ ( sqrt @ Y2 ) ) ) ).
% real_le_rsqrt
thf(fact_8022_sqrt__le__D,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ Y2 )
=> ( ord_less_eq_real @ X @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sqrt_le_D
thf(fact_8023_exp__le,axiom,
ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% exp_le
thf(fact_8024_tan__60,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% tan_60
thf(fact_8025_exp__divide__power__eq,axiom,
! [N: nat,X: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
= ( exp_complex @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_8026_exp__divide__power__eq,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
= ( exp_real @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_8027_tanh__altdef,axiom,
( tanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_8028_tanh__altdef,axiom,
( tanh_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_8029_real__sqrt__unique,axiom,
! [Y2: real,X: real] :
( ( ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( sqrt @ X )
= Y2 ) ) ) ).
% real_sqrt_unique
thf(fact_8030_real__le__lsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ Y2 ) ) ) ) ).
% real_le_lsqrt
thf(fact_8031_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less_real @ zero_zero_real @ U )
=> ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_8032_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y2: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= Y2 )
=> ( X = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_8033_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y2: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= X )
=> ( Y2 = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_8034_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_8035_real__sqrt__sum__squares__ge2,axiom,
! [Y2: real,X: real] : ( ord_less_eq_real @ Y2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_8036_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_8037_exp__half__le2,axiom,
ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% exp_half_le2
thf(fact_8038_sqrt__ge__absD,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y2 ) )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) ).
% sqrt_ge_absD
thf(fact_8039_cos__45,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_45
thf(fact_8040_sin__45,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_45
thf(fact_8041_exp__double,axiom,
! [Z: complex] :
( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
= ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% exp_double
thf(fact_8042_exp__double,axiom,
! [Z: real] :
( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
= ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% exp_double
thf(fact_8043_real__less__lsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sqrt @ X ) @ Y2 ) ) ) ) ).
% real_less_lsqrt
thf(fact_8044_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y2 ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_8045_tan__30,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% tan_30
thf(fact_8046_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_8047_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y2 ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_8048_real__sqrt__ge__abs2,axiom,
! [Y2: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_8049_real__sqrt__ge__abs1,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_8050_ln__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( sqrt @ X ) )
= ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% ln_sqrt
thf(fact_8051_cos__30,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_30
thf(fact_8052_sin__60,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_60
thf(fact_8053_exp__bound__half,axiom,
! [Z: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_8054_exp__bound__half,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_8055_arsinh__real__aux,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% arsinh_real_aux
thf(fact_8056_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_8057_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( sqrt @ X ) @ N )
= ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_8058_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y2: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8059_arith__geo__mean__sqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_8060_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_8061_cos__x__y__le__one,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% cos_x_y_le_one
thf(fact_8062_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y2: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_8063_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_8064_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_8065_cos__arctan,axiom,
! [X: real] :
( ( cos_real @ ( arctan @ X ) )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_8066_sin__arctan,axiom,
! [X: real] :
( ( sin_real @ ( arctan @ X ) )
= ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_8067_exp__bound__lemma,axiom,
! [Z: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_8068_exp__bound__lemma,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_8069_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_8070_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y2: real] :
( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_8071_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_8072_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
=> ( ( sin_real @ X )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_8073_arctan__half,axiom,
( arctan
= ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_8074_tanh__real__altdef,axiom,
( tanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_8075_cos__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( cos_real @ X )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_8076_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( arcosh_real @ X )
= ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_8077_arsinh__real__def,axiom,
( arsinh_real
= ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_8078_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_8079_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_8080_sin__arccos__abs,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( sin_real @ ( arccos @ Y2 ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_8081_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_8082_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_8083_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_8084_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_8085_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_8086_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_8087_cos__arccos,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y2 ) )
= Y2 ) ) ) ).
% cos_arccos
thf(fact_8088_sin__arcsin,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( sin_real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ).
% sin_arcsin
thf(fact_8089_arccos__0,axiom,
( ( arccos @ zero_zero_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arccos_0
thf(fact_8090_arcsin__1,axiom,
( ( arcsin @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arcsin_1
thf(fact_8091_arcsin__minus__1,axiom,
( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arcsin_minus_1
thf(fact_8092_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_8093_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_8094_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_8095_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_8096_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_8097_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_8098_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_8099_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] :
( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% Suc_times_binomial_add
thf(fact_8100_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_8101_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_8102_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_8103_arccos__le__arccos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y2 ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_8104_arccos__le__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_8105_arccos__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
& ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y2 ) )
= ( X = Y2 ) ) ) ).
% arccos_eq_iff
thf(fact_8106_arcsin__le__arcsin,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_8107_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( arcsin @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_8108_arcsin__le__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% arcsin_le_mono
thf(fact_8109_arcsin__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% arcsin_eq_iff
thf(fact_8110_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_8111_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_8112_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_8113_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_8114_binomial__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_mono
thf(fact_8115_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_8116_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% binomial_maximum
thf(fact_8117_binomial__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_8118_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_8119_arccos__lbound,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) ) ) ) ).
% arccos_lbound
thf(fact_8120_arccos__less__arccos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_real @ ( arccos @ Y2 ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_8121_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_8122_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_8123_arccos__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_8124_arccos__ubound,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_8125_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( arccos @ ( cos_real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_8126_arcsin__less__arcsin,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_8127_arcsin__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ).
% arcsin_less_mono
thf(fact_8128_cos__arccos__abs,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y2 ) )
= Y2 ) ) ).
% cos_arccos_abs
thf(fact_8129_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
=> ( ( arccos @ ( cos_real @ Theta ) )
= ( abs_abs_real @ Theta ) ) ) ).
% arccos_cos_eq_abs
thf(fact_8130_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_8131_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_8132_binomial__strict__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_strict_mono
thf(fact_8133_binomial__strict__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_8134_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_8135_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_8136_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8137_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8138_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8139_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_8140_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_8141_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_8142_arccos__bounded,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) )
& ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_8143_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( arccos @ ( cos_real @ X ) )
= ( uminus_uminus_real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_8144_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( arccos @ ( uminus_uminus_real @ X ) )
= ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_8145_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_8146_arccos,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) )
& ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi )
& ( ( cos_real @ ( arccos @ Y2 ) )
= Y2 ) ) ) ) ).
% arccos
thf(fact_8147_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arccos @ ( uminus_uminus_real @ X ) )
= ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_8148_arccos__le__pi2,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_8149_arcsin__lt__bounded,axiom,
! [Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_real @ Y2 @ one_one_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_8150_arcsin__lbound,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) ) ) ) ).
% arcsin_lbound
thf(fact_8151_arcsin__ubound,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_8152_arcsin__bounded,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_8153_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( arcsin @ ( sin_real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_8154_arcsin,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ) ).
% arcsin
thf(fact_8155_arcsin__pi,axiom,
! [Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq_real @ ( arcsin @ Y2 ) @ pi )
& ( ( sin_real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ) ).
% arcsin_pi
thf(fact_8156_arcsin__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y2 )
= ( ord_less_eq_real @ X @ ( sin_real @ Y2 ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_8157_le__arcsin__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ Y2 @ ( arcsin @ X ) )
= ( ord_less_eq_real @ ( sin_real @ Y2 ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_8158_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos_real @ Theta ) )
!= ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_8159_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_8160_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( sin_real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_8161_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8162_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) @ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8163_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) @ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8164_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8165_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] :
( if_complex
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8166_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] :
( if_int
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) )
@ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8167_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] :
( if_rat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) )
@ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8168_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] :
( if_real
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8169_pochhammer__double,axiom,
! [Z: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_8170_pochhammer__double,axiom,
! [Z: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_8171_pochhammer__double,axiom,
! [Z: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_8172_of__nat__code,axiom,
( semiri8010041392384452111omplex
= ( ^ [N2: nat] :
( semiri2816024913162550771omplex
@ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
@ N2
@ zero_zero_complex ) ) ) ).
% of_nat_code
thf(fact_8173_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_8174_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( semiri7260567687927622513x_real
@ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
@ N2
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_8175_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_8176_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
@ N2
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_8177_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y2: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y2 )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y2
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y2
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu3: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ( ~ Y2
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y2
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ~ Y2
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_8178_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8179_atMost__iff,axiom,
! [I2: set_int,K: set_int] :
( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K ) )
= ( ord_less_eq_set_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8180_atMost__iff,axiom,
! [I2: rat,K: rat] :
( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K ) )
= ( ord_less_eq_rat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8181_atMost__iff,axiom,
! [I2: num,K: num] :
( ( member_num @ I2 @ ( set_ord_atMost_num @ K ) )
= ( ord_less_eq_num @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8182_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8183_atMost__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8184_atMost__subset__iff,axiom,
! [X: set_int,Y2: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y2 ) )
= ( ord_less_eq_set_int @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_8185_atMost__subset__iff,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y2 ) )
= ( ord_less_eq_rat @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_8186_atMost__subset__iff,axiom,
! [X: num,Y2: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y2 ) )
= ( ord_less_eq_num @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_8187_atMost__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_8188_atMost__subset__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_8189_Icc__subset__Iic__iff,axiom,
! [L2: set_int,H2: set_int,H3: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L2 @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
= ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
| ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8190_Icc__subset__Iic__iff,axiom,
! [L2: rat,H2: rat,H3: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L2 @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
= ( ~ ( ord_less_eq_rat @ L2 @ H2 )
| ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8191_Icc__subset__Iic__iff,axiom,
! [L2: num,H2: num,H3: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L2 @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
= ( ~ ( ord_less_eq_num @ L2 @ H2 )
| ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8192_Icc__subset__Iic__iff,axiom,
! [L2: nat,H2: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L2 @ H2 )
| ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8193_Icc__subset__Iic__iff,axiom,
! [L2: int,H2: int,H3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L2 @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
= ( ~ ( ord_less_eq_int @ L2 @ H2 )
| ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8194_Icc__subset__Iic__iff,axiom,
! [L2: real,H2: real,H3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L2 @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
= ( ~ ( ord_less_eq_real @ L2 @ H2 )
| ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8195_sum_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8196_sum_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8197_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8198_sum_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8199_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X2: real] : ( ord_less_eq_real @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8200_atMost__def,axiom,
( set_or58775011639299419et_int
= ( ^ [U2: set_int] :
( collect_set_int
@ ^ [X2: set_int] : ( ord_less_eq_set_int @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8201_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X2: rat] : ( ord_less_eq_rat @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8202_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X2: num] : ( ord_less_eq_num @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8203_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8204_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8205_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_8206_pochhammer__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_8207_pochhammer__pos,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_8208_pochhammer__pos,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_8209_pochhammer__pos,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_8210_pochhammer__eq__0__mono,axiom,
! [A: complex,N: nat,M: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s2602460028002588243omplex @ A @ M )
= zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_8211_pochhammer__eq__0__mono,axiom,
! [A: real,N: nat,M: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ M )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_8212_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ M )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_8213_pochhammer__neq__0__mono,axiom,
! [A: complex,M: nat,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ M )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s2602460028002588243omplex @ A @ N )
!= zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_8214_pochhammer__neq__0__mono,axiom,
! [A: real,M: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ M )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_8215_pochhammer__neq__0__mono,axiom,
! [A: rat,M: nat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ M )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ N )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_8216_not__Iic__le__Icc,axiom,
! [H2: int,L3: int,H3: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% not_Iic_le_Icc
thf(fact_8217_not__Iic__le__Icc,axiom,
! [H2: real,L3: real,H3: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% not_Iic_le_Icc
thf(fact_8218_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S4: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_8219_pochhammer__nonneg,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_8220_pochhammer__nonneg,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_8221_pochhammer__nonneg,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_8222_pochhammer__nonneg,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_8223_pochhammer__binomial__sum,axiom,
! [A: int,B: int,N: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8224_pochhammer__binomial__sum,axiom,
! [A: rat,B: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8225_pochhammer__binomial__sum,axiom,
! [A: real,B: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8226_Iic__subset__Iio__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8227_Iic__subset__Iio__iff,axiom,
! [A: num,B: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
= ( ord_less_num @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8228_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8229_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8230_Iic__subset__Iio__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8231_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_8232_pochhammer__rec,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_8233_pochhammer__rec,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_8234_pochhammer__rec,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_8235_pochhammer__rec,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_8236_pochhammer__rec,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_8237_pochhammer__rec_H,axiom,
! [Z: int,N: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
= ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_8238_pochhammer__rec_H,axiom,
! [Z: real,N: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
= ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_8239_pochhammer__rec_H,axiom,
! [Z: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_8240_pochhammer__rec_H,axiom,
! [Z: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
= ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_8241_pochhammer__Suc,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_8242_pochhammer__Suc,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_8243_pochhammer__Suc,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_8244_pochhammer__Suc,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_8245_pochhammer__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_8246_pochhammer__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_8247_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_8248_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_8249_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_8250_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_8251_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_8252_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_8253_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_8254_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_8255_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_8256_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_8257_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_8258_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
!= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_8259_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
!= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_8260_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_8261_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
!= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_8262_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_8263_sum_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8264_sum_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8265_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8266_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8267_sum__telescope,axiom,
! [F: nat > rat,I2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_8268_sum__telescope,axiom,
! [F: nat > int,I2: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_8269_sum__telescope,axiom,
! [F: nat > real,I2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_8270_pochhammer__product_H,axiom,
! [Z: int,N: nat,M: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_8271_pochhammer__product_H,axiom,
! [Z: real,N: nat,M: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_8272_pochhammer__product_H,axiom,
! [Z: nat,N: nat,M: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_8273_pochhammer__product_H,axiom,
! [Z: rat,N: nat,M: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_8274_polyfun__eq__coeffs,axiom,
! [C: nat > complex,N: nat,D: nat > complex] :
( ( ! [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8275_polyfun__eq__coeffs,axiom,
! [C: nat > real,N: nat,D: nat > real] :
( ( ! [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8276_bounded__imp__summable,axiom,
! [A: nat > int,B4: int] :
( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B4 )
=> ( summable_int @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8277_bounded__imp__summable,axiom,
! [A: nat > nat,B4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B4 )
=> ( summable_nat @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8278_bounded__imp__summable,axiom,
! [A: nat > real,B4: real] :
( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B4 )
=> ( summable_real @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8279_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8280_sum_Onested__swap_H,axiom,
! [A: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8281_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_8282_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_8283_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_8284_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > complex,N: nat,K: nat] :
( ! [W2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8285_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N: nat,K: nat] :
( ! [W2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8286_polyfun__eq__0,axiom,
! [C: nat > complex,N: nat] :
( ( ! [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_8287_polyfun__eq__0,axiom,
! [C: nat > real,N: nat] :
( ( ! [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_8288_sum_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8289_sum_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8290_sum_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8291_sum_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8292_sum__up__index__split,axiom,
! [F: nat > rat,M: nat,N: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8293_sum__up__index__split,axiom,
! [F: nat > int,M: nat,N: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8294_sum__up__index__split,axiom,
! [F: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8295_sum__up__index__split,axiom,
! [F: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8296_pochhammer__product,axiom,
! [M: nat,N: nat,Z: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4660882817536571857er_int @ Z @ N )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_8297_pochhammer__product,axiom,
! [M: nat,N: nat,Z: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s7457072308508201937r_real @ Z @ N )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_8298_pochhammer__product,axiom,
! [M: nat,N: nat,Z: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4663373288045622133er_nat @ Z @ N )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_8299_pochhammer__product,axiom,
! [M: nat,N: nat,Z: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4028243227959126397er_rat @ Z @ N )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_8300_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_8301_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_8302_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_8303_vandermonde,axiom,
! [M: nat,N: nat,R2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R2 ) )
= ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_8304_sum__gp__basic,axiom,
! [X: complex,N: nat] :
( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8305_sum__gp__basic,axiom,
! [X: rat,N: nat] :
( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8306_sum__gp__basic,axiom,
! [X: int,N: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8307_sum__gp__basic,axiom,
! [X: real,N: nat] :
( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8308_polyfun__roots__finite,axiom,
! [C: nat > complex,K: nat,N: nat] :
( ( ( C @ K )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_8309_polyfun__roots__finite,axiom,
! [C: nat > real,K: nat,N: nat] :
( ( ( C @ K )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_8310_polyfun__finite__roots,axiom,
! [C: nat > complex,N: nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( C @ I3 )
!= zero_zero_complex ) ) ) ) ).
% polyfun_finite_roots
thf(fact_8311_polyfun__finite__roots,axiom,
! [C: nat > real,N: nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( C @ I3 )
!= zero_zero_real ) ) ) ) ).
% polyfun_finite_roots
thf(fact_8312_polyfun__linear__factor__root,axiom,
! [C: nat > complex,A: complex,N: nat] :
( ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ~ ! [B2: nat > complex] :
~ ! [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B2 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8313_polyfun__linear__factor__root,axiom,
! [C: nat > rat,A: rat,N: nat] :
( ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat )
=> ~ ! [B2: nat > rat] :
~ ! [Z4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B2 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8314_polyfun__linear__factor__root,axiom,
! [C: nat > int,A: int,N: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int )
=> ~ ! [B2: nat > int] :
~ ! [Z4: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( minus_minus_int @ Z4 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B2 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8315_polyfun__linear__factor__root,axiom,
! [C: nat > real,A: real,N: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ~ ! [B2: nat > real] :
~ ! [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( minus_minus_real @ Z4 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B2 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8316_polyfun__linear__factor,axiom,
! [C: nat > complex,N: nat,A: complex] :
? [B2: nat > complex] :
! [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_complex
@ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B2 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8317_polyfun__linear__factor,axiom,
! [C: nat > rat,N: nat,A: rat] :
? [B2: nat > rat] :
! [Z4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat
@ ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B2 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8318_polyfun__linear__factor,axiom,
! [C: nat > int,N: nat,A: int] :
? [B2: nat > int] :
! [Z4: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int
@ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B2 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8319_polyfun__linear__factor,axiom,
! [C: nat > real,N: nat,A: real] :
? [B2: nat > real] :
! [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real
@ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B2 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8320_sum__power__shift,axiom,
! [M: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8321_sum__power__shift,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8322_sum__power__shift,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8323_sum__power__shift,axiom,
! [M: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8324_pochhammer__absorb__comp,axiom,
! [R2: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
= ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_8325_pochhammer__absorb__comp,axiom,
! [R2: code_integer,K: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
= ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_8326_pochhammer__absorb__comp,axiom,
! [R2: int,K: nat] :
( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
= ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_8327_pochhammer__absorb__comp,axiom,
! [R2: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
= ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_8328_pochhammer__absorb__comp,axiom,
! [R2: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
= ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_8329_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% pochhammer_same
thf(fact_8330_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% pochhammer_same
thf(fact_8331_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% pochhammer_same
thf(fact_8332_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% pochhammer_same
thf(fact_8333_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% pochhammer_same
thf(fact_8334_sum_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_8335_sum_Otriangle__reindex,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_8336_summable__Cauchy__product,axiom,
! [A: nat > complex,B: nat > complex] :
( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
=> ( summable_complex
@ ^ [K3: nat] :
( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_8337_summable__Cauchy__product,axiom,
! [A: nat > real,B: nat > real] :
( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
=> ( summable_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_8338_choose__row__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% choose_row_sum
thf(fact_8339_Cauchy__product,axiom,
! [A: nat > complex,B: nat > complex] :
( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
=> ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) )
= ( suminf_complex
@ ^ [K3: nat] :
( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_8340_Cauchy__product,axiom,
! [A: nat > real,B: nat > real] :
( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
=> ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) )
= ( suminf_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_8341_binomial,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial
thf(fact_8342_sum_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8343_sum_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8344_sum_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8345_sum_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8346_polynomial__product,axiom,
! [M: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
( ! [I4: nat] :
( ( ord_less_nat @ M @ I4 )
=> ( ( A @ I4 )
= zero_zero_complex ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_complex ) )
=> ( ( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [R5: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_complex @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8347_polynomial__product,axiom,
! [M: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
( ! [I4: nat] :
( ( ord_less_nat @ M @ I4 )
=> ( ( A @ I4 )
= zero_zero_rat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_rat ) )
=> ( ( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2906978787729119204at_rat
@ ^ [R5: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_rat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8348_polynomial__product,axiom,
! [M: nat,A: nat > int,N: nat,B: nat > int,X: int] :
( ! [I4: nat] :
( ( ord_less_nat @ M @ I4 )
=> ( ( A @ I4 )
= zero_zero_int ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_int ) )
=> ( ( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3539618377306564664at_int
@ ^ [R5: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_int @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8349_polynomial__product,axiom,
! [M: nat,A: nat > real,N: nat,B: nat > real,X: real] :
( ! [I4: nat] :
( ( ord_less_nat @ M @ I4 )
=> ( ( A @ I4 )
= zero_zero_real ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_real ) )
=> ( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [R5: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_real @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8350_polyfun__eq__const,axiom,
! [C: nat > complex,N: nat,K: complex] :
( ( ! [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C @ X2 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_8351_polyfun__eq__const,axiom,
! [C: nat > real,N: nat,K: real] :
( ( ! [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C @ X2 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_8352_binomial__ring,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8353_binomial__ring,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8354_binomial__ring,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8355_binomial__ring,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8356_binomial__ring,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8357_pochhammer__minus_H,axiom,
! [B: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_8358_pochhammer__minus_H,axiom,
! [B: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_8359_pochhammer__minus_H,axiom,
! [B: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_8360_pochhammer__minus_H,axiom,
! [B: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_8361_pochhammer__minus_H,axiom,
! [B: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_8362_pochhammer__minus,axiom,
! [B: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_8363_pochhammer__minus,axiom,
! [B: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_8364_pochhammer__minus,axiom,
! [B: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_8365_pochhammer__minus,axiom,
! [B: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_8366_pochhammer__minus,axiom,
! [B: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_8367_polynomial__product__nat,axiom,
! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
( ! [I4: nat] :
( ( ord_less_nat @ M @ I4 )
=> ( ( A @ I4 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R5: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_nat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_8368_choose__square__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_8369_Cauchy__product__sums,axiom,
! [A: nat > complex,B: nat > complex] :
( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
=> ( sums_complex
@ ^ [K3: nat] :
( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_8370_Cauchy__product__sums,axiom,
! [A: nat > real,B: nat > real] :
( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
=> ( sums_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_8371_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8372_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8373_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8374_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8375_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8376_root__polyfun,axiom,
! [N: nat,Z: int,A: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_int @ Z @ N )
= A )
= ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ) ).
% root_polyfun
thf(fact_8377_root__polyfun,axiom,
! [N: nat,Z: complex,A: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_complex @ Z @ N )
= A )
= ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ).
% root_polyfun
thf(fact_8378_root__polyfun,axiom,
! [N: nat,Z: code_integer,A: code_integer] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_8256067586552552935nteger @ Z @ N )
= A )
= ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I3 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ) ).
% root_polyfun
thf(fact_8379_root__polyfun,axiom,
! [N: nat,Z: rat,A: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_rat @ Z @ N )
= A )
= ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I3 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ) ).
% root_polyfun
thf(fact_8380_root__polyfun,axiom,
! [N: nat,Z: real,A: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_real @ Z @ N )
= A )
= ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ).
% root_polyfun
thf(fact_8381_sum__gp0,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8382_sum__gp0,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8383_sum__gp0,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8384_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_linear_sum
thf(fact_8385_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_linear_sum
thf(fact_8386_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_linear_sum
thf(fact_8387_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_linear_sum
thf(fact_8388_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_linear_sum
thf(fact_8389_polyfun__diff__alt,axiom,
! [N: nat,A: nat > complex,X: complex,Y2: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y2 )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y2 @ K3 ) ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8390_polyfun__diff__alt,axiom,
! [N: nat,A: nat > rat,X: rat,Y2: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y2 )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y2 @ K3 ) ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8391_polyfun__diff__alt,axiom,
! [N: nat,A: nat > int,X: int,Y2: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y2 )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y2 @ K3 ) ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8392_polyfun__diff__alt,axiom,
! [N: nat,A: nat > real,X: real,Y2: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y2 )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y2 @ K3 ) ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8393_pochhammer__code,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A4: complex,N2: nat] :
( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
@ ( set_fo1517530859248394432omplex
@ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_complex ) ) ) ) ).
% pochhammer_code
thf(fact_8394_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A4: int,N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_8395_pochhammer__code,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A4: real,N2: nat] :
( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
@ ( set_fo3111899725591712190t_real
@ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_real ) ) ) ) ).
% pochhammer_code
thf(fact_8396_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A4: rat,N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_8397_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A4: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_8398_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_8399_choose__linear__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% choose_linear_sum
thf(fact_8400_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu3: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_8401_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_8402_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_sum
thf(fact_8403_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_sum
thf(fact_8404_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_sum
thf(fact_8405_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_sum
thf(fact_8406_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_sum
thf(fact_8407_polyfun__extremal__lemma,axiom,
! [E: real,C: nat > complex,N: nat] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: real] :
! [Z4: complex] :
( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z4 ) )
=> ( ord_less_eq_real
@ ( real_V1022390504157884413omplex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
@ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8408_polyfun__extremal__lemma,axiom,
! [E: real,C: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: real] :
! [Z4: real] :
( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z4 ) )
=> ( ord_less_eq_real
@ ( real_V7735802525324610683m_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
@ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8409_polyfun__diff,axiom,
! [N: nat,A: nat > complex,X: complex,Y2: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y2 )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8410_polyfun__diff,axiom,
! [N: nat,A: nat > rat,X: rat,Y2: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y2 )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8411_polyfun__diff,axiom,
! [N: nat,A: nat > int,X: int,Y2: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y2 )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8412_polyfun__diff,axiom,
! [N: nat,A: nat > real,X: real,Y2: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y2 )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8413_fact__double,axiom,
! [N: nat] :
( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% fact_double
thf(fact_8414_fact__double,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_double
thf(fact_8415_fact__double,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_double
thf(fact_8416_sin__x__sin__y,axiom,
! [X: real,Y2: real] :
( sums_real
@ ^ [P5: nat] :
( groups6591440286371151544t_real
@ ^ [N2: nat] :
( if_real
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ P5 @ N2 ) ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ P5 ) )
@ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y2 ) ) ) ).
% sin_x_sin_y
thf(fact_8417_sin__x__sin__y,axiom,
! [X: complex,Y2: complex] :
( sums_complex
@ ^ [P5: nat] :
( groups2073611262835488442omplex
@ ^ [N2: nat] :
( if_complex
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ P5 @ N2 ) ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ P5 ) )
@ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y2 ) ) ) ).
% sin_x_sin_y
thf(fact_8418_sums__cos__x__plus__y,axiom,
! [X: real,Y2: real] :
( sums_real
@ ^ [P5: nat] :
( groups6591440286371151544t_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ P5 ) )
@ ( cos_real @ ( plus_plus_real @ X @ Y2 ) ) ) ).
% sums_cos_x_plus_y
thf(fact_8419_sums__cos__x__plus__y,axiom,
! [X: complex,Y2: complex] :
( sums_complex
@ ^ [P5: nat] :
( groups2073611262835488442omplex
@ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ P5 ) )
@ ( cos_complex @ ( plus_plus_complex @ X @ Y2 ) ) ) ).
% sums_cos_x_plus_y
thf(fact_8420_cos__x__cos__y,axiom,
! [X: real,Y2: real] :
( sums_real
@ ^ [P5: nat] :
( groups6591440286371151544t_real
@ ^ [N2: nat] :
( if_real
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ P5 @ N2 ) ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ P5 ) )
@ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y2 ) ) ) ).
% cos_x_cos_y
thf(fact_8421_cos__x__cos__y,axiom,
! [X: complex,Y2: complex] :
( sums_complex
@ ^ [P5: nat] :
( groups2073611262835488442omplex
@ ^ [N2: nat] :
( if_complex
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ P5 @ N2 ) ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ P5 ) )
@ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y2 ) ) ) ).
% cos_x_cos_y
thf(fact_8422_pochhammer__times__pochhammer__half,axiom,
! [Z: complex,N: nat] :
( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8423_pochhammer__times__pochhammer__half,axiom,
! [Z: real,N: nat] :
( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups129246275422532515t_real
@ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8424_pochhammer__times__pochhammer__half,axiom,
! [Z: rat,N: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8425_gbinomial__partial__row__sum,axiom,
! [A: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8426_gbinomial__partial__row__sum,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8427_gbinomial__partial__row__sum,axiom,
! [A: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8428_of__nat__id,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] : N2 ) ) ).
% of_nat_id
thf(fact_8429_mult__scaleR__right,axiom,
! [X: real,A: real,Y2: real] :
( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y2 ) )
= ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y2 ) ) ) ).
% mult_scaleR_right
thf(fact_8430_mult__scaleR__right,axiom,
! [X: complex,A: real,Y2: complex] :
( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y2 ) )
= ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y2 ) ) ) ).
% mult_scaleR_right
thf(fact_8431_mult__scaleR__left,axiom,
! [A: real,X: real,Y2: real] :
( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y2 )
= ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y2 ) ) ) ).
% mult_scaleR_left
thf(fact_8432_mult__scaleR__left,axiom,
! [A: real,X: complex,Y2: complex] :
( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y2 )
= ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y2 ) ) ) ).
% mult_scaleR_left
thf(fact_8433_scaleR__scaleR,axiom,
! [A: real,B: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% scaleR_scaleR
thf(fact_8434_scaleR__scaleR,axiom,
! [A: real,B: real,X: complex] :
( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
= ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).
% scaleR_scaleR
thf(fact_8435_scaleR__eq__iff,axiom,
! [B: real,U: real,A: real] :
( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
= ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
= ( ( A = B )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_8436_scaleR__eq__iff,axiom,
! [B: complex,U: real,A: complex] :
( ( ( plus_plus_complex @ B @ ( real_V2046097035970521341omplex @ U @ A ) )
= ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B ) ) )
= ( ( A = B )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_8437_scaleR__power,axiom,
! [X: real,Y2: real,N: nat] :
( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y2 ) @ N )
= ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) ) ) ).
% scaleR_power
thf(fact_8438_scaleR__power,axiom,
! [X: real,Y2: complex,N: nat] :
( ( power_power_complex @ ( real_V2046097035970521341omplex @ X @ Y2 ) @ N )
= ( real_V2046097035970521341omplex @ ( power_power_real @ X @ N ) @ ( power_power_complex @ Y2 @ N ) ) ) ).
% scaleR_power
thf(fact_8439_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
= zero_zero_complex ) ).
% gbinomial_0(2)
thf(fact_8440_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
= zero_zero_real ) ).
% gbinomial_0(2)
thf(fact_8441_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
= zero_zero_rat ) ).
% gbinomial_0(2)
thf(fact_8442_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_8443_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_8444_scaleR__collapse,axiom,
! [U: real,A: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_8445_scaleR__collapse,axiom,
! [U: real,A: complex] :
( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_8446_prod_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_8447_prod_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_8448_prod_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_8449_prod_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_8450_prod_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8451_prod_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8452_prod_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8453_prod_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8454_norm__scaleR,axiom,
! [A: real,X: real] :
( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% norm_scaleR
thf(fact_8455_norm__scaleR,axiom,
! [A: real,X: complex] :
( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% norm_scaleR
thf(fact_8456_scaleR__times,axiom,
! [U: num,W: num,A: real] :
( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% scaleR_times
thf(fact_8457_scaleR__times,axiom,
! [U: num,W: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% scaleR_times
thf(fact_8458_inverse__scaleR__times,axiom,
! [V: num,W: num,A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% inverse_scaleR_times
thf(fact_8459_inverse__scaleR__times,axiom,
! [V: num,W: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% inverse_scaleR_times
thf(fact_8460_fraction__scaleR__times,axiom,
! [U: num,V: num,W: num,A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% fraction_scaleR_times
thf(fact_8461_fraction__scaleR__times,axiom,
! [U: num,V: num,W: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% fraction_scaleR_times
thf(fact_8462_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8463_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8464_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8465_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8466_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8467_scaleR__half__double,axiom,
! [A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
= A ) ).
% scaleR_half_double
thf(fact_8468_scaleR__half__double,axiom,
! [A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
= A ) ).
% scaleR_half_double
thf(fact_8469_scaleR__right__distrib,axiom,
! [A: real,X: real,Y2: real] :
( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y2 ) ) ) ).
% scaleR_right_distrib
thf(fact_8470_scaleR__right__distrib,axiom,
! [A: real,X: complex,Y2: complex] :
( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y2 ) )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y2 ) ) ) ).
% scaleR_right_distrib
thf(fact_8471_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_8472_prod_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).
% prod.distrib
thf(fact_8473_prod_Odistrib,axiom,
! [G: nat > int,H2: nat > int,A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).
% prod.distrib
thf(fact_8474_prod_Odistrib,axiom,
! [G: int > int,H2: int > int,A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).
% prod.distrib
thf(fact_8475_prod__power__distrib,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N )
= ( groups708209901874060359at_nat
@ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_8476_prod__power__distrib,axiom,
! [F: nat > int,A2: set_nat,N: nat] :
( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N )
= ( groups705719431365010083at_int
@ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_8477_prod__power__distrib,axiom,
! [F: int > int,A2: set_int,N: nat] :
( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N )
= ( groups1705073143266064639nt_int
@ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_8478_mod__prod__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_8479_mod__prod__eq,axiom,
! [F: nat > int,A: int,A2: set_nat] :
( ( modulo_modulo_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_8480_mod__prod__eq,axiom,
! [F: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups1705073143266064639nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_8481_prod__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8482_prod__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8483_prod__nonneg,axiom,
! [A2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8484_prod__mono,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8485_prod__mono,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8486_prod__mono,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ! [I4: int] :
( ( member_int @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8487_prod__mono,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ! [I4: complex] :
( ( member_complex @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8488_prod__mono,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8489_prod__mono,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8490_prod__mono,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ! [I4: int] :
( ( member_int @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8491_prod__mono,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ! [I4: complex] :
( ( member_complex @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8492_prod__mono,axiom,
! [A2: set_real,F: real > nat,G: real > nat] :
( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
& ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8493_prod__mono,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ! [I4: int] :
( ( member_int @ I4 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
& ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8494_prod__pos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8495_prod__pos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8496_prod__pos,axiom,
! [A2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8497_prod__ge__1,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8498_prod__ge__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8499_prod__ge__1,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8500_prod__ge__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8501_prod__ge__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8502_prod__ge__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8503_prod__ge__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8504_prod__ge__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8505_prod__ge__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8506_prod__ge__1,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8507_scaleR__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_8508_scaleR__left__distrib,axiom,
! [A: real,B: real,X: complex] :
( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_8509_scaleR__left_Oadd,axiom,
! [X: real,Y2: real,Xa2: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y2 ) @ Xa2 )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y2 @ Xa2 ) ) ) ).
% scaleR_left.add
thf(fact_8510_scaleR__left_Oadd,axiom,
! [X: real,Y2: real,Xa2: complex] :
( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X @ Y2 ) @ Xa2 )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X @ Xa2 ) @ ( real_V2046097035970521341omplex @ Y2 @ Xa2 ) ) ) ).
% scaleR_left.add
thf(fact_8511_complex__scaleR,axiom,
! [R2: real,A: real,B: real] :
( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
= ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).
% complex_scaleR
thf(fact_8512_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_8513_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_8514_power__sum,axiom,
! [C: real,F: nat > nat,A2: set_nat] :
( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [A4: nat] : ( power_power_real @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8515_power__sum,axiom,
! [C: complex,F: nat > nat,A2: set_nat] :
( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6464643781859351333omplex
@ ^ [A4: nat] : ( power_power_complex @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8516_power__sum,axiom,
! [C: nat,F: nat > nat,A2: set_nat] :
( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [A4: nat] : ( power_power_nat @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8517_power__sum,axiom,
! [C: int,F: nat > nat,A2: set_nat] :
( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [A4: nat] : ( power_power_int @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8518_power__sum,axiom,
! [C: int,F: int > nat,A2: set_int] :
( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [A4: int] : ( power_power_int @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8519_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_8520_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M: nat,K: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_8521_prod__le__1,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8522_prod__le__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8523_prod__le__1,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8524_prod__le__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8525_prod__le__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8526_prod__le__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8527_prod__le__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8528_prod__le__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8529_prod__le__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8530_prod__le__1,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8531_prod_Orelated,axiom,
! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups6464643781859351333omplex @ H2 @ S3 ) @ ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8532_prod_Orelated,axiom,
! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups7440179247065528705omplex @ H2 @ S3 ) @ ( groups7440179247065528705omplex @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8533_prod_Orelated,axiom,
! [R: complex > complex > $o,S3: set_complex,H2: complex > complex,G: complex > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3708469109370488835omplex @ H2 @ S3 ) @ ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8534_prod_Orelated,axiom,
! [R: real > real > $o,S3: set_nat,H2: nat > real,G: nat > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups129246275422532515t_real @ H2 @ S3 ) @ ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8535_prod_Orelated,axiom,
! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups2316167850115554303t_real @ H2 @ S3 ) @ ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8536_prod_Orelated,axiom,
! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups766887009212190081x_real @ H2 @ S3 ) @ ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8537_prod_Orelated,axiom,
! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups73079841787564623at_rat @ H2 @ S3 ) @ ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8538_prod_Orelated,axiom,
! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups1072433553688619179nt_rat @ H2 @ S3 ) @ ( groups1072433553688619179nt_rat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8539_prod_Orelated,axiom,
! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups225925009352817453ex_rat @ H2 @ S3 ) @ ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8540_prod_Orelated,axiom,
! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups1707563613775114915nt_nat @ H2 @ S3 ) @ ( groups1707563613775114915nt_nat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8541_prod__dvd__prod__subset,axiom,
! [B4: set_complex,A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8542_prod__dvd__prod__subset,axiom,
! [B4: set_complex,A2: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8543_prod__dvd__prod__subset,axiom,
! [B4: set_nat,A2: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8544_prod__dvd__prod__subset,axiom,
! [B4: set_complex,A2: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8545_prod__dvd__prod__subset,axiom,
! [B4: set_int,A2: set_int,F: int > nat] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8546_prod__dvd__prod__subset,axiom,
! [B4: set_int,A2: set_int,F: int > code_integer] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8547_prod__dvd__prod__subset,axiom,
! [B4: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8548_prod__dvd__prod__subset,axiom,
! [B4: set_nat,A2: set_nat,F: nat > int] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8549_prod__dvd__prod__subset,axiom,
! [B4: set_int,A2: set_int,F: int > int] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B4 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8550_prod__dvd__prod__subset2,axiom,
! [B4: set_real,A2: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8551_prod__dvd__prod__subset2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8552_prod__dvd__prod__subset2,axiom,
! [B4: set_real,A2: set_real,F: real > int,G: real > int] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8553_prod__dvd__prod__subset2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8554_prod__dvd__prod__subset2,axiom,
! [B4: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8555_prod__dvd__prod__subset2,axiom,
! [B4: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8556_prod__dvd__prod__subset2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8557_prod__dvd__prod__subset2,axiom,
! [B4: set_int,A2: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8558_prod__dvd__prod__subset2,axiom,
! [B4: set_int,A2: set_int,F: int > code_integer,G: int > code_integer] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8559_prod__dvd__prod__subset2,axiom,
! [B4: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8560_scaleR__right__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_8561_scaleR__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_8562_scaleR__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_8563_scaleR__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_8564_scaleR__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_8565_scaleR__left__mono,axiom,
! [X: real,Y2: real,A: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y2 ) ) ) ) ).
% scaleR_left_mono
thf(fact_8566_scaleR__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_8567_gbinomial__Suc__Suc,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
= ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8568_gbinomial__Suc__Suc,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
= ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8569_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8570_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_8571_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_8572_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_8573_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
= ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_8574_prod_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_8575_prod_Onat__diff__reindex,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_8576_prod_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_8577_prod_OatLeastAtMost__rev,axiom,
! [G: nat > int,N: nat,M: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_8578_gbinomial__Suc,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ A @ ( suc @ K ) )
= ( divide1717551699836669952omplex
@ ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8579_gbinomial__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8580_gbinomial__Suc,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ A @ ( suc @ K ) )
= ( divide_divide_real
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8581_gbinomial__Suc,axiom,
! [A: nat,K: nat] :
( ( gbinomial_nat @ A @ ( suc @ K ) )
= ( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8582_gbinomial__Suc,axiom,
! [A: int,K: nat] :
( ( gbinomial_int @ A @ ( suc @ K ) )
= ( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8583_less__1__prod2,axiom,
! [I5: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8584_less__1__prod2,axiom,
! [I5: set_nat,I2: nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8585_less__1__prod2,axiom,
! [I5: set_int,I2: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8586_less__1__prod2,axiom,
! [I5: set_complex,I2: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8587_less__1__prod2,axiom,
! [I5: set_real,I2: real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8588_less__1__prod2,axiom,
! [I5: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8589_less__1__prod2,axiom,
! [I5: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8590_less__1__prod2,axiom,
! [I5: set_complex,I2: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8591_less__1__prod2,axiom,
! [I5: set_real,I2: real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8592_less__1__prod2,axiom,
! [I5: set_complex,I2: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8593_less__1__prod,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8594_less__1__prod,axiom,
! [I5: set_nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8595_less__1__prod,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8596_less__1__prod,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8597_less__1__prod,axiom,
! [I5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8598_less__1__prod,axiom,
! [I5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8599_less__1__prod,axiom,
! [I5: set_int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8600_less__1__prod,axiom,
! [I5: set_real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8601_less__1__prod,axiom,
! [I5: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8602_less__1__prod,axiom,
! [I5: set_real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8603_prod_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups766887009212190081x_real @ G @ A2 )
= ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups766887009212190081x_real @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8604_prod_Osubset__diff,axiom,
! [B4: set_nat,A2: set_nat,G: nat > real] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups129246275422532515t_real @ G @ A2 )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ B4 ) ) @ ( groups129246275422532515t_real @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8605_prod_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups225925009352817453ex_rat @ G @ A2 )
= ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups225925009352817453ex_rat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8606_prod_Osubset__diff,axiom,
! [B4: set_nat,A2: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G @ A2 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8607_prod_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups861055069439313189ex_nat @ G @ A2 )
= ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups861055069439313189ex_nat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8608_prod_Osubset__diff,axiom,
! [B4: set_complex,A2: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B4 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups858564598930262913ex_int @ G @ A2 )
= ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) @ ( groups858564598930262913ex_int @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8609_prod_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups2316167850115554303t_real @ G @ A2 )
= ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups2316167850115554303t_real @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8610_prod_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G @ A2 )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups1072433553688619179nt_rat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8611_prod_Osubset__diff,axiom,
! [B4: set_int,A2: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G @ A2 )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ B4 ) ) @ ( groups1707563613775114915nt_nat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8612_prod_Osubset__diff,axiom,
! [B4: set_nat,A2: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G @ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8613_prod_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ A2 )
= ( groups713298508707869441omplex @ H2 @ B4 ) )
= ( ( groups713298508707869441omplex @ G @ C4 )
= ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8614_prod_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ A2 )
= ( groups3708469109370488835omplex @ H2 @ B4 ) )
= ( ( groups3708469109370488835omplex @ G @ C4 )
= ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8615_prod_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ A2 )
= ( groups1681761925125756287l_real @ H2 @ B4 ) )
= ( ( groups1681761925125756287l_real @ G @ C4 )
= ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8616_prod_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ A2 )
= ( groups766887009212190081x_real @ H2 @ B4 ) )
= ( ( groups766887009212190081x_real @ G @ C4 )
= ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8617_prod_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ A2 )
= ( groups4061424788464935467al_rat @ H2 @ B4 ) )
= ( ( groups4061424788464935467al_rat @ G @ C4 )
= ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8618_prod_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ A2 )
= ( groups225925009352817453ex_rat @ H2 @ B4 ) )
= ( ( groups225925009352817453ex_rat @ G @ C4 )
= ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8619_prod_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ A2 )
= ( groups4696554848551431203al_nat @ H2 @ B4 ) )
= ( ( groups4696554848551431203al_nat @ G @ C4 )
= ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8620_prod_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ A2 )
= ( groups861055069439313189ex_nat @ H2 @ B4 ) )
= ( ( groups861055069439313189ex_nat @ G @ C4 )
= ( groups861055069439313189ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8621_prod_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ A2 )
= ( groups4694064378042380927al_int @ H2 @ B4 ) )
= ( ( groups4694064378042380927al_int @ G @ C4 )
= ( groups4694064378042380927al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8622_prod_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ A2 )
= ( groups858564598930262913ex_int @ H2 @ B4 ) )
= ( ( groups858564598930262913ex_int @ G @ C4 )
= ( groups858564598930262913ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8623_prod_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ C4 )
= ( groups713298508707869441omplex @ H2 @ C4 ) )
=> ( ( groups713298508707869441omplex @ G @ A2 )
= ( groups713298508707869441omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8624_prod_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ C4 )
= ( groups3708469109370488835omplex @ H2 @ C4 ) )
=> ( ( groups3708469109370488835omplex @ G @ A2 )
= ( groups3708469109370488835omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8625_prod_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ C4 )
= ( groups1681761925125756287l_real @ H2 @ C4 ) )
=> ( ( groups1681761925125756287l_real @ G @ A2 )
= ( groups1681761925125756287l_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8626_prod_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ C4 )
= ( groups766887009212190081x_real @ H2 @ C4 ) )
=> ( ( groups766887009212190081x_real @ G @ A2 )
= ( groups766887009212190081x_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8627_prod_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ C4 )
= ( groups4061424788464935467al_rat @ H2 @ C4 ) )
=> ( ( groups4061424788464935467al_rat @ G @ A2 )
= ( groups4061424788464935467al_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8628_prod_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ C4 )
= ( groups225925009352817453ex_rat @ H2 @ C4 ) )
=> ( ( groups225925009352817453ex_rat @ G @ A2 )
= ( groups225925009352817453ex_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8629_prod_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ C4 )
= ( groups4696554848551431203al_nat @ H2 @ C4 ) )
=> ( ( groups4696554848551431203al_nat @ G @ A2 )
= ( groups4696554848551431203al_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8630_prod_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ C4 )
= ( groups861055069439313189ex_nat @ H2 @ C4 ) )
=> ( ( groups861055069439313189ex_nat @ G @ A2 )
= ( groups861055069439313189ex_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8631_prod_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B4: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A2 @ C4 )
=> ( ( ord_less_eq_set_real @ B4 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ C4 )
= ( groups4694064378042380927al_int @ H2 @ C4 ) )
=> ( ( groups4694064378042380927al_int @ G @ A2 )
= ( groups4694064378042380927al_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8632_prod_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B4: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B4 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ C4 )
= ( groups858564598930262913ex_int @ H2 @ C4 ) )
=> ( ( groups858564598930262913ex_int @ G @ A2 )
= ( groups858564598930262913ex_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8633_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8634_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ S3 )
= ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8635_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ S3 )
= ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8636_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ S3 )
= ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8637_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S3: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ S3 )
= ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8638_prod_Omono__neutral__left,axiom,
! [T3: set_nat,S3: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ S3 )
= ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8639_prod_Omono__neutral__left,axiom,
! [T3: set_nat,S3: set_nat,G: nat > real] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ S3 )
= ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8640_prod_Omono__neutral__left,axiom,
! [T3: set_nat,S3: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ S3 )
= ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8641_prod_Omono__neutral__left,axiom,
! [T3: set_int,S3: set_int,G: int > complex] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups7440179247065528705omplex @ G @ S3 )
= ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8642_prod_Omono__neutral__left,axiom,
! [T3: set_int,S3: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ S3 )
= ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8643_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ T3 )
= ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8644_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ T3 )
= ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8645_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ T3 )
= ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8646_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ T3 )
= ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8647_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ T3 )
= ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8648_prod_Omono__neutral__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ T3 )
= ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8649_prod_Omono__neutral__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > real] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ T3 )
= ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8650_prod_Omono__neutral__right,axiom,
! [T3: set_nat,S3: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S3 @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ T3 )
= ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8651_prod_Omono__neutral__right,axiom,
! [T3: set_int,S3: set_int,G: int > complex] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups7440179247065528705omplex @ G @ T3 )
= ( groups7440179247065528705omplex @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8652_prod_Omono__neutral__right,axiom,
! [T3: set_int,S3: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S3 @ T3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ T3 )
= ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8653_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_complex ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S3 )
= ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8654_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_complex ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8655_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S3 )
= ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8656_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_real ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups766887009212190081x_real @ G @ S3 )
= ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8657_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_rat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ S3 )
= ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8658_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_rat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ S3 )
= ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8659_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S3 )
= ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8660_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S3 )
= ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8661_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_int ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ S3 )
= ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8662_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_int ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ S3 )
= ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8663_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ T3 )
= ( groups713298508707869441omplex @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8664_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ T3 )
= ( groups3708469109370488835omplex @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8665_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ T3 )
= ( groups1681761925125756287l_real @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8666_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups766887009212190081x_real @ G @ T3 )
= ( groups766887009212190081x_real @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8667_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ T3 )
= ( groups4061424788464935467al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8668_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ T3 )
= ( groups225925009352817453ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8669_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ T3 )
= ( groups4696554848551431203al_nat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8670_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ T3 )
= ( groups861055069439313189ex_nat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8671_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S3 @ T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ T3 )
= ( groups4694064378042380927al_int @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8672_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ T3 )
= ( groups858564598930262913ex_int @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8673_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8674_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8675_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8676_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8677_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8678_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8679_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8680_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8681_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8682_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8683_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8684_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8685_scaleR__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( A = zero_zero_real ) ) ) ).
% scaleR_le_0_iff
thf(fact_8686_zero__le__scaleR__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( A = zero_zero_real ) ) ) ).
% zero_le_scaleR_iff
thf(fact_8687_scaleR__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_8688_scaleR__nonpos__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_8689_scaleR__nonneg__nonpos,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_8690_scaleR__nonneg__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_8691_split__scaleR__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_8692_split__scaleR__neg__le,axiom,
! [A: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_8693_scaleR__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_8694_scaleR__mono,axiom,
! [A: real,B: real,X: real,Y2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y2 ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_8695_scaleR__left__le__one__le,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% scaleR_left_le_one_le
thf(fact_8696_scaleR__2,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
= ( plus_plus_real @ X @ X ) ) ).
% scaleR_2
thf(fact_8697_scaleR__2,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
= ( plus_plus_complex @ X @ X ) ) ).
% scaleR_2
thf(fact_8698_gbinomial__addition__formula,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ A @ ( suc @ K ) )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8699_gbinomial__addition__formula,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ A @ ( suc @ K ) )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8700_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8701_gbinomial__absorb__comp,axiom,
! [A: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
= ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8702_gbinomial__absorb__comp,axiom,
! [A: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
= ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8703_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8704_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_8705_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: rat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_8706_gbinomial__mult__1,axiom,
! [A: real,K: nat] :
( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_8707_gbinomial__mult__1,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_8708_gbinomial__mult__1_H,axiom,
! [A: real,K: nat] :
( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_8709_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_8710_prod_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8711_prod_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8712_prod_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8713_prod_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8714_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ M )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8715_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ M )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8716_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ M )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8717_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ M )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8718_prod_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8719_prod_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8720_prod_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8721_prod_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8722_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_8723_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_8724_prod_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8725_prod_Onested__swap_H,axiom,
! [A: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8726_prod__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
@ A
@ B
@ one_one_complex ) ) ).
% prod_atLeastAtMost_code
thf(fact_8727_prod__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
@ A
@ B
@ one_one_real ) ) ).
% prod_atLeastAtMost_code
thf(fact_8728_prod__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
@ A
@ B
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_8729_prod__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
@ A
@ B
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_8730_prod__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
@ A
@ B
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_8731_prod__mono__strict,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_complex )
=> ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8732_prod__mono__strict,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8733_prod__mono__strict,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8734_prod__mono__strict,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
& ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_real )
=> ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8735_prod__mono__strict,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_complex )
=> ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8736_prod__mono__strict,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8737_prod__mono__strict,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8738_prod__mono__strict,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
& ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_real )
=> ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8739_prod__mono__strict,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
& ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_complex )
=> ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8740_prod__mono__strict,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
& ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8741_even__prod__iff,axiom,
! [A2: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8742_even__prod__iff,axiom,
! [A2: set_int,F: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3827104343326376752nteger @ F @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8743_even__prod__iff,axiom,
! [A2: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A2 ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8744_even__prod__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8745_even__prod__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A2 ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8746_even__prod__iff,axiom,
! [A2: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A2 ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8747_even__prod__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8748_even__prod__iff,axiom,
! [A2: set_nat,F: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8749_even__prod__iff,axiom,
! [A2: set_int,F: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8750_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8751_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8752_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8753_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8754_sin__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) )
@ ( sin_real @ X ) ) ).
% sin_converges
thf(fact_8755_sin__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) )
@ ( sin_complex @ X ) ) ).
% sin_converges
thf(fact_8756_sin__def,axiom,
( sin_real
= ( ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% sin_def
thf(fact_8757_sin__def,axiom,
( sin_complex
= ( ^ [X2: complex] :
( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% sin_def
thf(fact_8758_cos__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) )
@ ( cos_real @ X ) ) ).
% cos_converges
thf(fact_8759_cos__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) )
@ ( cos_complex @ X ) ) ).
% cos_converges
thf(fact_8760_cos__def,axiom,
( cos_real
= ( ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% cos_def
thf(fact_8761_cos__def,axiom,
( cos_complex
= ( ^ [X2: complex] :
( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% cos_def
thf(fact_8762_summable__norm__sin,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% summable_norm_sin
thf(fact_8763_summable__norm__sin,axiom,
! [X: complex] :
( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% summable_norm_sin
thf(fact_8764_summable__norm__cos,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% summable_norm_cos
thf(fact_8765_summable__norm__cos,axiom,
! [X: complex] :
( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% summable_norm_cos
thf(fact_8766_Suc__times__gbinomial,axiom,
! [K: nat,A: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
= ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8767_Suc__times__gbinomial,axiom,
! [K: nat,A: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
= ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8768_Suc__times__gbinomial,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
= ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8769_gbinomial__absorption,axiom,
! [K: nat,A: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
= ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8770_gbinomial__absorption,axiom,
! [K: nat,A: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
= ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8771_gbinomial__absorption,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8772_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: real] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
= ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_8773_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: rat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_8774_norm__prod__diff,axiom,
! [I5: set_real,Z: real > real,W: real > real] :
( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8775_norm__prod__diff,axiom,
! [I5: set_int,Z: int > real,W: int > real] :
( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8776_norm__prod__diff,axiom,
! [I5: set_complex,Z: complex > real,W: complex > real] :
( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I5 ) @ ( groups766887009212190081x_real @ W @ I5 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8777_norm__prod__diff,axiom,
! [I5: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > real,W: product_prod_nat_nat > real] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups6036352826371341000t_real @ Z @ I5 ) @ ( groups6036352826371341000t_real @ W @ I5 ) ) )
@ ( groups4567486121110086003t_real
@ ^ [I3: product_prod_nat_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8778_norm__prod__diff,axiom,
! [I5: set_real,Z: real > complex,W: real > complex] :
( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8779_norm__prod__diff,axiom,
! [I5: set_int,Z: int > complex,W: int > complex] :
( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8780_norm__prod__diff,axiom,
! [I5: set_complex,Z: complex > complex,W: complex > complex] :
( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I5 ) @ ( groups3708469109370488835omplex @ W @ I5 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8781_norm__prod__diff,axiom,
! [I5: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > complex,W: product_prod_nat_nat > complex] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups8110221916422527690omplex @ Z @ I5 ) @ ( groups8110221916422527690omplex @ W @ I5 ) ) )
@ ( groups4567486121110086003t_real
@ ^ [I3: product_prod_nat_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8782_norm__prod__diff,axiom,
! [I5: set_nat,Z: nat > real,W: nat > real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8783_norm__prod__diff,axiom,
! [I5: set_nat,Z: nat > complex,W: nat > complex] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8784_prod_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8785_prod_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8786_prod_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8787_prod_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8788_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1408675320244567234ct_nat @ M )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_8789_prod__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > real] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8790_prod__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8791_prod__mono2,axiom,
! [B4: set_nat,A2: set_nat,F: nat > real] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8792_prod__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > rat] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8793_prod__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8794_prod__mono2,axiom,
! [B4: set_nat,A2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8795_prod__mono2,axiom,
! [B4: set_real,A2: set_real,F: real > int] :
( ( finite_finite_real @ B4 )
=> ( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B4 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8796_prod__mono2,axiom,
! [B4: set_complex,A2: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B4 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B4 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8797_prod__mono2,axiom,
! [B4: set_int,A2: set_int,F: int > real] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8798_prod__mono2,axiom,
! [B4: set_int,A2: set_int,F: int > rat] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8799_gbinomial__parallel__sum,axiom,
! [A: complex,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8800_gbinomial__parallel__sum,axiom,
! [A: rat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8801_gbinomial__parallel__sum,axiom,
! [A: real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8802_sin__minus__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N2 ) ) )
@ ( sin_real @ X ) ) ).
% sin_minus_converges
thf(fact_8803_sin__minus__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N2 ) ) )
@ ( sin_complex @ X ) ) ).
% sin_minus_converges
thf(fact_8804_cos__minus__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N2 ) )
@ ( cos_real @ X ) ) ).
% cos_minus_converges
thf(fact_8805_cos__minus__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N2 ) )
@ ( cos_complex @ X ) ) ).
% cos_minus_converges
thf(fact_8806_gbinomial__factors,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_8807_gbinomial__factors,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_8808_gbinomial__factors,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_8809_gbinomial__rec,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8810_gbinomial__rec,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8811_gbinomial__rec,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8812_gbinomial__negated__upper,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8813_gbinomial__negated__upper,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8814_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8815_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_8816_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_8817_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_8818_pochhammer__Suc__prod,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8819_pochhammer__Suc__prod,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8820_pochhammer__Suc__prod,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8821_pochhammer__Suc__prod,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8822_pochhammer__prod__rev,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A4: real,N2: nat] :
( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8823_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A4: rat,N2: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8824_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A4: nat,N2: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8825_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A4: int,N2: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8826_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_8827_gbinomial__minus,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8828_gbinomial__minus,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8829_gbinomial__minus,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8830_prod_Oin__pairs,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8831_prod_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8832_prod_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8833_prod_Oin__pairs,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8834_prod_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8835_prod_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8836_prod_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8837_prod_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8838_gbinomial__reduce__nat,axiom,
! [K: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_complex @ A @ K )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8839_gbinomial__reduce__nat,axiom,
! [K: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_real @ A @ K )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8840_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A @ K )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8841_pochhammer__Suc__prod__rev,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8842_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8843_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8844_pochhammer__Suc__prod__rev,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8845_gbinomial__pochhammer,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8846_gbinomial__pochhammer,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8847_gbinomial__pochhammer,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8848_gbinomial__pochhammer_H,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8849_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8850_gbinomial__pochhammer_H,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8851_gbinomial__sum__lower__neg,axiom,
! [A: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8852_gbinomial__sum__lower__neg,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8853_gbinomial__sum__lower__neg,axiom,
! [A: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8854_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8855_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8856_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real @ ( sin_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) )
@ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_8857_cot__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% cot_less_zero
thf(fact_8858_i__even__power,axiom,
! [N: nat] :
( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% i_even_power
thf(fact_8859_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_8860_divide__i,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ imaginary_unit )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_8861_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% complex_i_mult_minus
thf(fact_8862_i__squared,axiom,
( ( times_times_complex @ imaginary_unit @ imaginary_unit )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% i_squared
thf(fact_8863_divide__numeral__i,axiom,
! [Z: complex,N: num] :
( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% divide_numeral_i
thf(fact_8864_log__le__cancel__iff,axiom,
! [A: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_8865_log__le__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_8866_one__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ A @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_8867_log__le__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_8868_zero__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_8869_cot__npi,axiom,
! [N: nat] :
( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% cot_npi
thf(fact_8870_log__pow__cancel,axiom,
! [A: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( power_power_real @ A @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_8871_cot__periodic,axiom,
! [X: real] :
( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( cot_real @ X ) ) ).
% cot_periodic
thf(fact_8872_power2__i,axiom,
( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power2_i
thf(fact_8873_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_8874_complex__i__not__numeral,axiom,
! [W: num] :
( imaginary_unit
!= ( numera6690914467698888265omplex @ W ) ) ).
% complex_i_not_numeral
thf(fact_8875_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X2: real,Y: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y ) ) ) ) ).
% divide_real_def
thf(fact_8876_i__times__eq__iff,axiom,
! [W: complex,Z: complex] :
( ( ( times_times_complex @ imaginary_unit @ W )
= Z )
= ( W
= ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% i_times_eq_iff
thf(fact_8877_complex__i__not__neg__numeral,axiom,
! [W: num] :
( imaginary_unit
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% complex_i_not_neg_numeral
thf(fact_8878_Complex__mult__i,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% Complex_mult_i
thf(fact_8879_i__mult__Complex,axiom,
! [A: real,B: real] :
( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
= ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% i_mult_Complex
thf(fact_8880_less__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% less_log_of_power
thf(fact_8881_log__of__power__eq,axiom,
! [M: nat,B: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_8882_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less_real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_8883_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [N2: nat] :
( ( N2 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_8884_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less_real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_8885_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( divide_divide_real @ ( sqrt @ X ) @ X )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_8886_prod__int__plus__eq,axiom,
! [I2: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X2: int] : X2
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_8887_log__mult,axiom,
! [A: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( log @ A @ ( times_times_real @ X @ Y2 ) )
= ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y2 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_8888_le__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% le_log_of_power
thf(fact_8889_log__base__pow,axiom,
! [A: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ ( power_power_real @ A @ N ) @ X )
= ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log_base_pow
thf(fact_8890_log__nat__power,axiom,
! [X: real,B: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% log_nat_power
thf(fact_8891_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_8892_log__of__power__less,axiom,
! [M: nat,B: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_8893_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_8894_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_8895_log__of__power__le,axiom,
! [M: nat,B: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_8896_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_8897_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( inverse_inverse_real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_8898_tan__cot,axiom,
! [X: real] :
( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
= ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% tan_cot
thf(fact_8899_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_8900_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_8901_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_8902_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_8903_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_8904_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_8905_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_8906_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_8907_tan__cot_H,axiom,
! [X: real] :
( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
= ( cot_real @ X ) ) ).
% tan_cot'
thf(fact_8908_Arg__minus__ii,axiom,
( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
= ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% Arg_minus_ii
thf(fact_8909_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_8910_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% Arg_ii
thf(fact_8911_ceiling__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
=> ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_8912_sinh__real__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% sinh_real_le_iff
thf(fact_8913_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_8914_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% sinh_real_nonpos_iff
thf(fact_8915_ceiling__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_8916_ceiling__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_8917_divide__complex__def,axiom,
( divide1717551699836669952omplex
= ( ^ [X2: complex,Y: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% divide_complex_def
thf(fact_8918_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_8919_complex__inverse,axiom,
! [A: real,B: real] :
( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
= ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_8920_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sinh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_ln_real
thf(fact_8921_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% ceiling_log2_div2
thf(fact_8922_cis__minus__pi__half,axiom,
( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% cis_minus_pi_half
thf(fact_8923_ceiling__log__eq__powr__iff,axiom,
! [X: real,B: real,K: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
= ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
& ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_8924_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_8925_powr__nonneg__iff,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_8926_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_8927_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr_real @ X @ one_one_real )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_8928_powr__le__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_8929_numeral__powr__numeral__real,axiom,
! [M: num,N: num] :
( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_8930_floor__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_8931_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_8932_cis__pi__half,axiom,
( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_8933_floor__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_8934_cis__2pi,axiom,
( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_complex ) ).
% cis_2pi
thf(fact_8935_floor__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_8936_square__powr__half,axiom,
! [X: real] :
( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( abs_abs_real @ X ) ) ).
% square_powr_half
thf(fact_8937_floor__minus__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_8938_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_8939_powr__powr,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B )
= ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% powr_powr
thf(fact_8940_powr__ge__pzero,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y2 ) ) ).
% powr_ge_pzero
thf(fact_8941_powr__mono2,axiom,
! [A: real,X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).
% powr_mono2
thf(fact_8942_powr__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_mono
thf(fact_8943_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_8944_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_8945_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% cosh_real_nonneg
thf(fact_8946_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% cosh_real_ge_1
thf(fact_8947_cis__mult,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% cis_mult
thf(fact_8948_powr__mono2_H,axiom,
! [A: real,X: real,Y2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ Y2 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_8949_powr__less__mono2,axiom,
! [A: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_8950_powr__le1,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_8951_powr__mono__both,axiom,
! [A: real,B: real,X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y2 @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_8952_ge__one__powr__ge__zero,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_8953_powr__divide,axiom,
! [X: real,Y2: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( divide_divide_real @ X @ Y2 ) @ A )
= ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).
% powr_divide
thf(fact_8954_powr__mult,axiom,
! [X: real,Y2: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( times_times_real @ X @ Y2 ) @ A )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).
% powr_mult
thf(fact_8955_inverse__powr,axiom,
! [Y2: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( inverse_inverse_real @ Y2 ) @ A )
= ( inverse_inverse_real @ ( powr_real @ Y2 @ A ) ) ) ) ).
% inverse_powr
thf(fact_8956_divide__powr__uminus,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
= ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% divide_powr_uminus
thf(fact_8957_ln__powr,axiom,
! [X: real,Y2: real] :
( ( X != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X @ Y2 ) )
= ( times_times_real @ Y2 @ ( ln_ln_real @ X ) ) ) ) ).
% ln_powr
thf(fact_8958_log__powr,axiom,
! [X: real,B: real,Y2: real] :
( ( X != zero_zero_real )
=> ( ( log @ B @ ( powr_real @ X @ Y2 ) )
= ( times_times_real @ Y2 @ ( log @ B @ X ) ) ) ) ).
% log_powr
thf(fact_8959_cosh__real__strict__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y2 ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_8960_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_8961_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_8962_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( arcosh_real @ ( cosh_real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_8963_floor__log__eq__powr__iff,axiom,
! [X: real,B: real,K: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
= K )
= ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
& ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_8964_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_8965_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_8966_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_8967_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_8968_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_8969_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_8970_DeMoivre,axiom,
! [A: real,N: nat] :
( ( power_power_complex @ ( cis @ A ) @ N )
= ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% DeMoivre
thf(fact_8971_powr__mult__base,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( powr_real @ X @ Y2 ) )
= ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y2 ) ) ) ) ).
% powr_mult_base
thf(fact_8972_le__log__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ ( log @ B @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_8973_log__le__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y2 )
= ( ord_less_eq_real @ X @ ( powr_real @ B @ Y2 ) ) ) ) ) ).
% log_le_iff
thf(fact_8974_le__powr__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y2 ) )
= ( ord_less_eq_real @ ( log @ B @ X ) @ Y2 ) ) ) ) ).
% le_powr_iff
thf(fact_8975_powr__le__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X )
= ( ord_less_eq_real @ Y2 @ ( log @ B @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_8976_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_8977_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% floor_divide_real_eq_div
thf(fact_8978_ln__powr__bound,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% ln_powr_bound
thf(fact_8979_ln__powr__bound2,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_8980_log__add__eq__powr,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ ( log @ B @ X ) @ Y2 )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y2 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_8981_add__log__eq__powr,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ Y2 @ ( log @ B @ X ) )
= ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y2 ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_8982_log__minus__eq__powr,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ ( log @ B @ X ) @ Y2 )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y2 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_8983_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_8984_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_8985_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( cosh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_ln_real
thf(fact_8986_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% floor_log2_div2
thf(fact_8987_floor__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
=> ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_8988_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_betw_nat_complex
@ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
@ ( set_ord_lessThan_nat @ N )
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= one_one_complex ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_8989_exp__pi__i,axiom,
( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i
thf(fact_8990_exp__pi__i_H,axiom,
( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i'
thf(fact_8991_exp__two__pi__i,axiom,
( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
= one_one_complex ) ).
% exp_two_pi_i
thf(fact_8992_exp__two__pi__i_H,axiom,
( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
= one_one_complex ) ).
% exp_two_pi_i'
thf(fact_8993_complex__exp__exists,axiom,
! [Z: complex] :
? [A3: complex,R3: real] :
( Z
= ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A3 ) ) ) ).
% complex_exp_exists
thf(fact_8994_Complex__mult__complex__of__real,axiom,
! [X: real,Y2: real,R2: real] :
( ( times_times_complex @ ( complex2 @ X @ Y2 ) @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y2 @ R2 ) ) ) ).
% Complex_mult_complex_of_real
thf(fact_8995_complex__of__real__mult__Complex,axiom,
! [R2: real,X: real,Y2: real] :
( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y2 ) )
= ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y2 ) ) ) ).
% complex_of_real_mult_Complex
thf(fact_8996_Complex__add__complex__of__real,axiom,
! [X: real,Y2: real,R2: real] :
( ( plus_plus_complex @ ( complex2 @ X @ Y2 ) @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ ( plus_plus_real @ X @ R2 ) @ Y2 ) ) ).
% Complex_add_complex_of_real
thf(fact_8997_complex__of__real__add__Complex,axiom,
! [R2: real,X: real,Y2: real] :
( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y2 ) )
= ( complex2 @ ( plus_plus_real @ R2 @ X ) @ Y2 ) ) ).
% complex_of_real_add_Complex
thf(fact_8998_cis__conv__exp,axiom,
( cis
= ( ^ [B3: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% cis_conv_exp
thf(fact_8999_complex__of__real__i,axiom,
! [R2: real] :
( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
= ( complex2 @ zero_zero_real @ R2 ) ) ).
% complex_of_real_i
thf(fact_9000_i__complex__of__real,axiom,
! [R2: real] :
( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ zero_zero_real @ R2 ) ) ).
% i_complex_of_real
thf(fact_9001_Complex__eq,axiom,
( complex2
= ( ^ [A4: real,B3: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% Complex_eq
thf(fact_9002_complex__split__polar,axiom,
! [Z: complex] :
? [R3: real,A3: real] :
( Z
= ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_9003_cmod__unit__one,axiom,
! [A: real] :
( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
= one_one_real ) ).
% cmod_unit_one
thf(fact_9004_cmod__complex__polar,axiom,
! [R2: real,A: real] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
= ( abs_abs_real @ R2 ) ) ).
% cmod_complex_polar
thf(fact_9005_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_9006_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D2: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z2: int] :
( ( ord_less_eq_int @ D2 @ Z2 )
& ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_9007_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D2: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z2: int] :
( ( ord_less_eq_int @ D2 @ Z6 )
& ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_9008_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I4: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I4 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I4 @ J2 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) )
=> ( P @ I4 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_9009_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z ) ).
% power2_csqrt
thf(fact_9010_of__real__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
= ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% of_real_sqrt
thf(fact_9011_arctan__def,axiom,
( arctan
= ( ^ [Y: real] :
( the_real
@ ^ [X2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
& ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X2 )
= Y ) ) ) ) ) ).
% arctan_def
thf(fact_9012_arcsin__def,axiom,
( arcsin
= ( ^ [Y: real] :
( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
& ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X2 )
= Y ) ) ) ) ) ).
% arcsin_def
thf(fact_9013_even__set__encode__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
= ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% even_set_encode_iff
thf(fact_9014_modulo__int__unfold,axiom,
! [L2: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L2 ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L2 ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L2 )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_9015_dvd__mult__sgn__iff,axiom,
! [L2: int,K: int,R2: int] :
( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L2 @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_9016_dvd__sgn__mult__iff,axiom,
! [L2: int,R2: int,K: int] :
( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
= ( ( dvd_dvd_int @ L2 @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_9017_mult__sgn__dvd__iff,axiom,
! [L2: int,R2: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R2 ) ) @ K )
= ( ( dvd_dvd_int @ L2 @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_9018_sgn__mult__dvd__iff,axiom,
! [R2: int,L2: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L2 ) @ K )
= ( ( dvd_dvd_int @ L2 @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_9019_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L4: int] :
( K
!= ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_sgnE
thf(fact_9020_sgn__mod,axiom,
! [L2: int,K: int] :
( ( L2 != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ L2 @ K )
=> ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
= ( sgn_sgn_int @ L2 ) ) ) ) ).
% sgn_mod
thf(fact_9021_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ln_ln_real @ X )
= ( the_real
@ ^ [X2: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_9022_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L2: int] :
( ( V != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
= ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_9023_div__dvd__sgn__abs,axiom,
! [L2: int,K: int] :
( ( dvd_dvd_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_9024_arccos__def,axiom,
( arccos
= ( ^ [Y: real] :
( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ pi )
& ( ( cos_real @ X2 )
= Y ) ) ) ) ) ).
% arccos_def
thf(fact_9025_eucl__rel__int__remainderI,axiom,
! [R2: int,L2: int,K: int,Q2: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L2 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L2 ) )
=> ( ( K
= ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_9026_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22 = zero_zero_int )
=> ( A32
!= ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
=> ( ( A22 != zero_zero_int )
=> ( A1
!= ( times_times_int @ Q3 @ A22 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32
= ( product_Pair_int_int @ Q3 @ R3 ) )
=> ( ( ( sgn_sgn_int @ R3 )
= ( sgn_sgn_int @ A22 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
=> ( A1
!= ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_9027_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23 = zero_zero_int )
& ( A33
= ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
| ? [L: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L )
& ( A33
= ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
& ( L != zero_zero_int )
& ( K3
= ( times_times_int @ Q4 @ L ) ) )
| ? [R5: int,L: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L )
& ( A33
= ( product_Pair_int_int @ Q4 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
& ( K3
= ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_9028_set__encode__def,axiom,
( nat_set_encode
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% set_encode_def
thf(fact_9029_pi__half,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X2 )
= zero_zero_real ) ) ) ) ).
% pi_half
thf(fact_9030_pi__def,axiom,
( pi
= ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X2 )
= zero_zero_real ) ) ) ) ) ).
% pi_def
thf(fact_9031_divide__int__unfold,axiom,
! [L2: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L2 ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L2 ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_9032_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K3: int,L: int] :
( if_int @ ( L = zero_zero_int ) @ K3
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L ) )
@ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L @ K3 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_9033_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K3: int,L: int] :
( if_int @ ( L = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_9034_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_9035_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_9036_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% sgn_le_0_iff
thf(fact_9037_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% zero_le_sgn_iff
thf(fact_9038_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_9039_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_9040_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_9041_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_9042_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_9043_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_9044_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_9045_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_9046_diff__nat__numeral,axiom,
! [V: num,V3: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_9047_nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( nat2 @ Y2 )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y2
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_9048_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( nat2 @ Y2 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_9049_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_9050_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_9051_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% nat_numeral_diff_1
thf(fact_9052_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_9053_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_9054_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_9055_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_9056_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_9057_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% nat_mask_eq
thf(fact_9058_nat__numeral__as__int,axiom,
( numeral_numeral_nat
= ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_9059_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_9060_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_9061_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_9062_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_9063_unset__bit__nat__def,axiom,
( bit_se4205575877204974255it_nat
= ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_9064_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_9065_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_9066_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_9067_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_9068_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_9069_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_9070_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_9071_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_9072_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_9073_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_9074_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_9075_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_9076_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_9077_nat__div__as__int,axiom,
( divide_divide_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_9078_sgn__real__def,axiom,
( sgn_sgn_real
= ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_real_def
thf(fact_9079_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_9080_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_9081_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_9082_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_9083_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_9084_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_9085_nat__mult__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_9086_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_9087_nat__diff__distrib_H,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y2 ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_9088_nat__diff__distrib,axiom,
! [Z7: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ord_less_eq_int @ Z7 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_9089_nat__abs__triangle__ineq,axiom,
! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_9090_nat__div__distrib_H,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y2 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib'
thf(fact_9091_nat__div__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y2 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib
thf(fact_9092_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_9093_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_9094_nat__mod__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( modulo_modulo_int @ X @ Y2 ) )
= ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_9095_div__abs__eq__div__nat,axiom,
! [K: int,L2: int] :
( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_9096_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_9097_le__nat__floor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_9098_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_9099_sgn__power__injE,axiom,
! [A: real,N: nat,X: real,B: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= X )
=> ( ( X
= ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ).
% sgn_power_injE
thf(fact_9100_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_9101_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_9102_nat__mult__distrib__neg,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_9103_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_9104_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_9105_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_9106_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_9107_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_9108_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_9109_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn_complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_9110_mask__nat__def,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% mask_nat_def
thf(fact_9111_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% mask_half_int
thf(fact_9112_mask__int__def,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% mask_int_def
thf(fact_9113_Arg__correct,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
=> ( ( ( sgn_sgn_complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_9114_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_9115_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( power_power_real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_9116_arctan__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
= ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% arctan_inverse
thf(fact_9117_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_9118_powr__int,axiom,
! [X: real,I2: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
= ( power_power_real @ X @ ( nat2 @ I2 ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_9119_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( ( K3 = zero_zero_int )
| ( L = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ L
@ ( if_int
@ ( L
= ( uminus_uminus_int @ one_one_int ) )
@ K3
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_9120_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ zero_zero_int )
= ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_9121_and__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
| ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% and_nonnegative_int_iff
thf(fact_9122_and__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% and_negative_int_iff
thf(fact_9123_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_9124_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_9125_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_9126_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_9127_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_9128_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_9129_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_9130_take__bit__diff,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% take_bit_diff
thf(fact_9131_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_9132_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_9133_take__bit__mult,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% take_bit_mult
thf(fact_9134_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% take_bit_minus
thf(fact_9135_concat__bit__take__bit__eq,axiom,
! [N: nat,B: int] :
( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
= ( bit_concat_bit @ N @ B ) ) ).
% concat_bit_take_bit_eq
thf(fact_9136_concat__bit__eq__iff,axiom,
! [N: nat,K: int,L2: int,R2: int,S2: int] :
( ( ( bit_concat_bit @ N @ K @ L2 )
= ( bit_concat_bit @ N @ R2 @ S2 ) )
= ( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2923211474154528505it_int @ N @ R2 ) )
& ( L2 = S2 ) ) ) ).
% concat_bit_eq_iff
thf(fact_9137_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_9138_AND__upper2_H,axiom,
! [Y2: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y2 ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_9139_AND__upper1_H,axiom,
! [Y2: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y2 @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_9140_AND__upper2,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y2 ) @ Y2 ) ) ).
% AND_upper2
thf(fact_9141_AND__upper1,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y2 ) @ X ) ) ).
% AND_upper1
thf(fact_9142_AND__lower,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y2 ) ) ) ).
% AND_lower
thf(fact_9143_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_9144_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_9145_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% take_bit_int_greater_self_iff
thf(fact_9146_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% not_take_bit_negative
thf(fact_9147_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_9148_and__less__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% and_less_eq
thf(fact_9149_AND__upper1_H_H,axiom,
! [Y2: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y2 @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_9150_AND__upper2_H_H,axiom,
! [Y2: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y2 ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_9151_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_9152_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_9153_even__and__iff__int,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% even_and_iff_int
thf(fact_9154_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2925701944663578781it_nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_9155_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_9156_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ M )
= M )
= ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_9157_take__bit__nat__def,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N2: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% take_bit_nat_def
thf(fact_9158_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_9159_take__bit__int__def,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N2: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% take_bit_int_def
thf(fact_9160_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_9161_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_9162_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_9163_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_9164_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_9165_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_9166_take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_9167_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_9168_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_9169_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_9170_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_9171_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_9172_and__int__rec,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_9173_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_9174_take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_9175_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_9176_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y2: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y2 )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y2
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y2
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_9177_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_9178_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% pred_numeral_inc
thf(fact_9179_and__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_9180_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% and_nat_numerals(3)
thf(fact_9181_and__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_9182_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_9183_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_Suc_0_eq
thf(fact_9184_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_0_and_eq
thf(fact_9185_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_9186_add__inc,axiom,
! [X: num,Y2: num] :
( ( plus_plus_num @ X @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus_num @ X @ Y2 ) ) ) ).
% add_inc
thf(fact_9187_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_9188_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_9189_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_9190_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_9191_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_9192_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_9193_and__nat__def,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% and_nat_def
thf(fact_9194_mult__inc,axiom,
! [X: num,Y2: num] :
( ( times_times_num @ X @ ( inc @ Y2 ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y2 ) @ X ) ) ).
% mult_inc
thf(fact_9195_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_9196_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_9197_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M6: nat,N2: nat] :
( if_nat
@ ( ( M6 = zero_zero_nat )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_9198_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M6: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_9199_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y2: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y2 )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y2
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y2
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_9200_and__int_Opsimps,axiom,
! [K: int,L2: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
=> ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L2 )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
& ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L2 )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_9201_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [K2: int,L4: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
=> ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K2 @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_9202_Arg__def,axiom,
( arg
= ( ^ [Z2: complex] :
( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
@ ( fChoice_real
@ ^ [A4: real] :
( ( ( sgn_sgn_complex @ Z2 )
= ( cis @ A4 ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
& ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_9203_set__encode__insert,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ N @ A2 )
=> ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% set_encode_insert
thf(fact_9204_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_9205_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_9206_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_9207_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_9208_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_9209_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_9210_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_9211_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_9212_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_9213_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
= ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_9214_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_9215_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( ( K3
= ( uminus_uminus_int @ one_one_int ) )
| ( L
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_9216_cis__multiple__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= one_one_complex ) ) ).
% cis_multiple_2pi
thf(fact_9217_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_9218_or__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% or_nonnegative_int_iff
thf(fact_9219_or__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
| ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% or_negative_int_iff
thf(fact_9220_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_9221_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
= ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_9222_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9223_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9224_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_9225_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_9226_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_9227_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_9228_bit__or__int__iff,axiom,
! [K: int,L2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
= ( ( bit_se1146084159140164899it_int @ K @ N )
| ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% bit_or_int_iff
thf(fact_9229_bit__and__int__iff,axiom,
! [K: int,L2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
= ( ( bit_se1146084159140164899it_int @ K @ N )
& ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% bit_and_int_iff
thf(fact_9230_OR__lower,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y2 ) ) ) ) ).
% OR_lower
thf(fact_9231_or__greater__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L2 )
=> ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% or_greater_eq
thf(fact_9232_plus__and__or,axiom,
! [X: int,Y2: int] :
( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y2 ) @ ( bit_se1409905431419307370or_int @ X @ Y2 ) )
= ( plus_plus_int @ X @ Y2 ) ) ).
% plus_and_or
thf(fact_9233_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_9234_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less_nat @ N @ M )
=> ( ( bit_se1146084159140164899it_int @ K @ N )
=> ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_9235_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
= ( ( ( ord_less_nat @ N @ M )
& ( bit_se1146084159140164899it_int @ K @ N ) )
| ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_9236_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( ( sin_real @ ( times_times_real @ X @ pi ) )
= zero_zero_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% sin_times_pi_eq_0
thf(fact_9237_signed__take__bit__eq__concat__bit,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,K3: int] : ( bit_concat_bit @ N2 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_9238_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ N3 @ M2 )
=> ( ( bit_se1146084159140164899it_int @ K @ M2 )
= ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_9239_bit__int__def,axiom,
( bit_se1146084159140164899it_int
= ( ^ [K3: int,N2: nat] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% bit_int_def
thf(fact_9240_OR__upper,axiom,
! [X: int,N: nat,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_9241_sin__integer__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= zero_zero_real ) ) ).
% sin_integer_2pi
thf(fact_9242_cos__integer__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= one_one_real ) ) ).
% cos_integer_2pi
thf(fact_9243_or__int__rec,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
| ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_9244_set__bit__eq,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N2: nat,K3: int] :
( plus_plus_int @ K3
@ ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ~ ( bit_se1146084159140164899it_int @ K3 @ N2 ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% set_bit_eq
thf(fact_9245_unset__bit__eq,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N2: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_9246_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_9247_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_9248_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_9249_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_9250_or__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).
% or_nat_numerals(2)
thf(fact_9251_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_9252_or__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).
% or_nat_numerals(1)
thf(fact_9253_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_9254_xor__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_9255_xor__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_9256_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_9257_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_9258_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_9259_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_9260_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_9261_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_9262_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one @ one )
= one ) ).
% or_not_num_neg.simps(1)
thf(fact_9263_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
= ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
thf(fact_9264_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_9265_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
= ( bit0 @ one ) ) ).
% or_not_num_neg.simps(4)
thf(fact_9266_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_9267_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
= one ) ).
% or_not_num_neg.simps(7)
thf(fact_9268_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_9269_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_9270_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_9271_or__not__num__neg_Osimps_I9_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_9272_or__nat__def,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% or_nat_def
thf(fact_9273_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_9274_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_9275_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_9276_or__not__num__neg_Oelims,axiom,
! [X: num,Xa2: num,Y2: num] :
( ( ( bit_or_not_num_neg @ X @ Xa2 )
= Y2 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y2 != one ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa2
= ( bit0 @ M5 ) )
=> ( Y2
!= ( bit1 @ M5 ) ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa2
= ( bit1 @ M5 ) )
=> ( Y2
!= ( bit1 @ M5 ) ) ) )
=> ( ( ? [N3: num] :
( X
= ( bit0 @ N3 ) )
=> ( ( Xa2 = one )
=> ( Y2
!= ( bit0 @ one ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit0 @ M5 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit1 @ M5 ) )
=> ( Y2
!= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
=> ( ( ? [N3: num] :
( X
= ( bit1 @ N3 ) )
=> ( ( Xa2 = one )
=> ( Y2 != one ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit0 @ M5 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit1 @ M5 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_9277_bit__nat__def,axiom,
( bit_se1148574629649215175it_nat
= ( ^ [M6: nat,N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% bit_nat_def
thf(fact_9278_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_9279_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_9280_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M6: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_9281_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_9282_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M6: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_9283_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_9284_or__not__num__neg_Opelims,axiom,
! [X: num,Xa2: num,Y2: num] :
( ( ( bit_or_not_num_neg @ X @ Xa2 )
= Y2 )
=> ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y2 = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa2
= ( bit0 @ M5 ) )
=> ( ( Y2
= ( bit1 @ M5 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa2
= ( bit1 @ M5 ) )
=> ( ( Y2
= ( bit1 @ M5 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( bit0 @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit0 @ M5 ) )
=> ( ( Y2
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit1 @ M5 ) )
=> ( ( Y2
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ( ( Xa2 = one )
=> ( ( Y2 = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit0 @ M5 ) )
=> ( ( Y2
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa2
= ( bit1 @ M5 ) )
=> ( ( Y2
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_9285_horner__sum__of__bool__2__less,axiom,
! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_9286_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_9287_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_9288_concat__bit__of__zero__1,axiom,
! [N: nat,L2: int] :
( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
= ( bit_se545348938243370406it_int @ N @ L2 ) ) ).
% concat_bit_of_zero_1
thf(fact_9289_xor__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
= ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_9290_xor__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
!= ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% xor_negative_int_iff
thf(fact_9291_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_9292_bit__xor__int__iff,axiom,
! [K: int,L2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
= ( ( bit_se1146084159140164899it_int @ K @ N )
!= ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% bit_xor_int_iff
thf(fact_9293_flip__bit__int__def,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N2: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% flip_bit_int_def
thf(fact_9294_push__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% push_bit_nat_eq
thf(fact_9295_XOR__lower,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y2 ) ) ) ) ).
% XOR_lower
thf(fact_9296_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M6: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_9297_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M6: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
thf(fact_9298_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_9299_xor__nat__def,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% xor_nat_def
thf(fact_9300_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_9301_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% concat_bit_eq
thf(fact_9302_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% concat_bit_def
thf(fact_9303_set__bit__int__def,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N2: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% set_bit_int_def
thf(fact_9304_push__bit__int__def,axiom,
( bit_se545348938243370406it_int
= ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_int_def
thf(fact_9305_push__bit__nat__def,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N2: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_nat_def
thf(fact_9306_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_9307_XOR__upper,axiom,
! [X: int,N: nat,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_9308_xor__int__rec,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_9309_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L )
@ ( if_int
@ ( L
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K3 )
@ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_9310_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_9311_VEBT_Osize_I3_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size(3)
thf(fact_9312_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% not_negative_int_iff
thf(fact_9313_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% not_nonnegative_int_iff
thf(fact_9314_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_9315_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_9316_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_9317_bit__not__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_not_int_iff
thf(fact_9318_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_9319_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_9320_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_9321_or__int__def,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% or_int_def
thf(fact_9322_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_9323_and__not__numerals_I1_J,axiom,
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= zero_zero_int ) ).
% and_not_numerals(1)
thf(fact_9324_or__not__numerals_I1_J,axiom,
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(1)
thf(fact_9325_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_9326_unset__bit__int__def,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N2: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_9327_xor__int__def,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).
% xor_int_def
thf(fact_9328_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_9329_not__int__div__2,axiom,
! [K: int] :
( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% not_int_div_2
thf(fact_9330_even__not__iff__int,axiom,
! [K: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% even_not_iff_int
thf(fact_9331_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= one_one_int ) ).
% and_not_numerals(2)
thf(fact_9332_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_9333_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_9334_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% or_not_numerals(4)
thf(fact_9335_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
= ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_9336_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_9337_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_9338_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_9339_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_9340_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_9341_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_fact
thf(fact_9342_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_9343_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_9344_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_9345_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_9346_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(7)
thf(fact_9347_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_9348_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_9349_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_9350_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_9351_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_9352_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_9353_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_9354_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_9355_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_9356_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_9357_sum__power2,axiom,
! [K: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_9358_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J2 ) ) ) )
=> ( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I4 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_9359_VEBT_Osize__gen_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size_gen(1)
thf(fact_9360_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9361_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X6: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M6: nat] :
( ( ord_less_eq_nat @ M9 @ M6 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M9 @ N2 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_9362_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_9363_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T @ D )
=> ( vEBT_invar_vebt @ T @ D ) ) ).
% valid_eq2
thf(fact_9364_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T @ D )
=> ( vEBT_VEBT_valid @ T @ D ) ) ).
% valid_eq1
thf(fact_9365_Code__Target__Int_Opositive__def,axiom,
code_Target_positive = numeral_numeral_int ).
% Code_Target_Int.positive_def
thf(fact_9366_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L: num] :
( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_9367_divmod__integer_H__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_9368_times__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(2)
thf(fact_9369_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(1)
thf(fact_9370_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
= K ) ).
% plus_integer_code(1)
thf(fact_9371_plus__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
= L2 ) ).
% plus_integer_code(2)
thf(fact_9372_less__eq__integer__code_I1_J,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% less_eq_integer_code(1)
thf(fact_9373_exhaustive__integer_H_Ocases,axiom,
! [X: produc8763457246119570046nteger] :
~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I4: code_integer] :
( X
!= ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I4 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_9374_full__exhaustive__integer_H_Ocases,axiom,
! [X: produc1908205239877642774nteger] :
~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I4: code_integer] :
( X
!= ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I4 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_9375_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_9376_Code__Numeral_Opositive__def,axiom,
code_positive = numera6620942414471956472nteger ).
% Code_Numeral.positive_def
thf(fact_9377_plus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus_int @ Xa2 @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_9378_times__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_9379_less__eq__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq_int @ Xa2 @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_9380_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% integer_of_num(3)
thf(fact_9381_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
@ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_9382_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% int_of_integer_numeral
thf(fact_9383_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa2 ) )
= ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% plus_integer.rep_eq
thf(fact_9384_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
= ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% times_integer.rep_eq
thf(fact_9385_less__eq__integer_Orep__eq,axiom,
( ord_le3102999989581377725nteger
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_9386_integer__less__eq__iff,axiom,
( ord_le3102999989581377725nteger
= ( ^ [K3: code_integer,L: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% integer_less_eq_iff
thf(fact_9387_integer__of__num__def,axiom,
code_integer_of_num = numera6620942414471956472nteger ).
% integer_of_num_def
thf(fact_9388_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one )
= one_one_Code_integer ) ).
% integer_of_num_triv(1)
thf(fact_9389_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_9390_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% integer_of_num_triv(2)
thf(fact_9391_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).
% divmod_integer_def
thf(fact_9392_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_9393_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_9394_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_9395_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
=> ( ( code_nat_of_integer @ K )
= zero_zero_nat ) ) ).
% nat_of_integer_non_positive
thf(fact_9396_nat__of__integer__code__post_I3_J,axiom,
! [K: num] :
( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_of_integer_code_post(3)
thf(fact_9397_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
@ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_9398_csqrt_Osimps_I1_J,axiom,
! [Z: complex] :
( ( re @ ( csqrt @ Z ) )
= ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_9399_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_9400_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_9401_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_9402_card__atLeastAtMost,axiom,
! [L2: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% card_atLeastAtMost
thf(fact_9403_complex__Re__numeral,axiom,
! [V: num] :
( ( re @ ( numera6690914467698888265omplex @ V ) )
= ( numeral_numeral_real @ V ) ) ).
% complex_Re_numeral
thf(fact_9404_card__atLeastAtMost__int,axiom,
! [L2: int,U: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_9405_Re__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_9406_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_Re_le_cmod
thf(fact_9407_plus__complex_Osimps_I1_J,axiom,
! [X: complex,Y2: complex] :
( ( re @ ( plus_plus_complex @ X @ Y2 ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y2 ) ) ) ).
% plus_complex.simps(1)
thf(fact_9408_scaleR__complex_Osimps_I1_J,axiom,
! [R2: real,X: complex] :
( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
= ( times_times_real @ R2 @ ( re @ X ) ) ) ).
% scaleR_complex.simps(1)
thf(fact_9409_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% abs_Re_le_cmod
thf(fact_9410_Re__csqrt,axiom,
! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% Re_csqrt
thf(fact_9411_card__less,axiom,
! [M7: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_9412_card__less__Suc,axiom,
! [M7: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_9413_card__less__Suc2,axiom,
! [M7: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_9414_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
=> ( A2
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_9415_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_9416_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% divmod_abs_code(6)
thf(fact_9417_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_9418_card__sum__le__nat__sum,axiom,
! [S3: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_9419_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_9420_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_9421_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
= ( ( re @ Z )
= ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_9422_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
= ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_9423_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% divmod_abs_def
thf(fact_9424_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_9425_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z2: complex] :
( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( times_times_real
@ ( if_real
@ ( ( im @ Z2 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z2 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_9426_csqrt_Osimps_I2_J,axiom,
! [Z: complex] :
( ( im @ ( csqrt @ Z ) )
= ( times_times_real
@ ( if_real
@ ( ( im @ Z )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_9427_Im__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= zero_zero_real )
=> ( ( im @ ( power_power_complex @ X @ N ) )
= zero_zero_real ) ) ).
% Im_power_real
thf(fact_9428_complex__Im__numeral,axiom,
! [V: num] :
( ( im @ ( numera6690914467698888265omplex @ V ) )
= zero_zero_real ) ).
% complex_Im_numeral
thf(fact_9429_Im__i__times,axiom,
! [Z: complex] :
( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
= ( re @ Z ) ) ).
% Im_i_times
thf(fact_9430_Re__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= zero_zero_real )
=> ( ( re @ ( power_power_complex @ X @ N ) )
= ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% Re_power_real
thf(fact_9431_Re__i__times,axiom,
! [Z: complex] :
( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
= ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% Re_i_times
thf(fact_9432_Im__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_9433_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( ( im @ X )
= zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
=> ( ( csqrt @ X )
= ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_9434_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
| ( ( ( im @ X )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
= ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_9435_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= zero_zero_real )
=> ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
=> ( ( csqrt @ X )
= ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_9436_plus__complex_Osimps_I2_J,axiom,
! [X: complex,Y2: complex] :
( ( im @ ( plus_plus_complex @ X @ Y2 ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y2 ) ) ) ).
% plus_complex.simps(2)
thf(fact_9437_scaleR__complex_Osimps_I2_J,axiom,
! [R2: real,X: complex] :
( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
= ( times_times_real @ R2 @ ( im @ X ) ) ) ).
% scaleR_complex.simps(2)
thf(fact_9438_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% abs_Im_le_cmod
thf(fact_9439_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y2: complex] :
( ( im @ ( times_times_complex @ X @ Y2 ) )
= ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y2 ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y2 ) ) ) ) ).
% times_complex.simps(2)
thf(fact_9440_cmod__Re__le__iff,axiom,
! [X: complex,Y2: complex] :
( ( ( im @ X )
= ( im @ Y2 ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) )
= ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y2 ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_9441_cmod__Im__le__iff,axiom,
! [X: complex,Y2: complex] :
( ( ( re @ X )
= ( re @ Y2 ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y2 ) )
= ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y2 ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_9442_times__complex_Osimps_I1_J,axiom,
! [X: complex,Y2: complex] :
( ( re @ ( times_times_complex @ X @ Y2 ) )
= ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y2 ) ) ) ) ).
% times_complex.simps(1)
thf(fact_9443_plus__complex_Ocode,axiom,
( plus_plus_complex
= ( ^ [X2: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ) ).
% plus_complex.code
thf(fact_9444_scaleR__complex_Ocode,axiom,
( real_V2046097035970521341omplex
= ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% scaleR_complex.code
thf(fact_9445_csqrt__principal,axiom,
! [Z: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
| ( ( ( re @ ( csqrt @ Z ) )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% csqrt_principal
thf(fact_9446_cmod__le,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% cmod_le
thf(fact_9447_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
= ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_9448_Re__exp,axiom,
! [Z: complex] :
( ( re @ ( exp_complex @ Z ) )
= ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% Re_exp
thf(fact_9449_Im__exp,axiom,
! [Z: complex] :
( ( im @ ( exp_complex @ Z ) )
= ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% Im_exp
thf(fact_9450_complex__eq,axiom,
! [A: complex] :
( A
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% complex_eq
thf(fact_9451_times__complex_Ocode,axiom,
( times_times_complex
= ( ^ [X2: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) ) ) ) ) ).
% times_complex.code
thf(fact_9452_exp__eq__polar,axiom,
( exp_complex
= ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).
% exp_eq_polar
thf(fact_9453_cmod__power2,axiom,
! [Z: complex] :
( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cmod_power2
thf(fact_9454_Im__power2,axiom,
! [X: complex] :
( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% Im_power2
thf(fact_9455_Re__power2,axiom,
! [X: complex] :
( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Re_power2
thf(fact_9456_complex__eq__0,axiom,
! [Z: complex] :
( ( Z = zero_zero_complex )
= ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ) ).
% complex_eq_0
thf(fact_9457_norm__complex__def,axiom,
( real_V1022390504157884413omplex
= ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_9458_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( invers8013647133539491842omplex @ X ) )
= ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_9459_complex__neq__0,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
= ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_9460_Re__divide,axiom,
! [X: complex,Y2: complex] :
( ( re @ ( divide1717551699836669952omplex @ X @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Re_divide
thf(fact_9461_csqrt__unique,axiom,
! [W: complex,Z: complex] :
( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z )
=> ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
| ( ( ( re @ W )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z )
= W ) ) ) ).
% csqrt_unique
thf(fact_9462_csqrt__square,axiom,
! [B: complex] :
( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
| ( ( ( re @ B )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
=> ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= B ) ) ).
% csqrt_square
thf(fact_9463_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( invers8013647133539491842omplex @ X ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_9464_Im__divide,axiom,
! [X: complex,Y2: complex] :
( ( im @ ( divide1717551699836669952omplex @ X @ Y2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Im_divide
thf(fact_9465_complex__abs__le__norm,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% complex_abs_le_norm
thf(fact_9466_complex__unit__circle,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ) ).
% complex_unit_circle
thf(fact_9467_inverse__complex_Ocode,axiom,
( invers8013647133539491842omplex
= ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_9468_Complex__divide,axiom,
( divide1717551699836669952omplex
= ( ^ [X2: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_9469_Im__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( ( member_complex @ R2 @ real_V2521375963428798218omplex )
=> ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
= ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_9470_imaginary__eq__real__iff,axiom,
! [Y2: complex,X: complex] :
( ( member_complex @ Y2 @ real_V2521375963428798218omplex )
=> ( ( member_complex @ X @ real_V2521375963428798218omplex )
=> ( ( ( times_times_complex @ imaginary_unit @ Y2 )
= X )
= ( ( X = zero_zero_complex )
& ( Y2 = zero_zero_complex ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_9471_real__eq__imaginary__iff,axiom,
! [Y2: complex,X: complex] :
( ( member_complex @ Y2 @ real_V2521375963428798218omplex )
=> ( ( member_complex @ X @ real_V2521375963428798218omplex )
=> ( ( X
= ( times_times_complex @ imaginary_unit @ Y2 ) )
= ( ( X = zero_zero_complex )
& ( Y2 = zero_zero_complex ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_9472_Re__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( ( member_complex @ R2 @ real_V2521375963428798218omplex )
=> ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
= ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_9473_complex__mult__cnj,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( cnj @ Z ) )
= ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_9474_cnj__add__mult__eq__Re,axiom,
! [Z: complex,W: complex] :
( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_9475_complex__div__cnj,axiom,
( divide1717551699836669952omplex
= ( ^ [A4: complex,B3: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B3 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_9476_complex__cnj__mult,axiom,
! [X: complex,Y2: complex] :
( ( cnj @ ( times_times_complex @ X @ Y2 ) )
= ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y2 ) ) ) ).
% complex_cnj_mult
thf(fact_9477_complex__cnj__power,axiom,
! [X: complex,N: nat] :
( ( cnj @ ( power_power_complex @ X @ N ) )
= ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
% complex_cnj_power
thf(fact_9478_complex__cnj__add,axiom,
! [X: complex,Y2: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y2 ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y2 ) ) ) ).
% complex_cnj_add
thf(fact_9479_complex__cnj__numeral,axiom,
! [W: num] :
( ( cnj @ ( numera6690914467698888265omplex @ W ) )
= ( numera6690914467698888265omplex @ W ) ) ).
% complex_cnj_numeral
thf(fact_9480_complex__cnj__neg__numeral,axiom,
! [W: num] :
( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% complex_cnj_neg_numeral
thf(fact_9481_complex__In__mult__cnj__zero,axiom,
! [Z: complex] :
( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
= zero_zero_real ) ).
% complex_In_mult_cnj_zero
thf(fact_9482_Re__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
= zero_zero_real )
= ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
= zero_zero_real ) ) ).
% Re_complex_div_eq_0
thf(fact_9483_Im__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
= zero_zero_real )
= ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
= zero_zero_real ) ) ).
% Im_complex_div_eq_0
thf(fact_9484_complex__mod__sqrt__Re__mult__cnj,axiom,
( real_V1022390504157884413omplex
= ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).
% complex_mod_sqrt_Re_mult_cnj
thf(fact_9485_Re__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_9486_Re__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Re_complex_div_lt_0
thf(fact_9487_Re__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Re_complex_div_le_0
thf(fact_9488_Re__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_9489_Im__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_9490_Im__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Im_complex_div_lt_0
thf(fact_9491_Im__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Im_complex_div_le_0
thf(fact_9492_Im__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_9493_complex__mod__mult__cnj,axiom,
! [Z: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_9494_complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
& ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_9495_complex__norm__square,axiom,
! [Z: complex] :
( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% complex_norm_square
thf(fact_9496_complex__add__cnj,axiom,
! [Z: complex] :
( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% complex_add_cnj
thf(fact_9497_complex__diff__cnj,axiom,
! [Z: complex] :
( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
= ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_9498_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu2: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_9499_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu2: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_9500_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_9501_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N @ M3 ) )
@ M ) ) ).
% max_Suc1
thf(fact_9502_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_9503_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K3: nat] : K3
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_9504_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_9505_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X2: real] :
( the_int
@ ^ [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X2 )
& ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_9506_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X2: rat] :
( the_int
@ ^ [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 )
& ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_9507_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_9508_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_9509_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% less_eq_rat_def
thf(fact_9510_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S: rat] :
( ( ord_less_rat @ zero_zero_rat @ S )
=> ! [T4: rat] :
( ( ord_less_rat @ zero_zero_rat @ T4 )
=> ( R2
!= ( plus_plus_rat @ S @ T4 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_9511_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y2: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y2 )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y2
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y2
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_9512_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y2: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y2 )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y2
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y2
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_9513_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B3 ) @ ( abs_abs_int @ A4 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_9514_normalize__negative,axiom,
! [Q2: int,P4: int] :
( ( ord_less_int @ Q2 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_9515_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% rat_one_code
thf(fact_9516_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral_rat @ K ) )
= ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% quotient_of_number(3)
thf(fact_9517_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% quotient_of_number(4)
thf(fact_9518_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% normalize_denom_zero
thf(fact_9519_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% quotient_of_number(5)
thf(fact_9520_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% rat_zero_code
thf(fact_9521_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_9522_diff__rat__def,axiom,
( minus_minus_rat
= ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_9523_rat__divide__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C3 @ B3 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_9524_rat__times__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B3 ) @ ( times_times_int @ C3 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_9525_quotient__of__div,axiom,
! [R2: rat,N: int,D: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ N @ D ) )
=> ( R2
= ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% quotient_of_div
thf(fact_9526_rat__plus__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_9527_rat__minus__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_9528_quotient__of__denom__pos,axiom,
! [R2: rat,P4: int,Q2: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ P4 @ Q2 ) )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% quotient_of_denom_pos
thf(fact_9529_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_9530_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs_rat @ P4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int] : ( product_Pair_int_int @ ( abs_abs_int @ A4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_9531_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P4: int,Q2: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P4 @ Q2 ) )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_9532_normalize__crossproduct,axiom,
! [Q2: int,S2: int,P4: int,R2: int] :
( ( Q2 != zero_zero_int )
=> ( ( S2 != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
= ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
=> ( ( times_times_int @ P4 @ S2 )
= ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_9533_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P5: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A4: int,C3: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D2: int] : ( ord_less_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C3 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_9534_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P5: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A4: int,C3: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C3 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_9535_quotient__of__int,axiom,
! [A: int] :
( ( quotient_of @ ( of_int @ A ) )
= ( product_Pair_int_int @ A @ one_one_int ) ) ).
% quotient_of_int
thf(fact_9536_drop__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_9537_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
= ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_9538_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_9539_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% drop_bit_negative_int_iff
thf(fact_9540_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% drop_bit_minus_one
thf(fact_9541_fst__divmod__nat,axiom,
! [M: nat,N: nat] :
( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
= ( divide_divide_nat @ M @ N ) ) ).
% fst_divmod_nat
thf(fact_9542_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_9543_drop__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_9544_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_9545_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K: int] :
( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
= ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_9546_drop__bit__int__def,axiom,
( bit_se8568078237143864401it_int
= ( ^ [N2: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% drop_bit_int_def
thf(fact_9547_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_9548_Frct__code__post_I6_J,axiom,
! [K: num,L2: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
= ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% Frct_code_post(6)
thf(fact_9549_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
= ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_9550_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_9551_drop__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% drop_bit_nat_eq
thf(fact_9552_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% rat_sgn_code
thf(fact_9553_drop__bit__nat__def,axiom,
( bit_se8570568707652914677it_nat
= ( ^ [N2: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_9554_Frct__code__post_I1_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
= zero_zero_rat ) ).
% Frct_code_post(1)
thf(fact_9555_Frct__code__post_I2_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
= zero_zero_rat ) ).
% Frct_code_post(2)
thf(fact_9556_Frct__code__post_I8_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(8)
thf(fact_9557_Frct__code__post_I7_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(7)
thf(fact_9558_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
% Frct_code_post(3)
thf(fact_9559_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
= ( numeral_numeral_rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_9560_bezw__non__0,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y2 )
=> ( ( bezw @ X @ Y2 )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_9561_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_9562_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y2: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y2 )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y2
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y2
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_9563_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y2: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y2 )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y2
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y2
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_9564_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_9565_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_9566_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_9567_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_9568_normalize__def,axiom,
( normalize
= ( ^ [P5: product_prod_int_int] :
( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_snd_int_int @ P5 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_9569_finite__enumerate,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ? [R3: nat > nat] :
( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
& ! [N7: nat] :
( ( ord_less_nat @ N7 @ ( finite_card_nat @ S3 ) )
=> ( member_nat @ ( R3 @ N7 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_9570_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_9571_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
= ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_9572_gcd__ge__0__int,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y2 ) ) ).
% gcd_ge_0_int
thf(fact_9573_bezout__int,axiom,
! [X: int,Y2: int] :
? [U3: int,V2: int] :
( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y2 ) )
= ( gcd_gcd_int @ X @ Y2 ) ) ).
% bezout_int
thf(fact_9574_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
= ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_9575_gcd__le2__int,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% gcd_le2_int
thf(fact_9576_gcd__le1__int,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% gcd_le1_int
thf(fact_9577_gcd__cases__int,axiom,
! [X: int,Y2: int,P: int > $o] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( P @ ( gcd_gcd_int @ X @ Y2 ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y2 ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y2 ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y2 ) ) ) ) )
=> ( P @ ( gcd_gcd_int @ X @ Y2 ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_9578_gcd__unique__int,axiom,
! [D: int,A: int,B: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ D @ A )
& ( dvd_dvd_int @ D @ B )
& ! [E3: int] :
( ( ( dvd_dvd_int @ E3 @ A )
& ( dvd_dvd_int @ E3 @ B ) )
=> ( dvd_dvd_int @ E3 @ D ) ) )
= ( D
= ( gcd_gcd_int @ A @ B ) ) ) ).
% gcd_unique_int
thf(fact_9579_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_9580_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_9581_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
= ( ( M != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
thf(fact_9582_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_9583_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% gcd_le2_nat
thf(fact_9584_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% gcd_le1_nat
thf(fact_9585_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_9586_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_9587_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X3: nat,Y3: nat] :
( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_nat
thf(fact_9588_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_9589_bezw__aux,axiom,
! [X: nat,Y2: nat] :
( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y2 ) )
= ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y2 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y2 ) ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) ) ).
% bezw_aux
thf(fact_9590_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y2: nat] :
( ( ( gcd_gcd_nat @ X @ Xa2 )
= Y2 )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y2 = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y2
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_9591_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_9592_card__greaterThanLessThan__int,axiom,
! [L2: int,U: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_9593_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_9594_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_9595_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_9596_real__root__eq__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y2 ) )
= ( X = Y2 ) ) ) ).
% real_root_eq_iff
thf(fact_9597_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_9598_real__root__less__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ).
% real_root_less_iff
thf(fact_9599_real__root__le__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% real_root_le_iff
thf(fact_9600_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_9601_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_9602_real__root__gt__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y2 ) )
= ( ord_less_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_gt_0_iff
thf(fact_9603_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_9604_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_9605_real__root__ge__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_ge_0_iff
thf(fact_9606_real__root__gt__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y2 ) )
= ( ord_less_real @ one_one_real @ Y2 ) ) ) ).
% real_root_gt_1_iff
thf(fact_9607_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_9608_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_9609_real__root__ge__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ one_one_real @ Y2 ) ) ) ).
% real_root_ge_1_iff
thf(fact_9610_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_9611_real__root__minus,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% real_root_minus
thf(fact_9612_real__root__divide,axiom,
! [N: nat,X: real,Y2: real] :
( ( root @ N @ ( divide_divide_real @ X @ Y2 ) )
= ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ).
% real_root_divide
thf(fact_9613_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ ( times_times_nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_9614_real__root__mult,axiom,
! [N: nat,X: real,Y2: real] :
( ( root @ N @ ( times_times_real @ X @ Y2 ) )
= ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ).
% real_root_mult
thf(fact_9615_real__root__commute,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ M @ ( root @ N @ X ) )
= ( root @ N @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_9616_real__root__inverse,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% real_root_inverse
thf(fact_9617_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_9618_real__root__less__mono,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_less_mono
thf(fact_9619_real__root__le__mono,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_le_mono
thf(fact_9620_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_9621_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_9622_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( sgn_sgn_real @ ( root @ N @ X ) )
= ( sgn_sgn_real @ X ) ) ) ).
% sgn_root
thf(fact_9623_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L2: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9624_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_9625_real__root__strict__decreasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_9626_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% sqrt_def
thf(fact_9627_root__abs__power,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y2 @ N ) ) )
= ( abs_abs_real @ Y2 ) ) ) ).
% root_abs_power
thf(fact_9628_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_9629_real__root__strict__increasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_9630_real__root__decreasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_9631_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_9632_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_9633_real__root__pos__unique,axiom,
! [N: nat,Y2: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( power_power_real @ Y2 @ N )
= X )
=> ( ( root @ N @ X )
= Y2 ) ) ) ) ).
% real_root_pos_unique
thf(fact_9634_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ).
% odd_real_root_pow
thf(fact_9635_odd__real__root__unique,axiom,
! [N: nat,Y2: real,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ( power_power_real @ Y2 @ N )
= X )
=> ( ( root @ N @ X )
= Y2 ) ) ) ).
% odd_real_root_unique
thf(fact_9636_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_9637_real__root__increasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_9638_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_9639_root__sgn__power,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) ) )
= Y2 ) ) ).
% root_sgn_power
thf(fact_9640_ln__root,axiom,
! [N: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_9641_log__root,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ B @ ( root @ N @ A ) )
= ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_9642_log__base__root,axiom,
! [N: nat,B: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log @ ( root @ N @ B ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% log_base_root
thf(fact_9643_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_real ) )
& ( ( ord_less_nat @ zero_zero_nat @ N )
=> ! [Y: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
= X )
=> ( P @ Y ) ) ) ) ) ).
% split_root
thf(fact_9644_bij__betw__nth__root__unity,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= one_one_complex ) )
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= C ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_9645_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_9646_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_9647_card__greaterThanLessThan,axiom,
! [L2: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% card_greaterThanLessThan
thf(fact_9648_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L2: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
= ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_9649_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_9650_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_9651_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
@ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9652_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K3 )
= ( sgn_sgn_Code_integer @ L ) )
@ ( code_divmod_abs @ K3 @ L )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_9653_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
@ Xa2
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_9654_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X3: nat,Y3: nat] :
( Z
!= ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_9655_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9656_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9657_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_9658_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9659_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
@ Xa2
@ X ) ) ).
% less_int.abs_eq
thf(fact_9660_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_9661_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
@ Xa2
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_9662_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
@ Xa2
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_9663_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_9664_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% Code_Numeral.negative_def
thf(fact_9665_Code__Target__Int_Onegative__def,axiom,
( code_Target_negative
= ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% Code_Target_Int.negative_def
thf(fact_9666_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y: nat,Z2: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9667_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y: nat,Z2: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9668_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_9669_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_9670_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_9671_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_9672_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_9673_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M6: nat,N2: nat] :
( N2
= ( suc @ M6 ) ) ) ) ) ).
% pred_nat_def
thf(fact_9674_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_9675_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_9676_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_9677_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_9678_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_9679_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_9680_pow_Osimps_I3_J,axiom,
! [X: num,Y2: num] :
( ( pow @ X @ ( bit1 @ Y2 ) )
= ( times_times_num @ ( sqr @ ( pow @ X @ Y2 ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_9681_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( Xa2 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_9682_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_9683_sqr_Osimps_I1_J,axiom,
( ( sqr @ one )
= one ) ).
% sqr.simps(1)
thf(fact_9684_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_9685_pow_Osimps_I2_J,axiom,
! [X: num,Y2: num] :
( ( pow @ X @ ( bit0 @ Y2 ) )
= ( sqr @ ( pow @ X @ Y2 ) ) ) ).
% pow.simps(2)
thf(fact_9686_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_9687_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_9688_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( Y2
= ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y2
= ( ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_9689_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( Xa2 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_9690_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y2: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( Y2
= ( Xa2 = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y2
= ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_9691_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_9692_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_9693_Sup__int__def,axiom,
( complete_Sup_Sup_int
= ( ^ [X6: set_int] :
( the_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X6 )
& ! [Y: int] :
( ( member_int @ Y @ X6 )
=> ( ord_less_eq_int @ Y @ X2 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_9694_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int
@ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_9695_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_9696_take__bit__num__simps_I1_J,axiom,
! [M: num] :
( ( bit_take_bit_num @ zero_zero_nat @ M )
= none_num ) ).
% take_bit_num_simps(1)
thf(fact_9697_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(2)
thf(fact_9698_take__bit__num__simps_I5_J,axiom,
! [R2: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(5)
thf(fact_9699_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9700_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_9701_take__bit__num__simps_I6_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_9702_take__bit__num__simps_I7_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_9703_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_9704_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_9705_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_9706_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
= ( case_nat_option_num @ none_num
@ ^ [N2: nat] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N2 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9707_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one @ one )
= none_num ) ).
% and_not_num.simps(1)
thf(fact_9708_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one )
= ( case_nat_option_num @ none_num
@ ^ [N2: nat] : ( some_num @ one )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9709_and__not__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(4)
thf(fact_9710_and__not__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one @ ( bit0 @ N ) )
= ( some_num @ one ) ) ).
% and_not_num.simps(2)
thf(fact_9711_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_9712_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_9713_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_9714_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one @ ( bit1 @ N ) )
= none_num ) ).
% and_not_num.simps(3)
thf(fact_9715_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
= ( case_nat_option_num @ none_num
@ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9716_and__not__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(7)
thf(fact_9717_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q2: num] :
( ( ( bit_and_not_num @ M @ N )
= ( some_num @ Q2 ) )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ Q2 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_9718_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_9719_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= none_num )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= zero_zero_int ) ) ).
% and_not_num_eq_None_iff
thf(fact_9720_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( 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
thf(fact_9721_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_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
thf(fact_9722_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M6: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_9723_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M6: num] :
( produc478579273971653890on_num
@ ^ [A4: nat,X2: num] :
( case_nat_option_num @ none_num
@ ^ [O: nat] :
( case_num_option_num @ ( some_num @ one )
@ ^ [P5: num] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P5 ) )
@ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
@ X2 )
@ A4 )
@ ( product_Pair_nat_num @ N2 @ M6 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_9724_Rats__eq__int__div__nat,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu2: real] :
? [I3: int,N2: nat] :
( ( Uu2
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( N2 != zero_zero_nat ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_9725_Rats__abs__iff,axiom,
! [X: real] :
( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
= ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% Rats_abs_iff
thf(fact_9726_Rats__no__bot__less,axiom,
! [X: real] :
? [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
& ( ord_less_real @ X3 @ X ) ) ).
% Rats_no_bot_less
thf(fact_9727_Rats__dense__in__real,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
& ( ord_less_real @ X @ X3 )
& ( ord_less_real @ X3 @ Y2 ) ) ) ).
% Rats_dense_in_real
thf(fact_9728_Rats__no__top__le,axiom,
! [X: real] :
? [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
& ( ord_less_eq_real @ X @ X3 ) ) ).
% Rats_no_top_le
thf(fact_9729_Rats__eq__int__div__int,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu2: real] :
? [I3: int,J3: int] :
( ( Uu2
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
& ( J3 != zero_zero_int ) ) ) ) ).
% Rats_eq_int_div_int
thf(fact_9730_and__not__num_Oelims,axiom,
! [X: num,Xa2: num,Y2: option_num] :
( ( ( bit_and_not_num @ X @ Xa2 )
= Y2 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y2 != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y2 != none_num ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa2 = one )
=> ( Y2
!= ( some_num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa2 = one )
=> ( Y2
!= ( some_num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_9731_xor__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% xor_num.simps(8)
thf(fact_9732_xor__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% xor_num.simps(6)
thf(fact_9733_xor__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% xor_num.simps(9)
thf(fact_9734_xor__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% xor_num.simps(5)
thf(fact_9735_and__not__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(5)
thf(fact_9736_and__not__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(6)
thf(fact_9737_and__not__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(9)
thf(fact_9738_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one @ one )
= none_num ) ).
% xor_num.simps(1)
thf(fact_9739_xor__num_Oelims,axiom,
! [X: num,Xa2: num,Y2: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
= Y2 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y2 != none_num ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( some_num @ ( bit1 @ N3 ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( some_num @ ( bit0 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa2 = one )
=> ( Y2
!= ( some_num @ ( bit1 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa2 = one )
=> ( Y2
!= ( some_num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_9740_xor__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% xor_num.simps(7)
thf(fact_9741_xor__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
= ( some_num @ ( bit1 @ M ) ) ) ).
% xor_num.simps(4)
thf(fact_9742_xor__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
= ( some_num @ ( bit0 @ N ) ) ) ).
% xor_num.simps(3)
thf(fact_9743_xor__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
= ( some_num @ ( bit1 @ N ) ) ) ).
% xor_num.simps(2)
thf(fact_9744_and__not__num_Opelims,axiom,
! [X: num,Xa2: num,Y2: option_num] :
( ( ( bit_and_not_num @ X @ Xa2 )
= Y2 )
=> ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y2 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( some_num @ ( bit0 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( some_num @ ( bit0 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_9745_xor__num_Opelims,axiom,
! [X: num,Xa2: num,Y2: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
= Y2 )
=> ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y2 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( some_num @ ( bit1 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( some_num @ ( bit0 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( some_num @ ( bit1 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( some_num @ ( bit0 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_9746_and__num_Oelims,axiom,
! [X: num,Xa2: num,Y2: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
= Y2 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y2
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y2 != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( some_num @ one ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit0 @ M5 ) )
=> ( ( Xa2 = one )
=> ( Y2 != none_num ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit1 @ M5 ) )
=> ( ( Xa2 = one )
=> ( Y2
!= ( some_num @ one ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y2
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y2
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_9747_xor__num__rel__dict,axiom,
bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% xor_num_rel_dict
thf(fact_9748_xor__num__dict,axiom,
bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% xor_num_dict
thf(fact_9749_and__num_Osimps_I1_J,axiom,
( ( bit_un7362597486090784418nd_num @ one @ one )
= ( some_num @ one ) ) ).
% and_num.simps(1)
thf(fact_9750_and__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(5)
thf(fact_9751_and__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
= ( some_num @ one ) ) ).
% and_num.simps(7)
thf(fact_9752_and__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
= ( some_num @ one ) ) ).
% and_num.simps(3)
thf(fact_9753_and__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
= none_num ) ).
% and_num.simps(4)
thf(fact_9754_and__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
= none_num ) ).
% and_num.simps(2)
thf(fact_9755_and__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(6)
thf(fact_9756_and__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(8)
thf(fact_9757_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_9758_and__num_Opelims,axiom,
! [X: num,Xa2: num,Y2: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
= Y2 )
=> ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa2 = one )
=> ( ( Y2 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa2 = one )
=> ( ( Y2
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y2
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y2
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_9759_and__num__dict,axiom,
bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% and_num_dict
thf(fact_9760_and__num__rel__dict,axiom,
bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% and_num_rel_dict
thf(fact_9761_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% rat_floor_lemma
thf(fact_9762_mult__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% mult_rat
thf(fact_9763_divide__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% divide_rat
thf(fact_9764_less__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% less_rat
thf(fact_9765_add__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% add_rat
thf(fact_9766_le__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% le_rat
thf(fact_9767_diff__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% diff_rat
thf(fact_9768_sgn__rat,axiom,
! [A: int,B: int] :
( ( sgn_sgn_rat @ ( fract @ A @ B ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% sgn_rat
thf(fact_9769_eq__rat_I1_J,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ( fract @ A @ B )
= ( fract @ C @ D ) )
= ( ( times_times_int @ A @ D )
= ( times_times_int @ C @ B ) ) ) ) ) ).
% eq_rat(1)
thf(fact_9770_mult__rat__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( fract @ A @ B ) ) ) ).
% mult_rat_cancel
thf(fact_9771_quotient__of__eq,axiom,
! [A: int,B: int,P4: int,Q2: int] :
( ( ( quotient_of @ ( fract @ A @ B ) )
= ( product_Pair_int_int @ P4 @ Q2 ) )
=> ( ( fract @ P4 @ Q2 )
= ( fract @ A @ B ) ) ) ).
% quotient_of_eq
thf(fact_9772_normalize__eq,axiom,
! [A: int,B: int,P4: int,Q2: int] :
( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ P4 @ Q2 ) )
=> ( ( fract @ P4 @ Q2 )
= ( fract @ A @ B ) ) ) ).
% normalize_eq
thf(fact_9773_rat__number__expand_I3_J,axiom,
( numeral_numeral_rat
= ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).
% rat_number_expand(3)
thf(fact_9774_rat__number__collapse_I3_J,axiom,
! [W: num] :
( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
= ( numeral_numeral_rat @ W ) ) ).
% rat_number_collapse(3)
thf(fact_9775_quotient__of__Fract,axiom,
! [A: int,B: int] :
( ( quotient_of @ ( fract @ A @ B ) )
= ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% quotient_of_Fract
thf(fact_9776_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
= ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% Fract_add_one
thf(fact_9777_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% Fract_le_zero_iff
thf(fact_9778_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_Fract_iff
thf(fact_9779_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% Fract_le_one_iff
thf(fact_9780_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% one_le_Fract_iff
thf(fact_9781_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
= ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% rat_number_expand(5)
thf(fact_9782_rat__number__collapse_I4_J,axiom,
! [W: num] :
( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% rat_number_collapse(4)
thf(fact_9783_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y2: nat,X: nat] :
( ( ( ord_less_nat @ C @ Y2 )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y2 ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y2 @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y2 )
=> ( ( ( ord_less_nat @ X @ Y2 )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y2 ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y2 ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_9784_positive__rat,axiom,
! [A: int,B: int] :
( ( positive @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% positive_rat
thf(fact_9785_bij__betw__Suc,axiom,
! [M7: set_nat,N5: set_nat] :
( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
= ( ( image_nat_nat @ suc @ M7 )
= N5 ) ) ).
% bij_betw_Suc
thf(fact_9786_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_9787_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_9788_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_9789_Rat_Opositive__add,axiom,
! [X: rat,Y2: rat] :
( ( positive @ X )
=> ( ( positive @ Y2 )
=> ( positive @ ( plus_plus_rat @ X @ Y2 ) ) ) ) ).
% Rat.positive_add
thf(fact_9790_Rat_Opositive__mult,axiom,
! [X: rat,Y2: rat] :
( ( positive @ X )
=> ( ( positive @ Y2 )
=> ( positive @ ( times_times_rat @ X @ Y2 ) ) ) ) ).
% Rat.positive_mult
thf(fact_9791_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_9792_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_9793_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9794_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9795_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_9796_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_9797_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_9798_Inf__real__def,axiom,
( comple4887499456419720421f_real
= ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% Inf_real_def
thf(fact_9799_finite__int__iff__bounded__le,axiom,
( finite_finite_int
= ( ^ [S4: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_9800_finite__int__iff__bounded,axiom,
( finite_finite_int
= ( ^ [S4: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_9801_suminf__eq__SUP__real,axiom,
! [X8: nat > real] :
( ( summable_real @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I4 ) )
=> ( ( suminf_real @ X8 )
= ( comple1385675409528146559p_real
@ ( image_nat_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
@ top_top_set_nat ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_9802_image__add__int__atLeastLessThan,axiom,
! [L2: int,U: int] :
( ( image_int_int
@ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
= ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9803_range__mod,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_nat
@ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% range_mod
thf(fact_9804_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_9805_UNIV__nat__eq,axiom,
( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% UNIV_nat_eq
thf(fact_9806_card__UNIV__bool,axiom,
( ( finite_card_o @ top_top_set_o )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card_UNIV_bool
thf(fact_9807_range__mult,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
& ( ( A != zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% range_mult
thf(fact_9808_root__def,axiom,
( root
= ( ^ [N2: nat,X2: real] :
( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_9809_card__UNIV__char,axiom,
( ( finite_card_char @ top_top_set_char )
= ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_9810_UNIV__char__of__nat,axiom,
( top_top_set_char
= ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_9811_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_9812_nat__of__char__less__256,axiom,
! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_9813_range__nat__of__char,axiom,
( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_9814_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_9815_String_Ochar__of__ascii__of,axiom,
! [C: char] :
( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
= ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% String.char_of_ascii_of
thf(fact_9816_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_9817_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D4
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D4
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( D4
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_9818_DERIV__neg__dec__right,axiom,
! [F: real > real,L2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L2 @ zero_zero_real )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_9819_DERIV__pos__inc__right,axiom,
! [F: real > real,L2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_9820_DERIV__pos__imp__increasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_9821_DERIV__neg__imp__decreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_9822_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L2: real,X: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
=> ( ( ord_less_real @ L2 @ zero_zero_real )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_9823_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L2: real,X: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
=> ( ( ord_less_real @ zero_zero_real @ L2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_9824_DERIV__nonpos__imp__nonincreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
=> ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_9825_DERIV__nonneg__imp__nondecreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_9826_deriv__nonneg__imp__mono,axiom,
! [A: real,B: real,G: real > real,G2: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_9827_DERIV__const__ratio__const,axiom,
! [A: real,B: real,F: real > real,K: real] :
( ( A != B )
=> ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_9828_MVT2,axiom,
! [A: real,B: real,F: real > real,F4: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_9829_DERIV__const__average,axiom,
! [A: real,B: real,V: real > real,K: real] :
( ( A != B )
=> ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_9830_DERIV__local__min,axiom,
! [F: real > real,L2: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( L2 = zero_zero_real ) ) ) ) ).
% DERIV_local_min
thf(fact_9831_DERIV__local__max,axiom,
! [F: real > real,L2: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( L2 = zero_zero_real ) ) ) ) ).
% DERIV_local_max
thf(fact_9832_DERIV__pow,axiom,
! [N: nat,X: real,S2: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( power_power_real @ X2 @ N )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).
% DERIV_pow
thf(fact_9833_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N: nat] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N )
@ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_fun_pow
thf(fact_9834_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( has_fi5821293074295781190e_real
@ ^ [Z2: real] : ( powr_real @ Z2 @ R2 )
@ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_9835_DERIV__series_H,axiom,
! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
( ! [N3: nat] :
( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( F @ X2 @ N3 )
@ ( F4 @ X0 @ N3 )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( summable_real @ ( F @ X3 ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( summable_real @ ( F4 @ X0 ) )
=> ( ( summable_real @ L5 )
=> ( ! [N3: nat,X3: real,Y3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N3 ) @ ( F @ Y3 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
@ ( suminf_real @ ( F4 @ X0 ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_9836_DERIV__log,axiom,
! [X: real,B: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_log
thf(fact_9837_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R2 )
@ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_9838_DERIV__powr,axiom,
! [G: real > real,M: real,X: real,F: real > real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
@ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_powr
thf(fact_9839_artanh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_9840_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_real_sqrt
thf(fact_9841_DERIV__arctan,axiom,
! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% DERIV_arctan
thf(fact_9842_arsinh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_9843_DERIV__real__sqrt__generic,axiom,
! [X: real,D4: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D4
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D4
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_9844_arcosh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] :
( ( ord_less_real @ one_one_real @ X )
=> ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_9845_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) )
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_9846_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_real_root
thf(fact_9847_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_arccos
thf(fact_9848_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_arcsin
thf(fact_9849_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ? [T4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_9850_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ? [T4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_9851_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( X != zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_9852_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ T4 )
& ( ord_less_real @ T4 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_9853_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_9854_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H2 @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ H2 @ T4 )
& ( ord_less_eq_real @ T4 @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ? [T4: real] :
( ( ord_less_real @ H2 @ T4 )
& ( ord_less_real @ T4 @ zero_zero_real )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_9855_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ? [T4: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
& ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_9856_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ? [T4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_9857_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ A @ T4 )
& ( ord_less_eq_real @ T4 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( X != C )
=> ? [T4: real] :
( ( ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ X @ T4 )
& ( ord_less_real @ T4 @ C ) ) )
& ( ~ ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ C @ T4 )
& ( ord_less_real @ T4 @ X ) ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_9858_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ A @ T4 )
& ( ord_less_eq_real @ T4 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_real @ C @ B )
=> ? [T4: real] :
( ( ord_less_real @ C @ T4 )
& ( ord_less_real @ T4 @ B )
& ( ( F @ B )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_9859_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ A @ T4 )
& ( ord_less_eq_real @ T4 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ? [T4: real] :
( ( ord_less_real @ A @ T4 )
& ( ord_less_real @ T4 @ C )
& ( ( F @ A )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_9860_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B4: real] :
( ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M2: nat,T5: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [U2: real] :
( minus_minus_real @ ( Diff @ M2 @ U2 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
@ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
@ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T5 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T5 @ P5 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
@ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T5 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_9861_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X9: real] :
( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
@ ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_arctan_series
thf(fact_9862_isCont__Lb__Ub,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
=> ? [L6: real,M8: real] :
( ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_eq_real @ X5 @ B ) )
=> ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
& ( ord_less_eq_real @ ( F @ X5 ) @ M8 ) ) )
& ! [Y4: real] :
( ( ( ord_less_eq_real @ L6 @ Y4 )
& ( ord_less_eq_real @ Y4 @ M8 ) )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y4 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_9863_isCont__real__sqrt,axiom,
! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_9864_isCont__real__root,axiom,
! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_9865_isCont__inverse__function2,axiom,
! [A: real,X: real,B: real,G: real > real,F: real > real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_9866_LIM__less__bound,axiom,
! [B: real,X: real,F: real > real] :
( ( ord_less_real @ B @ X )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_9867_isCont__inverse__function,axiom,
! [D: real,X: real,G: real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_9868_GMVT_H,axiom,
! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ? [C2: real] :
( ( ord_less_real @ A @ C2 )
& ( ord_less_real @ C2 @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C2 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_9869_LIM__cos__div__sin,axiom,
( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% LIM_cos_div_sin
thf(fact_9870_summable__Leibniz_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
=> ! [N7: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_9871_summable__Leibniz_I2_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
=> ! [N7: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_9872_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_9873_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_9874_mult__nat__left__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% mult_nat_left_at_top
thf(fact_9875_monoseq__convergent,axiom,
! [X8: nat > real,B4: real] :
( ( topolo6980174941875973593q_real @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I4 ) ) @ B4 )
=> ~ ! [L6: real] :
~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% monoseq_convergent
thf(fact_9876_LIMSEQ__root,axiom,
( filterlim_nat_real
@ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_root
thf(fact_9877_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim_nat_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
=> ? [L4: real] :
( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L4 )
& ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
& ! [N7: nat] : ( ord_less_eq_real @ L4 @ ( G @ N7 ) )
& ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_9878_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_real @ R3 @ ( X8 @ N3 ) ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_9879_LIMSEQ__root__const,axiom,
! [C: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( root @ N2 @ C )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ) ).
% LIMSEQ_root_const
thf(fact_9880_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_9881_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_9882_increasing__LIMSEQ,axiom,
! [F: nat > real,L2: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L2 )
=> ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N7: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N7 ) @ E2 ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_9883_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_9884_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_9885_LIMSEQ__abs__realpow__zero2,axiom,
! [C: real] :
( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_9886_LIMSEQ__abs__realpow__zero,axiom,
! [C: real] :
( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_9887_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_9888_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9889_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_nat ) ).
% tendsto_exp_limit_sequentially
thf(fact_9890_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9891_summable__Leibniz_I1_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_9892_summable,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% summable
thf(fact_9893_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ~ ! [K2: nat > int] :
~ ( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ Theta2 )
@ at_top_nat ) ) ).
% cos_diff_limit_1
thf(fact_9894_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ? [K2: nat > int] :
( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% cos_limit_1
thf(fact_9895_summable__Leibniz_I4_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(4)
thf(fact_9896_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% zeroseq_arctan_series
thf(fact_9897_summable__Leibniz_H_I2_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_9898_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_9899_sums__alternating__upper__lower,axiom,
! [A: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ? [L4: real] :
( ! [N7: nat] :
( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
@ L4 )
& ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat )
& ! [N7: nat] :
( ord_less_eq_real @ L4
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
& ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_9900_summable__Leibniz_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(5)
thf(fact_9901_summable__Leibniz_H_I4_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_9902_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_9903_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_Suc
thf(fact_9904_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat
@ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_9905_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat @ P @ at_top_nat )
=> ( eventually_nat
@ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_9906_le__sequentially,axiom,
! [F5: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
= ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_9907_eventually__sequentiallyI,axiom,
! [C: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( P @ X3 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_9908_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N6: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_9909_real__bounded__linear,axiom,
( real_V5970128139526366754l_real
= ( ^ [F3: real > real] :
? [C3: real] :
( F3
= ( ^ [X2: real] : ( times_times_real @ X2 @ C3 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_9910_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
= ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_9911_sqrt__at__top,axiom,
filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% sqrt_at_top
thf(fact_9912_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_real ) ).
% tendsto_power_div_exp_0
thf(fact_9913_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y ) ) @ Y )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_real ) ).
% tendsto_exp_limit_at_top
thf(fact_9914_filterlim__tan__at__left,axiom,
filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% filterlim_tan_at_left
thf(fact_9915_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq_real @ B @ X3 )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_real @ Y4 @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_9916_tendsto__arctan__at__top,axiom,
filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% tendsto_arctan_at_top
thf(fact_9917_at__top__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% at_top_le_at_infinity
thf(fact_9918_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9919_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: real > real,F5: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
@ at_top_real
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_9920_at__bot__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% at_bot_le_at_infinity
thf(fact_9921_DERIV__pos__imp__increasing__at__bot,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y4 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_9922_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: real > real,F5: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
@ at_bot_real
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_9923_tendsto__arctan__at__bot,axiom,
filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% tendsto_arctan_at_bot
thf(fact_9924_Bseq__eq__bounded,axiom,
! [F: nat > real,A: real,B: real] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( bfun_nat_real @ F @ at_top_nat ) ) ).
% Bseq_eq_bounded
thf(fact_9925_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% Bseq_realpow
thf(fact_9926_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_9927_filterlim__tan__at__right,axiom,
filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% filterlim_tan_at_right
thf(fact_9928_eventually__at__right__to__0,axiom,
! [P: real > $o,A: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
= ( eventually_real
@ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_9929_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_9930_decseq__bounded,axiom,
! [X8: nat > real,B4: real] :
( ( order_9091379641038594480t_real @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I4 ) )
=> ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% decseq_bounded
thf(fact_9931_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_9932_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_9933_decseq__convergent,axiom,
! [X8: nat > real,B4: real] :
( ( order_9091379641038594480t_real @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I4 ) )
=> ~ ! [L6: real] :
( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
=> ~ ! [I: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I ) ) ) ) ) ).
% decseq_convergent
thf(fact_9934_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_9935_GMVT,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
=> ( ! [X3: real] :
( ( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B ) )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
=> ( ! [X3: real] :
( ( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B ) )
=> ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ? [G_c: real,F_c: real,C2: real] :
( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
& ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
& ( ord_less_real @ A @ C2 )
& ( ord_less_real @ C2 @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_9936_MVT,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X3: real] :
( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_9937_continuous__on__arcosh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_9938_continuous__image__closed__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [C2: real,D3: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C2 @ D3 ) )
& ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_9939_continuous__on__arcosh,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% continuous_on_arcosh
thf(fact_9940_continuous__on__artanh,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% continuous_on_artanh
thf(fact_9941_DERIV__isconst2,axiom,
! [A: real,B: real,F: real > real,X: real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X3: real] :
( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( F @ X )
= ( F @ A ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_9942_upto_Opelims,axiom,
! [X: int,Xa2: int,Y2: list_int] :
( ( ( upto @ X @ Xa2 )
= Y2 )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y2
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y2 = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_9943_nth__upto,axiom,
! [I2: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I2 @ J ) @ K )
= ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_9944_length__upto,axiom,
! [I2: int,J: int] :
( ( size_size_list_int @ ( upto @ I2 @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_9945_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_9946_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_9947_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_9948_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_9949_atLeastAtMost__upto,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_9950_upto__split2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_9951_upto__rec1,axiom,
! [I2: int,J: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_9952_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_9953_upto_Oelims,axiom,
! [X: int,Xa2: int,Y2: list_int] :
( ( ( upto @ X @ Xa2 )
= Y2 )
=> ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y2
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y2 = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9954_upto__split1,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_9955_upto__rec2,axiom,
! [I2: int,J: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_9956_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_9957_greaterThanAtMost__upto,axiom,
( set_or6656581121297822940st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_9958_upto__split3,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_9959_greaterThanLessThan__upto,axiom,
( set_or5832277885323065728an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_9960_upto_Opsimps,axiom,
! [I2: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
=> ( ( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_9961_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_9962_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_9963_incseq__bounded,axiom,
! [X8: nat > real,B4: real] :
( ( order_mono_nat_real @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B4 )
=> ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% incseq_bounded
thf(fact_9964_incseq__convergent,axiom,
! [X8: nat > real,B4: real] :
( ( order_mono_nat_real @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B4 )
=> ~ ! [L6: real] :
( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
=> ~ ! [I: nat] : ( ord_less_eq_real @ ( X8 @ I ) @ L6 ) ) ) ) ).
% incseq_convergent
thf(fact_9965_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( order_mono_nat_nat
@ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_9966_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F: nat > real,G: nat > nat] :
( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
=> ( ( order_mono_nat_real @ F )
=> ( ( order_5726023648592871131at_nat @ G )
=> ( ( bfun_nat_real
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ at_top_nat )
= ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
thf(fact_9967_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_9968_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_real_real
@ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_9969_inj__on__diff__nat,axiom,
! [N5: set_nat,K: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N5 )
=> ( ord_less_eq_nat @ K @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_9970_inj__Suc,axiom,
! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_9971_summable__reindex,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
=> ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% summable_reindex
thf(fact_9972_suminf__reindex__mono,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
=> ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_9973_inj__on__char__of__nat,axiom,
inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_9974_suminf__reindex,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
=> ( ! [X3: nat] :
( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
= ( suminf_real @ F ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_9975_powr__real__of__int_H,axiom,
! [X: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( X != zero_zero_real )
| ( ord_less_int @ zero_zero_int @ N ) )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( power_int_real @ X @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_9976_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_9977_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_9978_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_9979_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_9980_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_9981_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_9982_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
= ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_9983_min__diff,axiom,
! [M: nat,I2: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I2 ) ) ).
% min_diff
thf(fact_9984_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_9985_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L2: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 )
= ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L2 @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_9986_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L2: int] :
( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
= ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L2 ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_9987_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M3: nat] : ( suc @ ( ord_min_nat @ N @ M3 ) )
@ M ) ) ).
% min_Suc1
thf(fact_9988_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M3: nat] : ( suc @ ( ord_min_nat @ M3 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_9989_min__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(2)
thf(fact_9990_min__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(3)
thf(fact_9991_inf__enat__def,axiom,
inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% inf_enat_def
thf(fact_9992_hd__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( hd_nat @ ( upt @ I2 @ J ) )
= I2 ) ) ).
% hd_upt
thf(fact_9993_upt__conv__Nil,axiom,
! [J: nat,I2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( upt @ I2 @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_9994_drop__upt,axiom,
! [M: nat,I2: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I2 @ J ) )
= ( upt @ ( plus_plus_nat @ I2 @ M ) @ J ) ) ).
% drop_upt
thf(fact_9995_length__upt,axiom,
! [I2: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I2 @ J ) )
= ( minus_minus_nat @ J @ I2 ) ) ).
% length_upt
thf(fact_9996_upt__eq__Nil__conv,axiom,
! [I2: nat,J: nat] :
( ( ( upt @ I2 @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_9997_nth__upt,axiom,
! [I2: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ K ) ) ) ).
% nth_upt
thf(fact_9998_take__upt,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I2 @ N ) )
= ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).
% take_upt
thf(fact_9999_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_10000_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_10001_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_10002_upt__Suc,axiom,
! [I2: nat,J: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_10003_upt__Suc__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_10004_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_10005_upt__conv__Cons,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( upt @ I2 @ J )
= ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_10006_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_10007_upt__add__eq__append,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_10008_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M6 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_10009_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M6 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_10010_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_10011_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M6 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_10012_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_10013_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X: nat,Xs2: list_nat] :
( ( ( upt @ I2 @ J )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ord_less_nat @ I2 @ J )
& ( I2 = X )
& ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_10014_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_10015_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_10016_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_10017_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_10018_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_10019_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_10020_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_10021_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_10022_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_10023_card__length__sum__list__rec,axiom,
! [M: nat,N5: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= M )
& ( ( groups4561878855575611511st_nat @ L )
= N5 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L )
= N5 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_10024_card__length__sum__list,axiom,
! [M: nat,N5: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= M )
& ( ( groups4561878855575611511st_nat @ L )
= N5 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% card_length_sum_list
thf(fact_10025_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_10026_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_10027_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I2: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_10028_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_10029_card__le__Suc__Max,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_10030_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M6 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_10031_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gcd_gcd_nat @ M @ N )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D2: nat] :
( ( dvd_dvd_nat @ D2 @ M )
& ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_10032_sup__enat__def,axiom,
sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% sup_enat_def
thf(fact_10033_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_10034_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_10035_less__eq,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
= ( ord_less_nat @ M @ N ) ) ).
% less_eq
thf(fact_10036_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_10037_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ N )
& ( ord_less_nat @ Y @ N )
& ( ord_less_eq_nat @ X2 @ Y ) ) ) ) )
= ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_10038_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_10039_wf__less,axiom,
wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% wf_less
thf(fact_10040_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
= ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% prod_encode_prod_decode_aux
thf(fact_10041_le__prod__encode__2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_2
thf(fact_10042_le__prod__encode__1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_1
thf(fact_10043_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M6: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N2 ) ) @ M6 ) ) ) ).
% prod_encode_def
thf(fact_10044_list__encode_Oelims,axiom,
! [X: list_nat,Y2: nat] :
( ( ( nat_list_encode @ X )
= Y2 )
=> ( ( ( X = nil_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs3 ) )
=> ( Y2
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_10045_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_10046_list__encode_Opelims,axiom,
! [X: list_nat,Y2: nat] :
( ( ( nat_list_encode @ X )
= Y2 )
=> ( ( accp_list_nat @ nat_list_encode_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( Y2
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_10047_Gcd__int__greater__eq__0,axiom,
! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_10048_Gcd__nat__set__eq__fold,axiom,
! [Xs2: list_nat] :
( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
= ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_10049_Gcd__int__set__eq__fold,axiom,
! [Xs2: list_int] :
( ( gcd_Gcd_int @ ( set_int2 @ Xs2 ) )
= ( fold_int_int @ gcd_gcd_int @ Xs2 @ zero_zero_int ) ) ).
% Gcd_int_set_eq_fold
thf(fact_10050_vanishes__mult__bounded,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ? [A7: rat] :
( ( ord_less_rat @ zero_zero_rat @ A7 )
& ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ A7 ) )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_10051_vanishes__const,axiom,
! [C: rat] :
( ( vanishes
@ ^ [N2: nat] : C )
= ( C = zero_zero_rat ) ) ).
% vanishes_const
thf(fact_10052_vanishes__diff,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% vanishes_diff
thf(fact_10053_vanishes__minus,axiom,
! [X8: nat > rat] :
( ( vanishes @ X8 )
=> ( vanishes
@ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).
% vanishes_minus
thf(fact_10054_vanishes__add,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% vanishes_add
thf(fact_10055_vanishesD,axiom,
! [X8: nat > rat,R2: rat] :
( ( vanishes @ X8 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N7 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_10056_vanishesI,axiom,
! [X8: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R3 ) ) )
=> ( vanishes @ X8 ) ) ).
% vanishesI
thf(fact_10057_vanishes__def,axiom,
( vanishes
= ( ^ [X6: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_10058_cauchy__def,axiom,
( cauchy
= ( ^ [X6: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [M6: nat] :
( ( ord_less_eq_nat @ K3 @ M6 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_10059_cauchy__inverse,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ( cauchy
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ).
% cauchy_inverse
thf(fact_10060_cauchy__mult,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( cauchy
@ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% cauchy_mult
thf(fact_10061_cauchy__minus,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( cauchy
@ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).
% cauchy_minus
thf(fact_10062_cauchy__const,axiom,
! [X: rat] :
( cauchy
@ ^ [N2: nat] : X ) ).
% cauchy_const
thf(fact_10063_cauchy__add,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( cauchy
@ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% cauchy_add
thf(fact_10064_cauchy__diff,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( cauchy
@ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% cauchy_diff
thf(fact_10065_cauchy__imp__bounded,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ? [B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ B2 )
& ! [N7: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N7 ) ) @ B2 ) ) ) ).
% cauchy_imp_bounded
thf(fact_10066_vanishes__diff__inverse,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ~ ( vanishes @ Y6 )
=> ( ( vanishes
@ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) )
=> ( vanishes
@ ^ [N2: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X8 @ N2 ) ) @ ( inverse_inverse_rat @ ( Y6 @ N2 ) ) ) ) ) ) ) ) ) ).
% vanishes_diff_inverse
thf(fact_10067_cauchy__not__vanishes__cases,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ B2 )
& ? [K2: nat] :
( ! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ ( X8 @ N7 ) ) ) )
| ! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ B2 @ ( X8 @ N7 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_10068_cauchy__not__vanishes,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ B2 )
& ? [K2: nat] :
! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ B2 @ ( abs_abs_rat @ ( X8 @ N7 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_10069_cauchyD,axiom,
! [X8: nat > rat,R2: rat] :
( ( cauchy @ X8 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M2 ) @ ( X8 @ N7 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_10070_cauchyI,axiom,
! [X8: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ K4 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X8 ) ) ).
% cauchyI
thf(fact_10071_le__Real,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y6 ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( plus_plus_rat @ ( Y6 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_10072_Real__induct,axiom,
! [P: real > $o,X: real] :
( ! [X10: nat > rat] :
( ( cauchy @ X10 )
=> ( P @ ( real2 @ X10 ) ) )
=> ( P @ X ) ) ).
% Real_induct
thf(fact_10073_of__int__Real,axiom,
( ring_1_of_int_real
= ( ^ [X2: int] :
( real2
@ ^ [N2: nat] : ( ring_1_of_int_rat @ X2 ) ) ) ) ).
% of_int_Real
thf(fact_10074_of__nat__Real,axiom,
( semiri5074537144036343181t_real
= ( ^ [X2: nat] :
( real2
@ ^ [N2: nat] : ( semiri681578069525770553at_rat @ X2 ) ) ) ) ).
% of_nat_Real
thf(fact_10075_zero__real__def,axiom,
( zero_zero_real
= ( real2
@ ^ [N2: nat] : zero_zero_rat ) ) ).
% zero_real_def
thf(fact_10076_one__real__def,axiom,
( one_one_real
= ( real2
@ ^ [N2: nat] : one_one_rat ) ) ).
% one_real_def
thf(fact_10077_minus__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( uminus_uminus_real @ ( real2 @ X8 ) )
= ( real2
@ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ) ).
% minus_Real
thf(fact_10078_add__Real,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ( plus_plus_real @ ( real2 @ X8 ) @ ( real2 @ Y6 ) )
= ( real2
@ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ) ).
% add_Real
thf(fact_10079_mult__Real,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ( times_times_real @ ( real2 @ X8 ) @ ( real2 @ Y6 ) )
= ( real2
@ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ) ).
% mult_Real
thf(fact_10080_diff__Real,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ( minus_minus_real @ ( real2 @ X8 ) @ ( real2 @ Y6 ) )
= ( real2
@ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ) ).
% diff_Real
thf(fact_10081_eq__Real,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ( ( real2 @ X8 )
= ( real2 @ Y6 ) )
= ( vanishes
@ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ) ).
% eq_Real
thf(fact_10082_inverse__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ( vanishes @ X8 )
=> ( ( inverse_inverse_real @ ( real2 @ X8 ) )
= zero_zero_real ) )
& ( ~ ( vanishes @ X8 )
=> ( ( inverse_inverse_real @ ( real2 @ X8 ) )
= ( real2
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% inverse_Real
thf(fact_10083_not__positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_eq_rat @ ( X8 @ N2 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_10084_Real_Opositive__minus,axiom,
! [X: real] :
( ~ ( positive2 @ X )
=> ( ( X != zero_zero_real )
=> ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% Real.positive_minus
thf(fact_10085_Real_Opositive__add,axiom,
! [X: real,Y2: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y2 )
=> ( positive2 @ ( plus_plus_real @ X @ Y2 ) ) ) ) ).
% Real.positive_add
thf(fact_10086_Real_Opositive__mult,axiom,
! [X: real,Y2: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y2 )
=> ( positive2 @ ( times_times_real @ X @ Y2 ) ) ) ) ).
% Real.positive_mult
thf(fact_10087_Real_Opositive__zero,axiom,
~ ( positive2 @ zero_zero_real ) ).
% Real.positive_zero
thf(fact_10088_less__real__def,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X2 ) ) ) ) ).
% less_real_def
thf(fact_10089_positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( positive2 @ ( real2 @ X8 ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_10090_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X2: real] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( rep_real @ X2 @ N2 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_10091_inverse__real_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( inverse_inverse_real @ ( real2 @ X ) )
= ( real2
@ ( if_nat_rat @ ( vanishes @ X )
@ ^ [N2: nat] : zero_zero_rat
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X @ N2 ) ) ) ) ) ) ).
% inverse_real.abs_eq
thf(fact_10092_realrel__refl,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( realrel @ X8 @ X8 ) ) ).
% realrel_refl
thf(fact_10093_zero__real_Orsp,axiom,
( realrel
@ ^ [N2: nat] : zero_zero_rat
@ ^ [N2: nat] : zero_zero_rat ) ).
% zero_real.rsp
thf(fact_10094_one__real_Orsp,axiom,
( realrel
@ ^ [N2: nat] : one_one_rat
@ ^ [N2: nat] : one_one_rat ) ).
% one_real.rsp
thf(fact_10095_real_Oabs__induct,axiom,
! [P: real > $o,X: real] :
( ! [Y3: nat > rat] :
( ( realrel @ Y3 @ Y3 )
=> ( P @ ( real2 @ Y3 ) ) )
=> ( P @ X ) ) ).
% real.abs_induct
thf(fact_10096_uminus__real_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( uminus_uminus_real @ ( real2 @ X ) )
= ( real2
@ ^ [N2: nat] : ( uminus_uminus_rat @ ( X @ N2 ) ) ) ) ) ).
% uminus_real.abs_eq
thf(fact_10097_plus__real_Oabs__eq,axiom,
! [Xa2: nat > rat,X: nat > rat] :
( ( realrel @ Xa2 @ Xa2 )
=> ( ( realrel @ X @ X )
=> ( ( plus_plus_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N2: nat] : ( plus_plus_rat @ ( Xa2 @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% plus_real.abs_eq
thf(fact_10098_times__real_Oabs__eq,axiom,
! [Xa2: nat > rat,X: nat > rat] :
( ( realrel @ Xa2 @ Xa2 )
=> ( ( realrel @ X @ X )
=> ( ( times_times_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N2: nat] : ( times_times_rat @ ( Xa2 @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% times_real.abs_eq
thf(fact_10099_realrelI,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y6 )
=> ( ( vanishes
@ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y6 @ N2 ) ) )
=> ( realrel @ X8 @ Y6 ) ) ) ) ).
% realrelI
thf(fact_10100_realrel__def,axiom,
( realrel
= ( ^ [X6: nat > rat,Y7: nat > rat] :
( ( cauchy @ X6 )
& ( cauchy @ Y7 )
& ( vanishes
@ ^ [N2: nat] : ( minus_minus_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% realrel_def
thf(fact_10101_Real_Opositive_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( positive2 @ ( real2 @ X ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_10102_inverse__real__def,axiom,
( inverse_inverse_real
= ( map_fu7146612038024189824t_real @ rep_real @ real2
@ ^ [X6: nat > rat] :
( if_nat_rat @ ( vanishes @ X6 )
@ ^ [N2: nat] : zero_zero_rat
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ) ).
% inverse_real_def
thf(fact_10103_cr__real__def,axiom,
( cr_real
= ( ^ [X2: nat > rat,Y: real] :
( ( realrel @ X2 @ X2 )
& ( ( real2 @ X2 )
= Y ) ) ) ) ).
% cr_real_def
thf(fact_10104_uminus__real__def,axiom,
( uminus_uminus_real
= ( map_fu7146612038024189824t_real @ rep_real @ real2
@ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ) ).
% uminus_real_def
thf(fact_10105_times__real__def,axiom,
( times_times_real
= ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ).
% times_real_def
thf(fact_10106_plus__real__def,axiom,
( plus_plus_real
= ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ).
% plus_real_def
thf(fact_10107_Real_Opositive_Orsp,axiom,
( bNF_re728719798268516973at_o_o @ realrel
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_10108_plus__real_Orsp,axiom,
( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ).
% plus_real.rsp
thf(fact_10109_uminus__real_Orsp,axiom,
( bNF_re895249473297799549at_rat @ realrel @ realrel
@ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
@ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ).
% uminus_real.rsp
thf(fact_10110_times__real_Orsp,axiom,
( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ).
% times_real.rsp
thf(fact_10111_less__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ord_less_nat
@ ord_less_nat ) ).
% less_natural.rsp
thf(fact_10112_divide__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
@ divide_divide_nat
@ divide_divide_nat ) ).
% divide_natural.rsp
thf(fact_10113_times__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ^ [Y5: int,Z5: int] : Y5 = Z5 )
@ times_times_int
@ times_times_int ) ).
% times_integer.rsp
thf(fact_10114_times__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
@ times_times_nat
@ times_times_nat ) ).
% times_natural.rsp
thf(fact_10115_Suc_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_10116_dup_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_10117_plus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
@ plus_plus_nat
@ plus_plus_nat ) ).
% plus_natural.rsp
thf(fact_10118_plus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ^ [Y5: int,Z5: int] : Y5 = Z5 )
@ plus_plus_int
@ plus_plus_int ) ).
% plus_integer.rsp
thf(fact_10119_sub_Orsp,axiom,
( bNF_re8402795839162346335um_int
@ ^ [Y5: num,Z5: num] : Y5 = Z5
@ ( bNF_re1822329894187522285nt_int
@ ^ [Y5: num,Z5: num] : Y5 = Z5
@ ^ [Y5: int,Z5: int] : Y5 = Z5 )
@ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) )
@ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% sub.rsp
thf(fact_10120_less__eq__integer_Orsp,axiom,
( bNF_re3403563459893282935_int_o
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ( bNF_re5089333283451836215nt_o_o
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ord_less_eq_int
@ ord_less_eq_int ) ).
% less_eq_integer.rsp
thf(fact_10121_less__eq__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ord_less_eq_nat
@ ord_less_eq_nat ) ).
% less_eq_natural.rsp
thf(fact_10122_inverse__real_Orsp,axiom,
( bNF_re895249473297799549at_rat @ realrel @ realrel
@ ^ [X6: nat > rat] :
( if_nat_rat @ ( vanishes @ X6 )
@ ^ [N2: nat] : zero_zero_rat
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
@ ^ [X6: nat > rat] :
( if_nat_rat @ ( vanishes @ X6 )
@ ^ [N2: nat] : zero_zero_rat
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ).
% inverse_real.rsp
thf(fact_10123_Real_Opositive_Otransfer,axiom,
( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_10124_real_Orel__eq__transfer,axiom,
( bNF_re4521903465945308077real_o @ pcr_real
@ ( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ realrel
@ ^ [Y5: real,Z5: real] : Y5 = Z5 ) ).
% real.rel_eq_transfer
thf(fact_10125_zero__real_Otransfer,axiom,
( pcr_real
@ ^ [N2: nat] : zero_zero_rat
@ zero_zero_real ) ).
% zero_real.transfer
thf(fact_10126_real_Opcr__cr__eq,axiom,
pcr_real = cr_real ).
% real.pcr_cr_eq
thf(fact_10127_one__real_Otransfer,axiom,
( pcr_real
@ ^ [N2: nat] : one_one_rat
@ one_one_real ) ).
% one_real.transfer
thf(fact_10128_cr__real__eq,axiom,
( pcr_real
= ( ^ [X2: nat > rat,Y: real] :
( ( cauchy @ X2 )
& ( ( real2 @ X2 )
= Y ) ) ) ) ).
% cr_real_eq
thf(fact_10129_uminus__real_Otransfer,axiom,
( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
@ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
@ uminus_uminus_real ) ).
% uminus_real.transfer
thf(fact_10130_plus__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) )
@ plus_plus_real ) ).
% plus_real.transfer
thf(fact_10131_times__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X6: nat > rat,Y7: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) )
@ times_times_real ) ).
% times_real.transfer
thf(fact_10132_inverse__real_Otransfer,axiom,
( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
@ ^ [X6: nat > rat] :
( if_nat_rat @ ( vanishes @ X6 )
@ ^ [N2: nat] : zero_zero_rat
@ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
@ inverse_inverse_real ) ).
% inverse_real.transfer
thf(fact_10133_plus__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ plus_plus_rat ) ).
% plus_rat.transfer
thf(fact_10134_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% one_rat.transfer
thf(fact_10135_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% zero_rat.transfer
thf(fact_10136_Fract_Otransfer,axiom,
( bNF_re3461391660133120880nt_rat
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ( bNF_re2214769303045360666nt_rat
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ pcr_rat )
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_10137_uminus__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
@ uminus_uminus_rat ) ).
% uminus_rat.transfer
thf(fact_10138_times__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ times_times_rat ) ).
% times_rat.transfer
thf(fact_10139_Rat_Opositive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_10140_inverse__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
@ inverse_inverse_rat ) ).
% inverse_rat.transfer
thf(fact_10141_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_10142_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% zero_int.transfer
thf(fact_10143_int__transfer,axiom,
( bNF_re6830278522597306478at_int
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ pcr_int
@ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
@ semiri1314217659103216013at_int ) ).
% int_transfer
thf(fact_10144_uminus__int_Otransfer,axiom,
( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) )
@ uminus_uminus_int ) ).
% uminus_int.transfer
thf(fact_10145_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% one_int.transfer
thf(fact_10146_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_10147_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_10148_plus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
@ plus_plus_int ) ).
% plus_int.transfer
thf(fact_10149_minus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
@ minus_minus_int ) ).
% minus_int.transfer
thf(fact_10150_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_10151_intrel__iff,axiom,
! [X: nat,Y2: nat,U: nat,V: nat] :
( ( intrel @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( product_Pair_nat_nat @ U @ V ) )
= ( ( plus_plus_nat @ X @ V )
= ( plus_plus_nat @ U @ Y2 ) ) ) ).
% intrel_iff
thf(fact_10152_uminus__int_Orsp,axiom,
( bNF_re2241393799969408733at_nat @ intrel @ intrel
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) )
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) ) ) ).
% uminus_int.rsp
thf(fact_10153_less__RealD,axiom,
! [Y6: nat > rat,X: real] :
( ( cauchy @ Y6 )
=> ( ( ord_less_real @ X @ ( real2 @ Y6 ) )
=> ? [N3: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y6 @ N3 ) ) ) ) ) ).
% less_RealD
thf(fact_10154_of__rat__Real,axiom,
( field_7254667332652039916t_real
= ( ^ [X2: rat] :
( real2
@ ^ [N2: nat] : X2 ) ) ) ).
% of_rat_Real
thf(fact_10155_zero__int_Orsp,axiom,
intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% zero_int.rsp
thf(fact_10156_of__rat__dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Q3: rat] :
( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q3 ) )
& ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y2 ) ) ) ).
% of_rat_dense
thf(fact_10157_Real__leI,axiom,
! [X8: nat > rat,Y2: real] :
( ( cauchy @ X8 )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X8 @ N3 ) ) @ Y2 )
=> ( ord_less_eq_real @ ( real2 @ X8 ) @ Y2 ) ) ) ).
% Real_leI
thf(fact_10158_le__RealI,axiom,
! [Y6: nat > rat,X: real] :
( ( cauchy @ Y6 )
=> ( ! [N3: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y6 @ N3 ) ) )
=> ( ord_less_eq_real @ X @ ( real2 @ Y6 ) ) ) ) ).
% le_RealI
thf(fact_10159_one__int_Orsp,axiom,
intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% one_int.rsp
thf(fact_10160_intrel__def,axiom,
( intrel
= ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] :
( ( plus_plus_nat @ X2 @ V4 )
= ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% intrel_def
thf(fact_10161_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% less_int.rsp
thf(fact_10162_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_10163_plus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_10164_minus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_10165_DeMoivre2,axiom,
! [R2: real,A: real,N: nat] :
( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
= ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% DeMoivre2
thf(fact_10166_Re__rcis,axiom,
! [R2: real,A: real] :
( ( re @ ( rcis @ R2 @ A ) )
= ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).
% Re_rcis
thf(fact_10167_Im__rcis,axiom,
! [R2: real,A: real] :
( ( im @ ( rcis @ R2 @ A ) )
= ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).
% Im_rcis
thf(fact_10168_le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ).
% le_enumerate
thf(fact_10169_rcis__mult,axiom,
! [R12: real,A: real,R23: real,B: real] :
( ( times_times_complex @ ( rcis @ R12 @ A ) @ ( rcis @ R23 @ B ) )
= ( rcis @ ( times_times_real @ R12 @ R23 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% rcis_mult
thf(fact_10170_rcis__def,axiom,
( rcis
= ( ^ [R5: real,A4: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A4 ) ) ) ) ).
% rcis_def
thf(fact_10171_finite__le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_10172_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_10173_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_10174_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M6: nat] : ( P @ ( suc @ M6 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_10175_Sup__real__def,axiom,
( comple1385675409528146559p_real
= ( ^ [X6: set_real] :
( ord_Least_real
@ ^ [Z2: real] :
! [X2: real] :
( ( member_real @ X2 @ X6 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ) ) ).
% Sup_real_def
thf(fact_10176_divmod__nat__code,axiom,
( divmod_nat
= ( ^ [M6: nat,N2: nat] :
( produc8678311845419106900er_nat @ code_nat_of_integer @ code_nat_of_integer
@ ( if_Pro6119634080678213985nteger
@ ( ( code_integer_of_nat @ M6 )
= zero_z3403309356797280102nteger )
@ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger
@ ( ( code_integer_of_nat @ N2 )
= zero_z3403309356797280102nteger )
@ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( code_integer_of_nat @ M6 ) )
@ ( code_divmod_abs @ ( code_integer_of_nat @ M6 ) @ ( code_integer_of_nat @ N2 ) ) ) ) ) ) ) ).
% divmod_nat_code
thf(fact_10177_eventually__prod__sequentially,axiom,
! [P: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N6: nat] :
! [M6: nat] :
( ( ord_less_eq_nat @ N6 @ M6 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( P @ ( product_Pair_nat_nat @ N2 @ M6 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_10178_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% integer_of_nat_numeral
thf(fact_10179_at__right__to__0,axiom,
! [A: real] :
( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
= ( filtermap_real_real
@ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% at_right_to_0
thf(fact_10180_plus__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% plus_rat.rsp
thf(fact_10181_ratrel__iff,axiom,
( ratrel
= ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
( ( ( product_snd_int_int @ X2 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
= ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_10182_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% zero_rat.rsp
thf(fact_10183_one__rat_Orsp,axiom,
ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% one_rat.rsp
thf(fact_10184_Fract_Orsp,axiom,
( bNF_re157797125943740599nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ( bNF_re6250860962936578807nt_int
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ ratrel )
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B3 ) )
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_10185_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
( ( ( product_snd_int_int @ X2 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
= ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_10186_uminus__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ).
% uminus_rat.rsp
thf(fact_10187_times__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% times_rat.rsp
thf(fact_10188_Rat_Opositive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_10189_inverse__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_10190_plus__rat_Oabs__eq,axiom,
! [Xa2: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_10191_one__rat__def,axiom,
( one_one_rat
= ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% one_rat_def
thf(fact_10192_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X2: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X2 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_10193_zero__rat__def,axiom,
( zero_zero_rat
= ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% zero_rat_def
thf(fact_10194_uminus__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_10195_times__rat_Oabs__eq,axiom,
! [Xa2: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X @ X )
=> ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_10196_Rat_Opositive_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% Rat.positive.abs_eq
thf(fact_10197_inverse__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_10198_inverse__rat__def,axiom,
( inverse_inverse_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_10199_uminus__rat__def,axiom,
( uminus_uminus_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% uminus_rat_def
thf(fact_10200_plus__rat__def,axiom,
( plus_plus_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% plus_rat_def
thf(fact_10201_times__rat__def,axiom,
( times_times_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% times_rat_def
thf(fact_10202_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
thf(fact_10203_Rat_Opositive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% Rat.positive_def
% Helper facts (40)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y2: num] :
( ( if_num @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y2: num] :
( ( if_num @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y2: rat] :
( ( if_rat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y2: rat] :
( ( if_rat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P: real > $o] :
( ( P @ ( fChoice_real @ P ) )
= ( ? [X6: real] : ( P @ X6 ) ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y2: complex] :
( ( if_complex @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y2: complex] :
( ( if_complex @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( if_Extended_enat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y2: code_integer] :
( ( if_Code_integer @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y2: code_integer] :
( ( if_Code_integer @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y2: set_int] :
( ( if_set_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y2: set_int] :
( ( if_set_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y2: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y2: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y2: list_int] :
( ( if_list_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y2: list_int] :
( ( if_list_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
! [X: nat > rat,Y2: nat > rat] :
( ( if_nat_rat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
! [X: nat > rat,Y2: nat > rat] :
( ( if_nat_rat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y2: option_nat] :
( ( if_option_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y2: option_nat] :
( ( if_option_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y2: option_num] :
( ( if_option_num @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y2: option_num] :
( ( if_option_num @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y2: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y2: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y2: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y2: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [U3: nat] :
( ~ ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U3 )
| ~ ( ord_less_nat @ U3 @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
%------------------------------------------------------------------------------