TPTP Problem File: ITP237^3.p
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%------------------------------------------------------------------------------
% File : ITP237^3 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Pred 00156_006356
% 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_00156_006356 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 11523 (5298 unt;1266 typ; 0 def)
% Number of atoms : 30253 (11880 equ; 0 cnn)
% Maximal formula atoms : 71 ( 2 avg)
% Number of connectives : 112977 (2701 ~; 490 |;2020 &;96176 @)
% ( 0 <=>;11590 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 6 avg)
% Number of types : 148 ( 147 usr)
% Number of type conns : 5136 (5136 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1122 (1119 usr; 71 con; 0-8 aty)
% Number of variables : 26327 (2481 ^;23040 !; 806 ?;26327 :)
% 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 22:30:40.420
%------------------------------------------------------------------------------
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thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_nat_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
list_int_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
option_VEBT_VEBT: $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_It__Int__Oint_M_Eo_J,type,
product_prod_int_o: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
list_Code_integer: $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__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $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__Option__Ooption_It__Int__Oint_J,type,
option_int: $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__Num__Onum_J,type,
list_num: $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__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Option__Ooption_I_Eo_J,type,
option_o: $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__String__Oliteral,type,
literal: $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 (1119)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
archim2889992004027027881ng_rat: rat > int ).
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_Ofrac_001t__Rat__Orat,type,
archimedean_frac_rat: rat > rat ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
archim2898591450579166408c_real: real > real ).
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_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
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_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_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__Code____Numeral__Ointeger_001_062_It__Int__Oint_M_Eo_J_001_062_It__Code____Numeral__Ointeger_M_Eo_J,type,
bNF_re6321650412969554871eger_o: ( int > code_integer > $o ) > ( ( int > $o ) > ( code_integer > $o ) > $o ) > ( int > int > $o ) > ( code_integer > code_integer > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re4711666741709854504_nat_o: ( int > code_integer > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( int > nat > $o ) > ( code_integer > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
bNF_re6574881592172037608er_o_o: ( int > code_integer > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( code_integer > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint,type,
bNF_re3804157879324367682nt_int: ( int > code_integer > $o ) > ( int > int > $o ) > ( int > int ) > ( code_integer > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re2807294637932363402at_nat: ( int > code_integer > $o ) > ( nat > nat > $o ) > ( int > nat ) > ( code_integer > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Num__Onum_001t__Num__Onum,type,
bNF_re6718328864250387230um_num: ( int > code_integer > $o ) > ( num > num > $o ) > ( int > num ) > ( code_integer > num ) > $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__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re3376528473927230327_nat_o: ( int > int > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( int > nat > $o ) > ( int > nat > $o ) > $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__Code____Numeral__Ointeger,type,
bNF_re982302072995117890nteger: ( int > int > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( int > code_integer ) > $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__Nat__Onat_001t__Nat__Onat,type,
bNF_re3715656647883201625at_nat: ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > ( int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
bNF_re7626690874201225453um_num: ( int > int > $o ) > ( num > num > $o ) > ( int > num ) > ( int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
bNF_re4935368626670024657nteger: ( nat > nat > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( nat > int > int ) > ( nat > code_integer > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $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__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re4153400068438556298nteger: ( nat > nat > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( nat > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $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__Code____Numeral__Ointeger_J,type,
bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $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__Code____Numeral__Ointeger,type,
bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $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_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__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__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_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__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_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__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
gbinom8545251970709558553nteger: code_integer > nat > code_integer ).
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_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
bit_ri7632146776885996613nteger: code_integer > code_integer ).
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 ).
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size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_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__Int__Oint,type,
list_update_int: list_int > nat > int > list_int ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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list_update_real: list_real > nat > real > list_real ).
thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
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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,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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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,
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thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
replicate_real: nat > real > list_real ).
thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(sy_c_List_Orotate1_001_Eo,type,
rotate1_o: list_o > list_o ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
rotate1_VEBT_VEBT: list_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__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_List_Ozip_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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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_Opred,type,
pred: nat > nat ).
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
semiri681578069525770553at_rat: nat > rat ).
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
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nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
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,
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thf(sy_c_NthRoot_Oroot,type,
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thf(sy_c_NthRoot_Osqrt,type,
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thf(sy_c_Num_OBitM,type,
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thf(sy_c_Num_Oinc,type,
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neg_nu3179335615603231917ec_rat: rat > rat ).
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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_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
abort_real: literal > ( product_unit > real ) > real ).
thf(sy_c_String_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
comm_s629917340098488124ar_nat: char > nat ).
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__Real__Oreal,type,
topolo6980174941875973593q_real: ( nat > real ) > $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_Oconvergent_001t__Real__Oreal,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
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_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
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_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_Opowr__real,type,
powr_real2: 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__Real__Oreal,type,
tanh_real: real > real ).
thf(sy_c_Transfer_Obi__total_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bi_tot1331153423839324337nteger: ( int > code_integer > $o ) > $o ).
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__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 ).
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accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
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accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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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,
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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 ).
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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_Omeasure_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
measur8038558561449204169at_nat: ( product_prod_nat_nat > nat ) > set_Pr8693737435421807431at_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,
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thf(sy_c_member_001_Eo,type,
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thf(sy_c_member_001t__Complex__Ocomplex,type,
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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__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,
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thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_I_Eo_J_J,type,
member9156582987741540206list_o: produc2617389633368699223list_o > set_Pr7490072032080894221list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member9189046780804443046st_nat: produc3676724955757786621st_nat > set_Pr5578615432719617117st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
member4376149543098372618T_VEBT: produc8504111982647392627T_VEBT > set_Pr5325845658263174057T_VEBT > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_I_Eo_J_J,type,
member3126162362653435956list_o: produc3962069817607390347list_o > set_Pr7508168486584781291list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member3703241499402361532st_int: produc7831203938951381541st_int > set_Pr4080907618048478043st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member6193324644334088288st_nat: produc1097915047028332489st_nat > set_Pr8894456036836396799st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $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__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_v_a____,type,
a: $o ).
thf(sy_v_b____,type,
b: $o ).
thf(sy_v_nat____,type,
nat3: nat ).
thf(sy_v_sucX____,type,
sucX: nat ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (10214)
thf(fact_0_True,axiom,
b ).
% True
thf(fact_1_calculation,axiom,
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ a @ b ) @ xa )
= ( some_nat @ one_one_nat ) ) ).
% calculation
thf(fact_2_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X22 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_3_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
= ( ? [A: $o,B: $o] :
( T
= ( vEBT_Leaf @ A @ B ) ) ) ) ).
% deg1Leaf
thf(fact_4_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
=> ? [A2: $o,B2: $o] :
( T
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ).
% deg_1_Leaf
thf(fact_5_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N = one_one_nat )
=> ? [A2: $o,B2: $o] :
( T
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ).
% deg_1_Leafy
thf(fact_6_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_7_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_8_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_9_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_10_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_11_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_12_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_13_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
= ( D = one_one_nat ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_14_euclidean__size__1,axiom,
( ( euclid6377331345833325938nteger @ one_one_Code_integer )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_15_euclidean__size__1,axiom,
( ( euclid4774559944035922753ze_int @ one_one_int )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_16_euclidean__size__1,axiom,
( ( euclid4777050414544973029ze_nat @ one_one_nat )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_17_fact__1,axiom,
( ( semiri5044797733671781792omplex @ one_one_nat )
= one_one_complex ) ).
% fact_1
thf(fact_18_fact__1,axiom,
( ( semiri773545260158071498ct_rat @ one_one_nat )
= one_one_rat ) ).
% fact_1
thf(fact_19_fact__1,axiom,
( ( semiri1406184849735516958ct_int @ one_one_nat )
= one_one_int ) ).
% fact_1
thf(fact_20_fact__1,axiom,
( ( semiri1408675320244567234ct_nat @ one_one_nat )
= one_one_nat ) ).
% fact_1
thf(fact_21_fact__1,axiom,
( ( semiri2265585572941072030t_real @ one_one_nat )
= one_one_real ) ).
% fact_1
thf(fact_22_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_23_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T @ D )
=> ( vEBT_VEBT_valid @ T @ D ) ) ).
% valid_eq1
thf(fact_24_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T @ D )
=> ( vEBT_invar_vebt @ T @ D ) ) ).
% valid_eq2
thf(fact_25_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_26_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_27_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_28_option_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( some_nat @ X2 )
= ( some_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_29_option_Oinject,axiom,
! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( some_P7363390416028606310at_nat @ X2 )
= ( some_P7363390416028606310at_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_30_option_Oinject,axiom,
! [X2: num,Y2: num] :
( ( ( some_num @ X2 )
= ( some_num @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_31_Leaf__0__not,axiom,
! [A3: $o,B3: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_32_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_33_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_34_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_35_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_0_not
thf(fact_36_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_tree_deg_neq_0
thf(fact_37_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_38_fact__0,axiom,
( ( semiri5044797733671781792omplex @ zero_zero_nat )
= one_one_complex ) ).
% fact_0
thf(fact_39_fact__0,axiom,
( ( semiri773545260158071498ct_rat @ zero_zero_nat )
= one_one_rat ) ).
% fact_0
thf(fact_40_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_41_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_42_fact__0,axiom,
( ( semiri2265585572941072030t_real @ zero_zero_nat )
= one_one_real ) ).
% fact_0
thf(fact_43_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_44_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_45_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_46_euclidean__size__eq__0__iff,axiom,
! [B3: code_integer] :
( ( ( euclid6377331345833325938nteger @ B3 )
= zero_zero_nat )
= ( B3 = zero_z3403309356797280102nteger ) ) ).
% euclidean_size_eq_0_iff
thf(fact_47_euclidean__size__eq__0__iff,axiom,
! [B3: int] :
( ( ( euclid4774559944035922753ze_int @ B3 )
= zero_zero_nat )
= ( B3 = zero_zero_int ) ) ).
% euclidean_size_eq_0_iff
thf(fact_48_euclidean__size__eq__0__iff,axiom,
! [B3: nat] :
( ( ( euclid4777050414544973029ze_nat @ B3 )
= zero_zero_nat )
= ( B3 = zero_zero_nat ) ) ).
% euclidean_size_eq_0_iff
thf(fact_49_size__0,axiom,
( ( euclid6377331345833325938nteger @ zero_z3403309356797280102nteger )
= zero_zero_nat ) ).
% size_0
thf(fact_50_size__0,axiom,
( ( euclid4774559944035922753ze_int @ zero_zero_int )
= zero_zero_nat ) ).
% size_0
thf(fact_51_size__0,axiom,
( ( euclid4777050414544973029ze_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% size_0
thf(fact_52_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_53_zero__reorient,axiom,
! [X: literal] :
( ( zero_zero_literal = X )
= ( X = zero_zero_literal ) ) ).
% zero_reorient
thf(fact_54_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_55_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_56_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_57_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_58_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X22 ) ) ).
% VEBT.distinct(1)
thf(fact_59_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X222: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X222 ) ) ) ).
% VEBT.exhaust
thf(fact_60_fact__nonzero,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ N )
!= zero_zero_rat ) ).
% fact_nonzero
thf(fact_61_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ N )
!= zero_zero_int ) ).
% fact_nonzero
thf(fact_62_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ N )
!= zero_zero_nat ) ).
% fact_nonzero
thf(fact_63_fact__nonzero,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ N )
!= zero_zero_real ) ).
% fact_nonzero
thf(fact_64_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( ( ( X = zero_zero_nat )
=> A3 )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B3 )
& ( X = one_one_nat ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_65_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_66_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_67_obtain__set__pred,axiom,
! [Z: nat,X: nat,A4: set_nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
=> ( ( finite_finite_nat @ A4 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_68_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X ) ) ).
% min_Null_member
thf(fact_69_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_70_mem__Collect__eq,axiom,
! [A3: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A3: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A3 @ ( collect_list_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A3 @ ( collect_set_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A3: int,P: int > $o] :
( ( member_int @ A3 @ ( collect_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A4: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_82_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_83_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_84_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_85_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_86_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_87_pred__none__empty,axiom,
! [Xs: set_nat,A3: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A3 @ X_1 )
=> ( ( finite_finite_nat @ Xs )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ Xs )
& ( ord_less_nat @ X5 @ A3 ) ) ) ) ).
% pred_none_empty
thf(fact_88_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_89_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_90_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_91_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_92_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_93_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_94_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S ) ) ) ).
% deg_SUcn_Node
thf(fact_95_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs2: set_nat,X3: nat] :
( ( member_nat @ X3 @ Xs2 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ Xs2 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% max_in_set_def
thf(fact_96_Suc,axiom,
( sucX
= ( suc @ nat3 ) ) ).
% Suc
thf(fact_97_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs2: set_nat,X3: nat] :
( ( member_nat @ X3 @ Xs2 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ Xs2 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% min_in_set_def
thf(fact_98_pred__member,axiom,
! [T: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
= ( ( vEBT_vebt_member @ T @ Y )
& ( ord_less_nat @ Y @ X )
& ! [Z2: nat] :
( ( ( vEBT_vebt_member @ T @ Z2 )
& ( ord_less_nat @ Z2 @ X ) )
=> ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ).
% pred_member
thf(fact_99_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_100_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_101__092_060open_0620_A_092_060le_062_Ax_A_092_060and_062_Ax_A_061_ASuc_AsucX_092_060close_062,axiom,
( ( ord_less_eq_nat @ zero_zero_nat @ xa )
& ( xa
= ( suc @ sucX ) ) ) ).
% \<open>0 \<le> x \<and> x = Suc sucX\<close>
thf(fact_102_fact__Suc__0,axiom,
( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
= one_one_complex ) ).
% fact_Suc_0
thf(fact_103_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_104_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_105_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_106_fact__Suc__0,axiom,
( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% fact_Suc_0
thf(fact_107_euclidean__size__greater__0__iff,axiom,
! [B3: code_integer] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid6377331345833325938nteger @ B3 ) )
= ( B3 != zero_z3403309356797280102nteger ) ) ).
% euclidean_size_greater_0_iff
thf(fact_108_euclidean__size__greater__0__iff,axiom,
! [B3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4774559944035922753ze_int @ B3 ) )
= ( B3 != zero_zero_int ) ) ).
% euclidean_size_greater_0_iff
thf(fact_109_euclidean__size__greater__0__iff,axiom,
! [B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4777050414544973029ze_nat @ B3 ) )
= ( B3 != zero_zero_nat ) ) ).
% euclidean_size_greater_0_iff
thf(fact_110_less__shift,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ).
% less_shift
thf(fact_111_lesseq__shift,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ).
% lesseq_shift
thf(fact_112_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_113_linorder__neqE__linordered__idom,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_114_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_115_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_116_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_self
thf(fact_117_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_118_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_119_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_120_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_121_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_122_finite__has__maximal2,axiom,
! [A4: set_real,A3: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ A3 @ X4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_123_finite__has__maximal2,axiom,
! [A4: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A3 @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ord_less_eq_set_nat @ A3 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_124_finite__has__maximal2,axiom,
! [A4: set_set_int,A3: set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( member_set_int @ A3 @ A4 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ( ord_less_eq_set_int @ A3 @ X4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_125_finite__has__maximal2,axiom,
! [A4: set_rat,A3: rat] :
( ( finite_finite_rat @ A4 )
=> ( ( member_rat @ A3 @ A4 )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ( ord_less_eq_rat @ A3 @ X4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_126_finite__has__maximal2,axiom,
! [A4: set_num,A3: num] :
( ( finite_finite_num @ A4 )
=> ( ( member_num @ A3 @ A4 )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ( ord_less_eq_num @ A3 @ X4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_127_finite__has__maximal2,axiom,
! [A4: set_nat,A3: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ A3 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_128_finite__has__maximal2,axiom,
! [A4: set_int,A3: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A3 @ A4 )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ A3 @ X4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_129_finite__has__minimal2,axiom,
! [A4: set_real,A3: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ X4 @ A3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_130_finite__has__minimal2,axiom,
! [A4: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A3 @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ord_less_eq_set_nat @ X4 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_131_finite__has__minimal2,axiom,
! [A4: set_set_int,A3: set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( member_set_int @ A3 @ A4 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ( ord_less_eq_set_int @ X4 @ A3 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_132_finite__has__minimal2,axiom,
! [A4: set_rat,A3: rat] :
( ( finite_finite_rat @ A4 )
=> ( ( member_rat @ A3 @ A4 )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ( ord_less_eq_rat @ X4 @ A3 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_133_finite__has__minimal2,axiom,
! [A4: set_num,A3: num] :
( ( finite_finite_num @ A4 )
=> ( ( member_num @ A3 @ A4 )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ( ord_less_eq_num @ X4 @ A3 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_134_finite__has__minimal2,axiom,
! [A4: set_nat,A3: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ X4 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_135_finite__has__minimal2,axiom,
! [A4: set_int,A3: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A3 @ A4 )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ X4 @ A3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_136_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_137_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_138_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_139_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_140_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs2: set_nat,X3: nat,Y3: nat] :
( ( member_nat @ Y3 @ Xs2 )
& ( ord_less_nat @ Y3 @ X3 )
& ! [Z2: nat] :
( ( member_nat @ Z2 @ Xs2 )
=> ( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_141_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_142_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_143_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_144_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_145_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_146_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_147_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_148_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_149_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_150_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [Va2: nat] :
( X
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_151_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_152_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_153_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_154_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_155_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_156_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_157_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_158_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_159_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_160_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_161_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_162_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_163_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_173_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_gt_zero
thf(fact_174_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_175_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_176_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_gt_zero
thf(fact_177_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% fact_not_neg
thf(fact_178_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_179_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_180_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% fact_not_neg
thf(fact_181_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_zero
thf(fact_182_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_zero
thf(fact_183_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_zero
thf(fact_184_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_zero
thf(fact_185_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_1
thf(fact_186_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_187_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_188_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_1
thf(fact_189_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_190_invar__vebt_Ointros_I1_J,axiom,
! [A3: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ).
% invar_vebt.intros(1)
thf(fact_191_greater__shift,axiom,
( ord_less_nat
= ( ^ [Y3: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ).
% greater_shift
thf(fact_192_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_193_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_194_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_195_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_196_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_197_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_198_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_199_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_200_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_201_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_202_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_203_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_204_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_205_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_206_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_207_finite__psubset__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_208_finite__psubset__induct,axiom,
! [A4: set_int,P: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [A5: set_int] :
( ( finite_finite_int @ A5 )
=> ( ! [B4: set_int] :
( ( ord_less_set_int @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_209_finite__psubset__induct,axiom,
! [A4: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [A5: set_complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ! [B4: set_complex] :
( ( ord_less_set_complex @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_210_rev__finite__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_211_rev__finite__subset,axiom,
! [B5: set_complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( finite3207457112153483333omplex @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_212_rev__finite__subset,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_213_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_214_infinite__super,axiom,
! [S2: set_complex,T2: set_complex] :
( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% infinite_super
thf(fact_215_infinite__super,axiom,
! [S2: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_216_finite__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_217_finite__subset,axiom,
! [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( finite3207457112153483333omplex @ A4 ) ) ) ).
% finite_subset
thf(fact_218_finite__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% finite_subset
thf(fact_219_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ zero_zero_nat )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_220_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_221_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_222_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_223_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_224_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_225_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_226_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_227_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_228_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_229_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_230_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_231_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_232_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_233_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_234_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_235_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_236_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_237_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_238_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_239_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_240_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_241_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_242_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_243_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_244_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_245_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_246_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_247_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_248_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_249_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_250_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_251_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_252_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_253_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_254_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_255_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_256_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_257_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_258_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_259_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_260_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_261_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_262_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_263_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_264_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_265_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_266_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_267_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_268_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_269_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_270_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_271_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_272_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_273_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_274_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_275_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_276_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_277_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_278_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_279_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_280_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_281_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_282_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_283_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_284_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_285_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_286_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_287_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_288_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_289_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_290_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_291_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_292_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_293_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_294_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_295_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_296_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_297_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_298_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_299_lift__Suc__mono__less,axiom,
! [F: nat > rat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_300_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_301_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_302_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_303_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_304_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_305_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_306_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_307_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_308_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_309_lift__Suc__mono__le,axiom,
! [F: nat > rat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_310_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_311_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_312_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_313_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_314_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_315_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_316_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_317_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_318_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_319_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_320_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_321_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_322_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_323_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_324_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_325_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_326_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_327_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_328_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_329_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_330_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_331_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_332_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_333_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_334_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_335_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_336_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= bot_bot_set_nat ) ).
% buildup_gives_empty
thf(fact_337_dual__order_Orefl,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_338_dual__order_Orefl,axiom,
! [A3: rat] : ( ord_less_eq_rat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_339_dual__order_Orefl,axiom,
! [A3: num] : ( ord_less_eq_num @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_340_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_341_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_342_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_343_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_344_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_345_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_346_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_347_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_348_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_349_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M6: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( member_nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_350_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_351_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M6: nat] :
? [N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ( member_nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_352_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_353_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M3: nat] :
( ( ord_less_nat @ K @ M3 )
=> ? [N7: nat] :
( ( ord_less_nat @ M3 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_354_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_355_bot_Oextremum,axiom,
! [A3: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A3 ) ).
% bot.extremum
thf(fact_356_bot_Oextremum,axiom,
! [A3: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A3 ) ).
% bot.extremum
thf(fact_357_bot_Oextremum,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% bot.extremum
thf(fact_358_bot_Oextremum,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).
% bot.extremum
thf(fact_359_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_360_bot_Oextremum__unique,axiom,
! [A3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ bot_bot_filter_nat )
= ( A3 = bot_bot_filter_nat ) ) ).
% bot.extremum_unique
thf(fact_361_bot_Oextremum__unique,axiom,
! [A3: set_real] :
( ( ord_less_eq_set_real @ A3 @ bot_bot_set_real )
= ( A3 = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_362_bot_Oextremum__unique,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_363_bot_Oextremum__unique,axiom,
! [A3: set_int] :
( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
= ( A3 = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_364_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_365_bot_Oextremum__uniqueI,axiom,
! [A3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ bot_bot_filter_nat )
=> ( A3 = bot_bot_filter_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_366_bot_Oextremum__uniqueI,axiom,
! [A3: set_real] :
( ( ord_less_eq_set_real @ A3 @ bot_bot_set_real )
=> ( A3 = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_367_bot_Oextremum__uniqueI,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
=> ( A3 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_368_bot_Oextremum__uniqueI,axiom,
! [A3: set_int] :
( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
=> ( A3 = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_369_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_370_bot_Oextremum__strict,axiom,
! [A3: filter_nat] :
~ ( ord_less_filter_nat @ A3 @ bot_bot_filter_nat ) ).
% bot.extremum_strict
thf(fact_371_bot_Oextremum__strict,axiom,
! [A3: set_real] :
~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_372_bot_Oextremum__strict,axiom,
! [A3: set_nat] :
~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_373_bot_Oextremum__strict,axiom,
! [A3: set_int] :
~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_374_bot_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_375_bot_Onot__eq__extremum,axiom,
! [A3: filter_nat] :
( ( A3 != bot_bot_filter_nat )
= ( ord_less_filter_nat @ bot_bot_filter_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_376_bot_Onot__eq__extremum,axiom,
! [A3: set_real] :
( ( A3 != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_377_bot_Onot__eq__extremum,axiom,
! [A3: set_nat] :
( ( A3 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_378_bot_Onot__eq__extremum,axiom,
! [A3: set_int] :
( ( A3 != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_379_bot_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_380_finite__transitivity__chain,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [X4: product_prod_nat_nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ? [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A4 )
& ( R @ X4 @ Y5 ) ) )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_381_finite__transitivity__chain,axiom,
! [A4: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ! [X4: set_nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: set_nat,Y4: set_nat,Z3: set_nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A4 )
& ( R @ X4 @ Y5 ) ) )
=> ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_382_finite__transitivity__chain,axiom,
! [A4: set_complex,R: complex > complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: complex,Y4: complex,Z3: complex] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ? [Y5: complex] :
( ( member_complex @ Y5 @ A4 )
& ( R @ X4 @ Y5 ) ) )
=> ( A4 = bot_bot_set_complex ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_383_finite__transitivity__chain,axiom,
! [A4: set_real,R: real > real > $o] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: real,Y4: real,Z3: real] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Y5: real] :
( ( member_real @ Y5 @ A4 )
& ( R @ X4 @ Y5 ) ) )
=> ( A4 = bot_bot_set_real ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_384_finite__transitivity__chain,axiom,
! [A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
& ( R @ X4 @ Y5 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_385_finite__transitivity__chain,axiom,
! [A4: set_int,R: int > int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: int,Y4: int,Z3: int] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ A4 )
& ( R @ X4 @ Y5 ) ) )
=> ( A4 = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_386_finite_OemptyI,axiom,
finite3207457112153483333omplex @ bot_bot_set_complex ).
% finite.emptyI
thf(fact_387_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_388_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_389_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_390_infinite__imp__nonempty,axiom,
! [S2: set_complex] :
( ~ ( finite3207457112153483333omplex @ S2 )
=> ( S2 != bot_bot_set_complex ) ) ).
% infinite_imp_nonempty
thf(fact_391_infinite__imp__nonempty,axiom,
! [S2: set_real] :
( ~ ( finite_finite_real @ S2 )
=> ( S2 != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_392_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_393_infinite__imp__nonempty,axiom,
! [S2: set_int] :
( ~ ( finite_finite_int @ S2 )
=> ( S2 != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_394_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_395_finite__has__minimal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_396_finite__has__minimal,axiom,
! [A4: set_set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( A4 != bot_bot_set_set_int )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_397_finite__has__minimal,axiom,
! [A4: set_rat] :
( ( finite_finite_rat @ A4 )
=> ( ( A4 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_398_finite__has__minimal,axiom,
! [A4: set_num] :
( ( finite_finite_num @ A4 )
=> ( ( A4 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_399_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_400_finite__has__minimal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_401_finite__has__maximal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_402_finite__has__maximal,axiom,
! [A4: set_set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( A4 != bot_bot_set_set_int )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_403_finite__has__maximal,axiom,
! [A4: set_rat] :
( ( finite_finite_rat @ A4 )
=> ( ( A4 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_404_finite__has__maximal,axiom,
! [A4: set_num] :
( ( finite_finite_num @ A4 )
=> ( ( A4 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_405_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_406_finite__has__maximal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_407_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_408_nle__le,axiom,
! [A3: rat,B3: rat] :
( ( ~ ( ord_less_eq_rat @ A3 @ B3 ) )
= ( ( ord_less_eq_rat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_409_nle__le,axiom,
! [A3: num,B3: num] :
( ( ~ ( ord_less_eq_num @ A3 @ B3 ) )
= ( ( ord_less_eq_num @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_410_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_411_nle__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_412_le__cases3,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_eq_rat @ X @ Z ) )
=> ( ( ( ord_less_eq_rat @ X @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X )
=> ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_413_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_414_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_415_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_416_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
= ( ^ [X3: set_int,Y3: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y3 )
& ( ord_less_eq_set_int @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_417_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
& ( ord_less_eq_rat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_418_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_419_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_420_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_421_ord__eq__le__trans,axiom,
! [A3: set_int,B3: set_int,C: set_int] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_int @ B3 @ C )
=> ( ord_less_eq_set_int @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_422_ord__eq__le__trans,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_eq_rat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_423_ord__eq__le__trans,axiom,
! [A3: num,B3: num,C: num] :
( ( A3 = B3 )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ord_less_eq_num @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_424_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_425_ord__eq__le__trans,axiom,
! [A3: int,B3: int,C: int] :
( ( A3 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_426_ord__le__eq__trans,axiom,
! [A3: set_int,B3: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_int @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_427_ord__le__eq__trans,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_rat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_428_ord__le__eq__trans,axiom,
! [A3: num,B3: num,C: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_num @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_429_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_430_ord__le__eq__trans,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_int @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_431_order__antisym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_432_order__antisym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_433_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_434_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_435_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_436_order_Otrans,axiom,
! [A3: set_int,B3: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ C )
=> ( ord_less_eq_set_int @ A3 @ C ) ) ) ).
% order.trans
thf(fact_437_order_Otrans,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_eq_rat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_438_order_Otrans,axiom,
! [A3: num,B3: num,C: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ord_less_eq_num @ A3 @ C ) ) ) ).
% order.trans
thf(fact_439_order_Otrans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_440_order_Otrans,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A3 @ C ) ) ) ).
% order.trans
thf(fact_441_order__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_442_order__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_eq_rat @ X @ Z ) ) ) ).
% order_trans
thf(fact_443_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_444_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_445_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_446_linorder__wlog,axiom,
! [P: rat > rat > $o,A3: rat,B3: rat] :
( ! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: rat,B2: rat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_447_linorder__wlog,axiom,
! [P: num > num > $o,A3: num,B3: num] :
( ! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_448_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_449_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_450_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
= ( ^ [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ B @ A )
& ( ord_less_eq_set_int @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_451_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
= ( ^ [A: rat,B: rat] :
( ( ord_less_eq_rat @ B @ A )
& ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_452_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
= ( ^ [A: num,B: num] :
( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_453_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_454_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_455_dual__order_Oantisym,axiom,
! [B3: set_int,A3: set_int] :
( ( ord_less_eq_set_int @ B3 @ A3 )
=> ( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_456_dual__order_Oantisym,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_457_dual__order_Oantisym,axiom,
! [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
=> ( ( ord_less_eq_num @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_458_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_459_dual__order_Oantisym,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_460_dual__order_Otrans,axiom,
! [B3: set_int,A3: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B3 @ A3 )
=> ( ( ord_less_eq_set_int @ C @ B3 )
=> ( ord_less_eq_set_int @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_461_dual__order_Otrans,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ C @ B3 )
=> ( ord_less_eq_rat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_462_dual__order_Otrans,axiom,
! [B3: num,A3: num,C: num] :
( ( ord_less_eq_num @ B3 @ A3 )
=> ( ( ord_less_eq_num @ C @ B3 )
=> ( ord_less_eq_num @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_463_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_464_dual__order_Otrans,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_465_antisym,axiom,
! [A3: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_466_antisym,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_467_antisym,axiom,
! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_eq_num @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_468_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_469_antisym,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_470_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
= ( ^ [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
& ( ord_less_eq_set_int @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_471_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
= ( ^ [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
& ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_472_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
= ( ^ [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
& ( ord_less_eq_num @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_473_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_474_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_475_order__subst1,axiom,
! [A3: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_476_order__subst1,axiom,
! [A3: rat,F: num > rat,B3: num,C: num] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_477_order__subst1,axiom,
! [A3: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_478_order__subst1,axiom,
! [A3: rat,F: int > rat,B3: int,C: int] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_479_order__subst1,axiom,
! [A3: num,F: rat > num,B3: rat,C: rat] :
( ( ord_less_eq_num @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_480_order__subst1,axiom,
! [A3: num,F: num > num,B3: num,C: num] :
( ( ord_less_eq_num @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_481_order__subst1,axiom,
! [A3: num,F: nat > num,B3: nat,C: nat] :
( ( ord_less_eq_num @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_482_order__subst1,axiom,
! [A3: num,F: int > num,B3: int,C: int] :
( ( ord_less_eq_num @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_483_order__subst1,axiom,
! [A3: nat,F: rat > nat,B3: rat,C: rat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_484_order__subst1,axiom,
! [A3: nat,F: num > nat,B3: num,C: num] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_485_order__subst2,axiom,
! [A3: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_486_order__subst2,axiom,
! [A3: rat,B3: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_487_order__subst2,axiom,
! [A3: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_488_order__subst2,axiom,
! [A3: rat,B3: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_489_order__subst2,axiom,
! [A3: num,B3: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_490_order__subst2,axiom,
! [A3: num,B3: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_491_order__subst2,axiom,
! [A3: num,B3: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_492_order__subst2,axiom,
! [A3: num,B3: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_493_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_494_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_495_order__eq__refl,axiom,
! [X: set_int,Y: set_int] :
( ( X = Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_496_order__eq__refl,axiom,
! [X: rat,Y: rat] :
( ( X = Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_497_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_498_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_499_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_500_linorder__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_linear
thf(fact_501_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_502_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_503_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_504_ord__eq__le__subst,axiom,
! [A3: rat,F: rat > rat,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_505_ord__eq__le__subst,axiom,
! [A3: num,F: rat > num,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_506_ord__eq__le__subst,axiom,
! [A3: nat,F: rat > nat,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_507_ord__eq__le__subst,axiom,
! [A3: int,F: rat > int,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_508_ord__eq__le__subst,axiom,
! [A3: rat,F: num > rat,B3: num,C: num] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_509_ord__eq__le__subst,axiom,
! [A3: num,F: num > num,B3: num,C: num] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_510_ord__eq__le__subst,axiom,
! [A3: nat,F: num > nat,B3: num,C: num] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_511_ord__eq__le__subst,axiom,
! [A3: int,F: num > int,B3: num,C: num] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_512_ord__eq__le__subst,axiom,
! [A3: rat,F: nat > rat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_513_ord__eq__le__subst,axiom,
! [A3: num,F: nat > num,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_514_ord__le__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_515_ord__le__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_516_ord__le__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_517_ord__le__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_518_ord__le__eq__subst,axiom,
! [A3: num,B3: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_519_ord__le__eq__subst,axiom,
! [A3: num,B3: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_ord__le__eq__subst,axiom,
! [A3: num,B3: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_521_ord__le__eq__subst,axiom,
! [A3: num,B3: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_522_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_523_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_524_linorder__le__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_525_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_526_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_527_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_528_order__antisym__conv,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_529_order__antisym__conv,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_530_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_531_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_532_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_533_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_534_lt__ex,axiom,
! [X: rat] :
? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% lt_ex
thf(fact_535_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_536_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_537_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_538_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_539_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_540_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_541_dense,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ? [Z3: rat] :
( ( ord_less_rat @ X @ Z3 )
& ( ord_less_rat @ Z3 @ Y ) ) ) ).
% dense
thf(fact_542_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_543_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_544_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_545_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_546_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_547_order_Oasym,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order.asym
thf(fact_548_order_Oasym,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ~ ( ord_less_rat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_549_order_Oasym,axiom,
! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ~ ( ord_less_num @ B3 @ A3 ) ) ).
% order.asym
thf(fact_550_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_551_order_Oasym,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ~ ( ord_less_int @ B3 @ A3 ) ) ).
% order.asym
thf(fact_552_ord__eq__less__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( A3 = B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_553_ord__eq__less__trans,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( A3 = B3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_554_ord__eq__less__trans,axiom,
! [A3: num,B3: num,C: num] :
( ( A3 = B3 )
=> ( ( ord_less_num @ B3 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_555_ord__eq__less__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_556_ord__eq__less__trans,axiom,
! [A3: int,B3: int,C: int] :
( ( A3 = B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_557_ord__less__eq__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_558_ord__less__eq__trans,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_rat @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_559_ord__less__eq__trans,axiom,
! [A3: num,B3: num,C: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_560_ord__less__eq__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_561_ord__less__eq__trans,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_562_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_563_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_564_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_rat @ Y @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_565_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_566_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_567_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_568_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_569_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_570_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_571_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_572_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_573_dual__order_Oasym,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_574_dual__order_Oasym,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ~ ( ord_less_rat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_575_dual__order_Oasym,axiom,
! [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
=> ~ ( ord_less_num @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_576_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_577_dual__order_Oasym,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ~ ( ord_less_int @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_578_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_579_dual__order_Oirrefl,axiom,
! [A3: rat] :
~ ( ord_less_rat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_580_dual__order_Oirrefl,axiom,
! [A3: num] :
~ ( ord_less_num @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_581_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_582_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_583_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ~ ( P3 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_584_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B3: real] :
( ! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real] : ( P @ A2 @ A2 )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_585_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A3: rat,B3: rat] :
( ! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: rat] : ( P @ A2 @ A2 )
=> ( ! [A2: rat,B2: rat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_586_linorder__less__wlog,axiom,
! [P: num > num > $o,A3: num,B3: num] :
( ! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num] : ( P @ A2 @ A2 )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_587_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_588_linorder__less__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_589_order_Ostrict__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_590_order_Ostrict__trans,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_591_order_Ostrict__trans,axiom,
! [A3: num,B3: num,C: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( ( ord_less_num @ B3 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_592_order_Ostrict__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_593_order_Ostrict__trans,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_594_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_595_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ( ord_less_rat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_596_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_597_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_598_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_599_dual__order_Ostrict__trans,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_600_dual__order_Ostrict__trans,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_less_rat @ C @ B3 )
=> ( ord_less_rat @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_601_dual__order_Ostrict__trans,axiom,
! [B3: num,A3: num,C: num] :
( ( ord_less_num @ B3 @ A3 )
=> ( ( ord_less_num @ C @ B3 )
=> ( ord_less_num @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_602_dual__order_Ostrict__trans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_603_dual__order_Ostrict__trans,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_604_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_605_order_Ostrict__implies__not__eq,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_606_order_Ostrict__implies__not__eq,axiom,
! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_607_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_608_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_609_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_610_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_611_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_612_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_613_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_614_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_615_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_616_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_617_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_618_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_619_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_620_order__less__asym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_621_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_622_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_623_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_624_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_625_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
= ( ( ord_less_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_626_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_627_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_628_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_629_order__less__asym_H,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_630_order__less__asym_H,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ~ ( ord_less_rat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_631_order__less__asym_H,axiom,
! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ~ ( ord_less_num @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_632_order__less__asym_H,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_633_order__less__asym_H,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ~ ( ord_less_int @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_634_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_635_order__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_636_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_637_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_638_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_639_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_640_ord__eq__less__subst,axiom,
! [A3: rat,F: real > rat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_641_ord__eq__less__subst,axiom,
! [A3: num,F: real > num,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_642_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_643_ord__eq__less__subst,axiom,
! [A3: int,F: real > int,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_644_ord__eq__less__subst,axiom,
! [A3: real,F: rat > real,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_645_ord__eq__less__subst,axiom,
! [A3: rat,F: rat > rat,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_646_ord__eq__less__subst,axiom,
! [A3: num,F: rat > num,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_647_ord__eq__less__subst,axiom,
! [A3: nat,F: rat > nat,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_648_ord__eq__less__subst,axiom,
! [A3: int,F: rat > int,B3: rat,C: rat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_649_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_650_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > rat,C: rat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_651_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > num,C: num] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_652_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_653_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > int,C: int] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_654_ord__less__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_655_ord__less__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_656_ord__less__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_657_ord__less__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_658_ord__less__eq__subst,axiom,
! [A3: rat,B3: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_659_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_660_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_661_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_662_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_663_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_664_order__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_665_order__less__subst1,axiom,
! [A3: real,F: rat > real,B3: rat,C: rat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_666_order__less__subst1,axiom,
! [A3: real,F: num > real,B3: num,C: num] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_667_order__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_668_order__less__subst1,axiom,
! [A3: real,F: int > real,B3: int,C: int] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_669_order__less__subst1,axiom,
! [A3: rat,F: real > rat,B3: real,C: real] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_670_order__less__subst1,axiom,
! [A3: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_671_order__less__subst1,axiom,
! [A3: rat,F: num > rat,B3: num,C: num] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_672_order__less__subst1,axiom,
! [A3: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_673_order__less__subst1,axiom,
! [A3: rat,F: int > rat,B3: int,C: int] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_674_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_675_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > rat,C: rat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_676_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > num,C: num] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_677_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_678_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > int,C: int] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_679_order__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_680_order__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_681_order__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_682_order__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_683_order__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_684_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_685_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_686_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_687_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_688_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_689_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_690_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_691_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_692_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_693_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_694_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_695_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_696_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_697_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_698_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_699_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_700_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_701_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_702_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_703_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_704_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_705_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_706_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_707_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_708_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_709_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_710_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_711_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_712_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_713_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_714_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_715_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( ( ( X = zero_zero_nat )
=> A3 )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B3 )
& ( X = one_one_nat ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_716_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_717_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_718_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_719_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_720_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_721_leD,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ~ ( ord_less_set_int @ X @ Y ) ) ).
% leD
thf(fact_722_leD,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_rat @ X @ Y ) ) ).
% leD
thf(fact_723_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_724_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_725_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_726_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_727_leI,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% leI
thf(fact_728_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_729_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_730_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_731_nless__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_732_nless__le,axiom,
! [A3: set_int,B3: set_int] :
( ( ~ ( ord_less_set_int @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_set_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_733_nless__le,axiom,
! [A3: rat,B3: rat] :
( ( ~ ( ord_less_rat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_rat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_734_nless__le,axiom,
! [A3: num,B3: num] :
( ( ~ ( ord_less_num @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_735_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_736_nless__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_int @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_737_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_738_antisym__conv1,axiom,
! [X: set_int,Y: set_int] :
( ~ ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_739_antisym__conv1,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_740_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_741_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_742_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_743_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_744_antisym__conv2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ~ ( ord_less_set_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_745_antisym__conv2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_746_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_747_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_748_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_749_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ord_less_eq_real @ Y @ X4 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_750_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X4: rat] :
( ( ord_less_rat @ Z @ X4 )
=> ( ord_less_eq_rat @ Y @ X4 ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_ge
thf(fact_751_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_752_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X4: rat] :
( ( ord_less_rat @ X4 @ Y )
=> ( ord_less_eq_rat @ X4 @ Z ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_le
thf(fact_753_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_754_less__le__not__le,axiom,
( ord_less_set_int
= ( ^ [X3: set_int,Y3: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y3 )
& ~ ( ord_less_eq_set_int @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_755_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
& ~ ( ord_less_eq_rat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_756_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_757_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_758_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_759_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_760_not__le__imp__less,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_rat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_761_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_762_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_763_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_764_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A: real,B: real] :
( ( ord_less_real @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_765_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [A: set_int,B: set_int] :
( ( ord_less_set_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_766_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_767_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A: num,B: num] :
( ( ord_less_num @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_768_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_769_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_less_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_770_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_771_order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_772_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_773_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_774_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_775_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_776_order_Ostrict__trans1,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_777_order_Ostrict__trans1,axiom,
! [A3: set_int,B3: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( ord_less_set_int @ B3 @ C )
=> ( ord_less_set_int @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_778_order_Ostrict__trans1,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_779_order_Ostrict__trans1,axiom,
! [A3: num,B3: num,C: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_num @ B3 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_780_order_Ostrict__trans1,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_781_order_Ostrict__trans1,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_782_order_Ostrict__trans2,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_783_order_Ostrict__trans2,axiom,
! [A3: set_int,B3: set_int,C: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ C )
=> ( ord_less_set_int @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_784_order_Ostrict__trans2,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_rat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_785_order_Ostrict__trans2,axiom,
! [A3: num,B3: num,C: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_786_order_Ostrict__trans2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_787_order_Ostrict__trans2,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_788_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ~ ( ord_less_eq_real @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_789_order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
& ~ ( ord_less_eq_set_int @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_790_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
& ~ ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_791_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
& ~ ( ord_less_eq_num @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_792_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ~ ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_793_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ~ ( ord_less_eq_int @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_794_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_795_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z @ W )
=> ( ( ord_less_rat @ W @ X )
=> ( ord_less_eq_rat @ Y @ W ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_796_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_797_dense__le__bounded,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ! [W: rat] :
( ( ord_less_rat @ X @ W )
=> ( ( ord_less_rat @ W @ Y )
=> ( ord_less_eq_rat @ W @ Z ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_798_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B: real,A: real] :
( ( ord_less_real @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_799_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [B: set_int,A: set_int] :
( ( ord_less_set_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_800_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_801_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B: num,A: num] :
( ( ord_less_num @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_802_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_803_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_less_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_804_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_805_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_806_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_807_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_808_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_809_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_810_dual__order_Ostrict__trans1,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_811_dual__order_Ostrict__trans1,axiom,
! [B3: set_int,A3: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B3 @ A3 )
=> ( ( ord_less_set_int @ C @ B3 )
=> ( ord_less_set_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_812_dual__order_Ostrict__trans1,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_rat @ C @ B3 )
=> ( ord_less_rat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_813_dual__order_Ostrict__trans1,axiom,
! [B3: num,A3: num,C: num] :
( ( ord_less_eq_num @ B3 @ A3 )
=> ( ( ord_less_num @ C @ B3 )
=> ( ord_less_num @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_814_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_815_dual__order_Ostrict__trans1,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_816_dual__order_Ostrict__trans2,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_817_dual__order_Ostrict__trans2,axiom,
! [B3: set_int,A3: set_int,C: set_int] :
( ( ord_less_set_int @ B3 @ A3 )
=> ( ( ord_less_eq_set_int @ C @ B3 )
=> ( ord_less_set_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_818_dual__order_Ostrict__trans2,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ C @ B3 )
=> ( ord_less_rat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_819_dual__order_Ostrict__trans2,axiom,
! [B3: num,A3: num,C: num] :
( ( ord_less_num @ B3 @ A3 )
=> ( ( ord_less_eq_num @ C @ B3 )
=> ( ord_less_num @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_820_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_821_dual__order_Ostrict__trans2,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_822_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
& ~ ( ord_less_eq_real @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_823_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
& ~ ( ord_less_eq_set_int @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_824_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
& ~ ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_825_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
& ~ ( ord_less_eq_num @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_826_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_827_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ~ ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_828_order_Ostrict__implies__order,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_829_order_Ostrict__implies__order,axiom,
! [A3: set_int,B3: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ord_less_eq_set_int @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_830_order_Ostrict__implies__order,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_831_order_Ostrict__implies__order,axiom,
! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( ord_less_eq_num @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_832_order_Ostrict__implies__order,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_833_order_Ostrict__implies__order,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_834_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_835_dual__order_Ostrict__implies__order,axiom,
! [B3: set_int,A3: set_int] :
( ( ord_less_set_int @ B3 @ A3 )
=> ( ord_less_eq_set_int @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_836_dual__order_Ostrict__implies__order,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ord_less_eq_rat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_837_dual__order_Ostrict__implies__order,axiom,
! [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
=> ( ord_less_eq_num @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_838_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_839_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_840_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_841_order__le__less,axiom,
( ord_less_eq_set_int
= ( ^ [X3: set_int,Y3: set_int] :
( ( ord_less_set_int @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_842_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_843_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_844_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_845_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_846_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_847_order__less__le,axiom,
( ord_less_set_int
= ( ^ [X3: set_int,Y3: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_848_order__less__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_849_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_850_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_851_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_852_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_853_linorder__not__le,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y ) )
= ( ord_less_rat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_854_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_855_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_856_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_857_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_858_linorder__not__less,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_859_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_860_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_861_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_862_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_863_order__less__imp__le,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_864_order__less__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_865_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_866_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_867_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_868_order__le__neq__trans,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_869_order__le__neq__trans,axiom,
! [A3: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_int @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_870_order__le__neq__trans,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_rat @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_871_order__le__neq__trans,axiom,
! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_num @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_872_order__le__neq__trans,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_873_order__le__neq__trans,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_874_order__neq__le__trans,axiom,
! [A3: real,B3: real] :
( ( A3 != B3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_875_order__neq__le__trans,axiom,
! [A3: set_int,B3: set_int] :
( ( A3 != B3 )
=> ( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ord_less_set_int @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_876_order__neq__le__trans,axiom,
! [A3: rat,B3: rat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_rat @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_877_order__neq__le__trans,axiom,
! [A3: num,B3: num] :
( ( A3 != B3 )
=> ( ( ord_less_eq_num @ A3 @ B3 )
=> ( ord_less_num @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_878_order__neq__le__trans,axiom,
! [A3: nat,B3: nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_879_order__neq__le__trans,axiom,
! [A3: int,B3: int] :
( ( A3 != B3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_880_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_881_order__le__less__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_882_order__le__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_883_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_884_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_885_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_886_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_887_order__less__le__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_888_order__less__le__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_889_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_890_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_891_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_892_order__le__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_893_order__le__less__subst1,axiom,
! [A3: real,F: rat > real,B3: rat,C: rat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_894_order__le__less__subst1,axiom,
! [A3: real,F: num > real,B3: num,C: num] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_895_order__le__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_896_order__le__less__subst1,axiom,
! [A3: real,F: int > real,B3: int,C: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_897_order__le__less__subst1,axiom,
! [A3: rat,F: real > rat,B3: real,C: real] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_898_order__le__less__subst1,axiom,
! [A3: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_899_order__le__less__subst1,axiom,
! [A3: rat,F: num > rat,B3: num,C: num] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_900_order__le__less__subst1,axiom,
! [A3: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_901_order__le__less__subst1,axiom,
! [A3: rat,F: int > rat,B3: int,C: int] :
( ( ord_less_eq_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_902_order__le__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_903_order__le__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_904_order__le__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_905_order__le__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_906_order__le__less__subst2,axiom,
! [A3: rat,B3: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_907_order__le__less__subst2,axiom,
! [A3: num,B3: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_908_order__le__less__subst2,axiom,
! [A3: num,B3: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_909_order__le__less__subst2,axiom,
! [A3: num,B3: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_910_order__le__less__subst2,axiom,
! [A3: num,B3: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_911_order__le__less__subst2,axiom,
! [A3: num,B3: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_912_order__less__le__subst1,axiom,
! [A3: real,F: rat > real,B3: rat,C: rat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_913_order__less__le__subst1,axiom,
! [A3: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_914_order__less__le__subst1,axiom,
! [A3: num,F: rat > num,B3: rat,C: rat] :
( ( ord_less_num @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_915_order__less__le__subst1,axiom,
! [A3: nat,F: rat > nat,B3: rat,C: rat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_916_order__less__le__subst1,axiom,
! [A3: int,F: rat > int,B3: rat,C: rat] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_917_order__less__le__subst1,axiom,
! [A3: real,F: num > real,B3: num,C: num] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_918_order__less__le__subst1,axiom,
! [A3: rat,F: num > rat,B3: num,C: num] :
( ( ord_less_rat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_919_order__less__le__subst1,axiom,
! [A3: num,F: num > num,B3: num,C: num] :
( ( ord_less_num @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_920_order__less__le__subst1,axiom,
! [A3: nat,F: num > nat,B3: num,C: num] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_921_order__less__le__subst1,axiom,
! [A3: int,F: num > int,B3: num,C: num] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_922_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_923_order__less__le__subst2,axiom,
! [A3: rat,B3: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_924_order__less__le__subst2,axiom,
! [A3: num,B3: num,F: num > real,C: real] :
( ( ord_less_num @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_925_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_926_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > real,C: real] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_927_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > rat,C: rat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_928_order__less__le__subst2,axiom,
! [A3: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_929_order__less__le__subst2,axiom,
! [A3: num,B3: num,F: num > rat,C: rat] :
( ( ord_less_num @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_930_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_931_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > rat,C: rat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_932_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_933_linorder__le__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_934_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_935_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_936_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_937_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_938_order__le__imp__less__or__eq,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_939_order__le__imp__less__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_940_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_941_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_942_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_943_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_944_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_945_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_946_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_947_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_948_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_949_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_950_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_951_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_952_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_953_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_954_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_955_psubsetI,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_int @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_956_subset__empty,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
= ( A4 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_957_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_958_subset__empty,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
= ( A4 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_959_empty__subsetI,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).
% empty_subsetI
thf(fact_960_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_961_empty__subsetI,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).
% empty_subsetI
thf(fact_962_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_963_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_964_ex__min__if__finite,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ S2 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S2 )
& ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_965_ex__min__if__finite,axiom,
! [S2: set_rat] :
( ( finite_finite_rat @ S2 )
=> ( ( S2 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ S2 )
& ~ ? [Xa: rat] :
( ( member_rat @ Xa @ S2 )
& ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_966_ex__min__if__finite,axiom,
! [S2: set_num] :
( ( finite_finite_num @ S2 )
=> ( ( S2 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ S2 )
& ~ ? [Xa: num] :
( ( member_num @ Xa @ S2 )
& ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_967_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_968_ex__min__if__finite,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ S2 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S2 )
& ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_969_infinite__growing,axiom,
! [X7: set_real] :
( ( X7 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X7 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X7 )
& ( ord_less_real @ X4 @ Xa ) ) )
=> ~ ( finite_finite_real @ X7 ) ) ) ).
% infinite_growing
thf(fact_970_infinite__growing,axiom,
! [X7: set_rat] :
( ( X7 != bot_bot_set_rat )
=> ( ! [X4: rat] :
( ( member_rat @ X4 @ X7 )
=> ? [Xa: rat] :
( ( member_rat @ Xa @ X7 )
& ( ord_less_rat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_rat @ X7 ) ) ) ).
% infinite_growing
thf(fact_971_infinite__growing,axiom,
! [X7: set_num] :
( ( X7 != bot_bot_set_num )
=> ( ! [X4: num] :
( ( member_num @ X4 @ X7 )
=> ? [Xa: num] :
( ( member_num @ Xa @ X7 )
& ( ord_less_num @ X4 @ Xa ) ) )
=> ~ ( finite_finite_num @ X7 ) ) ) ).
% infinite_growing
thf(fact_972_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( ord_less_nat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_973_infinite__growing,axiom,
! [X7: set_int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X7 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X7 )
& ( ord_less_int @ X4 @ Xa ) ) )
=> ~ ( finite_finite_int @ X7 ) ) ) ).
% infinite_growing
thf(fact_974_subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( member8440522571783428010at_nat @ X4 @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_975_subsetI,axiom,
! [A4: set_real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( member_real @ X4 @ B5 ) )
=> ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% subsetI
thf(fact_976_subsetI,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( member_set_nat @ X4 @ B5 ) )
=> ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_977_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_978_subsetI,axiom,
! [A4: set_int,B5: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( member_int @ X4 @ B5 ) )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% subsetI
thf(fact_979_subset__antisym,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_980_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_981_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_982_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_983_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_984_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_985_all__not__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ! [X3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X3 @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_986_all__not__in__conv,axiom,
! [A4: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A4 ) )
= ( A4 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_987_all__not__in__conv,axiom,
! [A4: set_real] :
( ( ! [X3: real] :
~ ( member_real @ X3 @ A4 ) )
= ( A4 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_988_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_989_all__not__in__conv,axiom,
! [A4: set_int] :
( ( ! [X3: int] :
~ ( member_int @ X3 @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_990_minminNull,axiom,
! [T: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T )
= none_nat )
=> ( vEBT_VEBT_minNull @ T ) ) ).
% minminNull
thf(fact_991_minNullmin,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T )
=> ( ( vEBT_vebt_mint @ T )
= none_nat ) ) ).
% minNullmin
thf(fact_992_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X3: list_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_993_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_994_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X3: real] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_995_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_996_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_997_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X3: list_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_998_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_999_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X3: real] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1000_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1001_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1002_not__Some__eq,axiom,
! [X: option_nat] :
( ( ! [Y3: nat] :
( X
!= ( some_nat @ Y3 ) ) )
= ( X = none_nat ) ) ).
% not_Some_eq
thf(fact_1003_not__Some__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( ! [Y3: product_prod_nat_nat] :
( X
!= ( some_P7363390416028606310at_nat @ Y3 ) ) )
= ( X = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
thf(fact_1004_not__Some__eq,axiom,
! [X: option_num] :
( ( ! [Y3: num] :
( X
!= ( some_num @ Y3 ) ) )
= ( X = none_num ) ) ).
% not_Some_eq
thf(fact_1005_not__None__eq,axiom,
! [X: option_nat] :
( ( X != none_nat )
= ( ? [Y3: nat] :
( X
= ( some_nat @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_1006_not__None__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( X != none_P5556105721700978146at_nat )
= ( ? [Y3: product_prod_nat_nat] :
( X
= ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_1007_not__None__eq,axiom,
! [X: option_num] :
( ( X != none_num )
= ( ? [Y3: num] :
( X
= ( some_num @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_1008_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_1009_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_1010_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_1011_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1012_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_1013_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1014_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A2: nat,B2: nat] :
( ( X
= ( some_nat @ A2 ) )
=> ( ( Y
= ( some_nat @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1015_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ! [A2: nat,B2: product_prod_nat_nat] :
( ( X
= ( some_nat @ A2 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1016_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P @ X @ Y ) )
=> ( ! [A2: nat,B2: num] :
( ( X
= ( some_nat @ A2 ) )
=> ( ( Y
= ( some_num @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1017_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A2: product_prod_nat_nat,B2: nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A2 ) )
=> ( ( Y
= ( some_nat @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1018_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A2 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1019_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P @ X @ Y ) )
=> ( ! [A2: product_prod_nat_nat,B2: num] :
( ( X
= ( some_P7363390416028606310at_nat @ A2 ) )
=> ( ( Y
= ( some_num @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1020_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A2: num,B2: nat] :
( ( X
= ( some_num @ A2 ) )
=> ( ( Y
= ( some_nat @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1021_combine__options__cases,axiom,
! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ! [A2: num,B2: product_prod_nat_nat] :
( ( X
= ( some_num @ A2 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1022_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
( ( ( X = none_num )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P @ X @ Y ) )
=> ( ! [A2: num,B2: num] :
( ( X
= ( some_num @ A2 ) )
=> ( ( Y
= ( some_num @ B2 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1023_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X6: option_nat] : ( P2 @ X6 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_1024_split__option__all,axiom,
( ( ^ [P2: option4927543243414619207at_nat > $o] :
! [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
= ( ^ [P3: option4927543243414619207at_nat > $o] :
( ( P3 @ none_P5556105721700978146at_nat )
& ! [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_1025_split__option__all,axiom,
( ( ^ [P2: option_num > $o] :
! [X6: option_num] : ( P2 @ X6 ) )
= ( ^ [P3: option_num > $o] :
( ( P3 @ none_num )
& ! [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_1026_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X6: option_nat] : ( P2 @ X6 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_1027_split__option__ex,axiom,
( ( ^ [P2: option4927543243414619207at_nat > $o] :
? [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
= ( ^ [P3: option4927543243414619207at_nat > $o] :
( ( P3 @ none_P5556105721700978146at_nat )
| ? [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_1028_split__option__ex,axiom,
( ( ^ [P2: option_num > $o] :
? [X6: option_num] : ( P2 @ X6 ) )
= ( ^ [P3: option_num > $o] :
( ( P3 @ none_num )
| ? [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_1029_option_Oexhaust,axiom,
! [Y: option_nat] :
( ( Y != none_nat )
=> ~ ! [X23: nat] :
( Y
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_1030_option_Oexhaust,axiom,
! [Y: option4927543243414619207at_nat] :
( ( Y != none_P5556105721700978146at_nat )
=> ~ ! [X23: product_prod_nat_nat] :
( Y
!= ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_1031_option_Oexhaust,axiom,
! [Y: option_num] :
( ( Y != none_num )
=> ~ ! [X23: num] :
( Y
!= ( some_num @ X23 ) ) ) ).
% option.exhaust
thf(fact_1032_option_OdiscI,axiom,
! [Option: option_nat,X2: nat] :
( ( Option
= ( some_nat @ X2 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_1033_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
( ( Option
= ( some_P7363390416028606310at_nat @ X2 ) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
thf(fact_1034_option_OdiscI,axiom,
! [Option: option_num,X2: num] :
( ( Option
= ( some_num @ X2 ) )
=> ( Option != none_num ) ) ).
% option.discI
thf(fact_1035_option_Odistinct_I1_J,axiom,
! [X2: nat] :
( none_nat
!= ( some_nat @ X2 ) ) ).
% option.distinct(1)
thf(fact_1036_option_Odistinct_I1_J,axiom,
! [X2: product_prod_nat_nat] :
( none_P5556105721700978146at_nat
!= ( some_P7363390416028606310at_nat @ X2 ) ) ).
% option.distinct(1)
thf(fact_1037_option_Odistinct_I1_J,axiom,
! [X2: num] :
( none_num
!= ( some_num @ X2 ) ) ).
% option.distinct(1)
thf(fact_1038_ex__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A4 ) )
= ( A4 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_1039_ex__in__conv,axiom,
! [A4: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A4 ) )
= ( A4 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_1040_ex__in__conv,axiom,
! [A4: set_real] :
( ( ? [X3: real] : ( member_real @ X3 @ A4 ) )
= ( A4 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_1041_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1042_ex__in__conv,axiom,
! [A4: set_int] :
( ( ? [X3: int] : ( member_int @ X3 @ A4 ) )
= ( A4 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_1043_equals0I,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [Y4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y4 @ A4 )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_1044_equals0I,axiom,
! [A4: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A4 )
=> ( A4 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_1045_equals0I,axiom,
! [A4: set_real] :
( ! [Y4: real] :
~ ( member_real @ Y4 @ A4 )
=> ( A4 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_1046_equals0I,axiom,
! [A4: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1047_equals0I,axiom,
! [A4: set_int] :
( ! [Y4: int] :
~ ( member_int @ Y4 @ A4 )
=> ( A4 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_1048_equals0D,axiom,
! [A4: set_Pr1261947904930325089at_nat,A3: product_prod_nat_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A3 @ A4 ) ) ).
% equals0D
thf(fact_1049_equals0D,axiom,
! [A4: set_set_nat,A3: set_nat] :
( ( A4 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A3 @ A4 ) ) ).
% equals0D
thf(fact_1050_equals0D,axiom,
! [A4: set_real,A3: real] :
( ( A4 = bot_bot_set_real )
=> ~ ( member_real @ A3 @ A4 ) ) ).
% equals0D
thf(fact_1051_equals0D,axiom,
! [A4: set_nat,A3: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A3 @ A4 ) ) ).
% equals0D
thf(fact_1052_equals0D,axiom,
! [A4: set_int,A3: int] :
( ( A4 = bot_bot_set_int )
=> ~ ( member_int @ A3 @ A4 ) ) ).
% equals0D
thf(fact_1053_emptyE,axiom,
! [A3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_1054_emptyE,axiom,
! [A3: set_nat] :
~ ( member_set_nat @ A3 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_1055_emptyE,axiom,
! [A3: real] :
~ ( member_real @ A3 @ bot_bot_set_real ) ).
% emptyE
thf(fact_1056_emptyE,axiom,
! [A3: nat] :
~ ( member_nat @ A3 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1057_emptyE,axiom,
! [A3: int] :
~ ( member_int @ A3 @ bot_bot_set_int ) ).
% emptyE
thf(fact_1058_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X3: real] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1059_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1060_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1061_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1062_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1063_set__eq__subset,axiom,
( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
= ( ^ [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_1064_subset__trans,axiom,
! [A4: set_int,B5: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C2 )
=> ( ord_less_eq_set_int @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_1065_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_1066_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1067_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1068_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1069_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_1070_subset__refl,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).
% subset_refl
thf(fact_1071_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T3 @ A6 )
=> ( member8440522571783428010at_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1072_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A6 )
=> ( member_real @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1073_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
! [T3: set_nat] :
( ( member_set_nat @ T3 @ A6 )
=> ( member_set_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1074_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1075_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T3: int] :
( ( member_int @ T3 @ A6 )
=> ( member_int @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1076_equalityD2,axiom,
! [A4: set_int,B5: set_int] :
( ( A4 = B5 )
=> ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_1077_equalityD1,axiom,
! [A4: set_int,B5: set_int] :
( ( A4 = B5 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_1078_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A6 )
=> ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1079_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A6 )
=> ( member_real @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1080_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
=> ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1081_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1082_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( member_int @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1083_equalityE,axiom,
! [A4: set_int,B5: set_int] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_1084_subsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1085_subsetD,axiom,
! [A4: set_real,B5: set_real,C: real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ C @ A4 )
=> ( member_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1086_subsetD,axiom,
! [A4: set_set_nat,B5: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ( member_set_nat @ C @ A4 )
=> ( member_set_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1087_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1088_subsetD,axiom,
! [A4: set_int,B5: set_int,C: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1089_in__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1090_in__mono,axiom,
! [A4: set_real,B5: set_real,X: real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ X @ A4 )
=> ( member_real @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1091_in__mono,axiom,
! [A4: set_set_nat,B5: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ( member_set_nat @ X @ A4 )
=> ( member_set_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1092_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1093_in__mono,axiom,
! [A4: set_int,B5: set_int,X: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ X @ A4 )
=> ( member_int @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1094_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_1095_psubsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1096_psubsetD,axiom,
! [A4: set_real,B5: set_real,C: real] :
( ( ord_less_set_real @ A4 @ B5 )
=> ( ( member_real @ C @ A4 )
=> ( member_real @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1097_psubsetD,axiom,
! [A4: set_set_nat,B5: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A4 @ B5 )
=> ( ( member_set_nat @ C @ A4 )
=> ( member_set_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1098_psubsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1099_psubsetD,axiom,
! [A4: set_int,B5: set_int,C: int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1100_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
= none_nat ) ).
% vebt_pred.simps(5)
thf(fact_1101_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
= none_nat ) ).
% vebt_pred.simps(6)
thf(fact_1102_vebt__pred_Osimps_I2_J,axiom,
! [A3: $o,Uw: $o] :
( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) )
= none_nat ) ) ) ).
% vebt_pred.simps(2)
thf(fact_1103_vebt__pred_Osimps_I3_J,axiom,
! [B3: $o,A3: $o,Va: nat] :
( ( B3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B3
=> ( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= none_nat ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_1104_not__psubset__empty,axiom,
! [A4: set_real] :
~ ( ord_less_set_real @ A4 @ bot_bot_set_real ) ).
% not_psubset_empty
thf(fact_1105_not__psubset__empty,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1106_not__psubset__empty,axiom,
! [A4: set_int] :
~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_1107_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_1108_subset__psubset__trans,axiom,
! [A4: set_int,B5: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_set_int @ B5 @ C2 )
=> ( ord_less_set_int @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1109_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_1110_psubset__subset__trans,axiom,
! [A4: set_int,B5: set_int,C2: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C2 )
=> ( ord_less_set_int @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1111_psubset__imp__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1112_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_1113_psubsetE,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_1114_vebt__maxt_Osimps_I1_J,axiom,
! [B3: $o,A3: $o] :
( ( B3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B3
=> ( ( A3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
= none_nat ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_1115_vebt__mint_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( A3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( B3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
= none_nat ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_1116_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_1117_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_1118_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_1119_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X5: complex] :
( ( member_complex @ X5 @ S2 )
& ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1120_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X5: real] :
( ( member_real @ X5 @ S2 )
& ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1121_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S2 )
& ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1122_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X5: int] :
( ( member_int @ X5 @ S2 )
& ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1123_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X5: complex] :
( ( member_complex @ X5 @ S2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1124_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X5: real] :
( ( member_real @ X5 @ S2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1125_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1126_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X5: int] :
( ( member_int @ X5 @ S2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1127_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X5: complex] :
( ( member_complex @ X5 @ S2 )
& ( ord_less_num @ ( F @ X5 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1128_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X5: real] :
( ( member_real @ X5 @ S2 )
& ( ord_less_num @ ( F @ X5 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1129_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1130_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1131_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1132_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1133_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1134_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1135_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > num] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1136_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > num] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1137_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1138_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1139_option_Osize_I3_J,axiom,
( ( size_size_option_nat @ none_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1140_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1141_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1142_option_Osize_I4_J,axiom,
! [X2: nat] :
( ( size_size_option_nat @ ( some_nat @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1143_option_Osize_I4_J,axiom,
! [X2: product_prod_nat_nat] :
( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1144_option_Osize_I4_J,axiom,
! [X2: num] :
( ( size_size_option_num @ ( some_num @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1145_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 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1146_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_1147_subset__emptyI,axiom,
! [A4: set_set_nat] :
( ! [X4: set_nat] :
~ ( member_set_nat @ X4 @ A4 )
=> ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_1148_subset__emptyI,axiom,
! [A4: set_real] :
( ! [X4: real] :
~ ( member_real @ X4 @ A4 )
=> ( ord_less_eq_set_real @ A4 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_1149_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1150_subset__emptyI,axiom,
! [A4: set_int] :
( ! [X4: int] :
~ ( member_int @ X4 @ A4 )
=> ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_1151_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1152_size__neq__size__imp__neq,axiom,
! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ X )
!= ( size_s6755466524823107622T_VEBT @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1153_size__neq__size__imp__neq,axiom,
! [X: list_o,Y: list_o] :
( ( ( size_size_list_o @ X )
!= ( size_size_list_o @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1154_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1155_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_1161_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_1162_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_1163_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_1164_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_1165_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ~ ( ( B2
=> ( Y
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( Y = none_nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat,Ma: nat] :
( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some_nat @ Ma ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_1166_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ~ ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( ( B2
=> ( Y
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( Y = none_nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat] :
( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some_nat @ Mi ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_1167_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1168_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1169_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_1170_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1171_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_1172_Collect__empty__eq__bot,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( P = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1173_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1174_Collect__empty__eq__bot,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( P = bot_bot_real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1175_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1176_Collect__empty__eq__bot,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( P = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1177_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1178_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1179_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1180_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_subrelI,axiom,
! [R2: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
( ! [X4: code_integer,Y4: code_integer] :
( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X4 @ Y4 ) @ R2 )
=> ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X4 @ Y4 ) @ S3 ) )
=> ( ord_le3725938330318615451nteger @ R2 @ S3 ) ) ).
% subrelI
thf(fact_1197_subrelI,axiom,
! [R2: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
( ! [X4: code_integer,Y4: $o] :
( ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y4 ) @ R2 )
=> ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y4 ) @ S3 ) )
=> ( ord_le8980329558974975238eger_o @ R2 @ S3 ) ) ).
% subrelI
thf(fact_1198_subrelI,axiom,
! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ S3 ) )
=> ( ord_le3000389064537975527at_nat @ R2 @ S3 ) ) ).
% subrelI
thf(fact_1199_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ! [X4: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S3 ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).
% subrelI
thf(fact_1200_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ! [X4: int,Y4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S3 ) )
=> ( ord_le2843351958646193337nt_int @ R2 @ S3 ) ) ).
% subrelI
thf(fact_1201_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1202_pinf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1203_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1204_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1205_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1206_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1207_pinf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1208_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1209_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1210_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1211_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1212_pinf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1213_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1214_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1215_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1216_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1217_pinf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1218_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1219_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1220_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1221_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1222_pinf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ~ ( ord_less_rat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1223_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ~ ( ord_less_num @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1224_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1225_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1226_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1227_pinf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ord_less_rat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1228_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ord_less_num @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1229_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1230_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1231_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1232_minf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1233_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1234_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1235_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1236_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1237_minf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1238_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1239_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1240_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1241_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1242_minf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1243_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1244_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1245_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1246_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1247_minf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1248_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1249_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1250_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1251_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_1252_minf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ord_less_rat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1253_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ord_less_num @ X5 @ T ) ) ).
% minf(5)
thf(fact_1254_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1255_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_1256_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_1257_minf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ~ ( ord_less_rat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1258_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ~ ( ord_less_num @ T @ X5 ) ) ).
% minf(7)
thf(fact_1259_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1260_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_1261_vebt__mint_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A2: $o,B2: $o] :
( X
!= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_mint.cases
thf(fact_1262_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi2: nat,Ma2: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
= ( ( X = Mi2 )
| ( X = Ma2 ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_1263_VEBT_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= zero_zero_nat ) ).
% VEBT.size(4)
thf(fact_1264_vebt__mint_Osimps_I3_J,axiom,
! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
= ( some_nat @ Mi2 ) ) ).
% vebt_mint.simps(3)
thf(fact_1265_vebt__maxt_Osimps_I3_J,axiom,
! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
= ( some_nat @ Ma2 ) ) ).
% vebt_maxt.simps(3)
thf(fact_1266_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_1267_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_1268_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_1269_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_1270_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_1271_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( ( B2
=> ( Y
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( Y = none_nat ) ) ) ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A2 @ B2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y = none_nat )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some_nat @ Ma ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_1272_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( ( B2
=> ( Y
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( Y = none_nat ) ) ) ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A2 @ B2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y = none_nat )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some_nat @ Mi ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_1273_vebt__pred_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
=> ( ! [A2: $o,Uw2: $o] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
=> ( ! [A2: $o,B2: $o,Va2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% vebt_pred.cases
thf(fact_1274_VEBT__internal_Omembermima_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X4 ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1275_vebt__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A2: $o,B2: $o,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ X4 ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_1276_complete__interval,axiom,
! [A3: real,B3: real,P: real > $o] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A3 @ C3 )
& ( ord_less_eq_real @ C3 @ B3 )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A3 @ X5 )
& ( ord_less_real @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D3: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_real @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1277_complete__interval,axiom,
! [A3: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A3 @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A3 @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A3 @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1278_complete__interval,axiom,
! [A3: int,B3: int,P: int > $o] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A3 @ C3 )
& ( ord_less_eq_int @ C3 @ B3 )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A3 @ X5 )
& ( ord_less_int @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A3 @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1279_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
= ( ord_less_real @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1280_verit__comp__simplify1_I3_J,axiom,
! [B7: rat,A7: rat] :
( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
= ( ord_less_rat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1281_verit__comp__simplify1_I3_J,axiom,
! [B7: num,A7: num] :
( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
= ( ord_less_num @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1282_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1283_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
= ( ord_less_int @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1284_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_1285_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1286_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_1287_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1288_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= zero_zero_nat ) ).
% VEBT.size_gen(2)
thf(fact_1289_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X3 )
| ( vEBT_VEBT_membermima @ T3 @ X3 ) ) ) ) ).
% both_member_options_def
thf(fact_1290_vebt__insert_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A2: $o,B2: $o,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ X4 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ X4 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X4 ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_1291_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_1292_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_1293_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_1294_maxbmo,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% maxbmo
thf(fact_1295_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc8306885398267862888on_nat] :
( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
( X
!= ( produc8929957630744042906on_nat @ Uu2 @ ( 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,A2: nat,B2: nat] :
( X
!= ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1296_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc5542196010084753463at_nat] :
( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
( X
!= ( produc2899441246263362727at_nat @ Uu2 @ ( 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,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( X
!= ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A2 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1297_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc1193250871479095198on_num] :
( ! [Uu2: num > num > num,Uv2: option_num] :
( X
!= ( produc5778274026573060048on_num @ Uu2 @ ( 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,A2: num,B2: num] :
( X
!= ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A2 ) @ ( some_num @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1298_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc2233624965454879586on_nat] :
( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
( X
!= ( produc4035269172776083154on_nat @ Uu2 @ ( 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,X4: nat,Y4: nat] :
( X
!= ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1299_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc5491161045314408544at_nat] :
( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
( X
!= ( produc3994169339658061776at_nat @ Uu2 @ ( 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,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( X
!= ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1300_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc7036089656553540234on_num] :
( ! [Uu2: num > num > $o,Uv2: option_num] :
( X
!= ( produc3576312749637752826on_num @ Uu2 @ ( 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,X4: num,Y4: num] :
( X
!= ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1301_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A2: nat,B2: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1302_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,D4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_1303_verit__comp__simplify1_I2_J,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_1304_verit__comp__simplify1_I2_J,axiom,
! [A3: rat] : ( ord_less_eq_rat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_1305_verit__comp__simplify1_I2_J,axiom,
! [A3: num] : ( ord_less_eq_num @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_1306_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_1307_verit__comp__simplify1_I2_J,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_1308_verit__la__disequality,axiom,
! [A3: rat,B3: rat] :
( ( A3 = B3 )
| ~ ( ord_less_eq_rat @ A3 @ B3 )
| ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_1309_verit__la__disequality,axiom,
! [A3: num,B3: num] :
( ( A3 = B3 )
| ~ ( ord_less_eq_num @ A3 @ B3 )
| ~ ( ord_less_eq_num @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_1310_verit__la__disequality,axiom,
! [A3: nat,B3: nat] :
( ( A3 = B3 )
| ~ ( ord_less_eq_nat @ A3 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_1311_verit__la__disequality,axiom,
! [A3: int,B3: int] :
( ( A3 = B3 )
| ~ ( ord_less_eq_int @ A3 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_1312_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1313_verit__comp__simplify1_I1_J,axiom,
! [A3: rat] :
~ ( ord_less_rat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1314_verit__comp__simplify1_I1_J,axiom,
! [A3: num] :
~ ( ord_less_num @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1315_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1316_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1317_ex__gt__or__lt,axiom,
! [A3: real] :
? [B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( ord_less_real @ B2 @ A3 ) ) ).
% ex_gt_or_lt
thf(fact_1318_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A2: $o,B2: $o,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ X4 ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ X4 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1319_mi__eq__ma__no__ch,axiom,
! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi2 = Ma2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_1320_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_1321_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_1322_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 ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_1323_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_1324_pair__lessI2,axiom,
! [A3: nat,B3: nat,S3: nat,T: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ S3 @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1325_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z ) ) ).
% pair_less_iff1
thf(fact_1326_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_1327_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = none_P5556105721700978146at_nat )
=> ( Y != none_P5556105721700978146at_nat ) )
=> ( ( ? [V2: product_prod_nat_nat] :
( Xa2
= ( some_P7363390416028606310at_nat @ V2 ) )
=> ( ( Xb = none_P5556105721700978146at_nat )
=> ( Y != none_P5556105721700978146at_nat ) ) )
=> ~ ! [A2: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ A2 ) )
=> ! [B2: product_prod_nat_nat] :
( ( Xb
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( Y
!= ( some_P7363390416028606310at_nat @ ( X @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1328_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: num > num > num,Xa2: option_num,Xb: option_num,Y: option_num] :
( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = none_num )
=> ( Y != none_num ) )
=> ( ( ? [V2: num] :
( Xa2
= ( some_num @ V2 ) )
=> ( ( Xb = none_num )
=> ( Y != none_num ) ) )
=> ~ ! [A2: num] :
( ( Xa2
= ( some_num @ A2 ) )
=> ! [B2: num] :
( ( Xb
= ( some_num @ B2 ) )
=> ( Y
!= ( some_num @ ( X @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1329_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = none_nat )
=> ( Y != none_nat ) )
=> ( ( ? [V2: nat] :
( Xa2
= ( some_nat @ V2 ) )
=> ( ( Xb = none_nat )
=> ( Y != none_nat ) ) )
=> ~ ! [A2: nat] :
( ( Xa2
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xb
= ( some_nat @ B2 ) )
=> ( Y
!= ( some_nat @ ( X @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1330_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_1331_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_1332_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_1333_set__encode__eq,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ( nat_set_encode @ A4 )
= ( nat_set_encode @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% set_encode_eq
thf(fact_1334_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) )
= ( some_P7363390416028606310at_nat @ ( F @ A3 @ B3 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1335_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: num > num > num,A3: num,B3: num] :
( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A3 ) @ ( some_num @ B3 ) )
= ( some_num @ ( F @ A3 @ B3 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1336_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: nat > nat > nat,A3: nat,B3: nat] :
( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) )
= ( some_nat @ ( F @ A3 @ B3 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1337_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ X )
= ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) ) ).
% vebt_insert.simps(2)
thf(fact_1338_set__encode__inf,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= zero_zero_nat ) ) ).
% set_encode_inf
thf(fact_1339_pair__lessI1,axiom,
! [A3: nat,B3: nat,S3: nat,T: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1340_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X )
= ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) ).
% vebt_insert.simps(3)
thf(fact_1341_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A3: $o,B3: $o] :
( ( ( X = zero_zero_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( X != one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_1342_List_Ofinite__set,axiom,
! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% List.finite_set
thf(fact_1343_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1344_List_Ofinite__set,axiom,
! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_1345_List_Ofinite__set,axiom,
! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% List.finite_set
thf(fact_1346_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1347_length__pos__if__in__set,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1348_length__pos__if__in__set,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1349_length__pos__if__in__set,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1350_length__pos__if__in__set,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1351_length__pos__if__in__set,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1352_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1353_pair__leqI2,axiom,
! [A3: nat,B3: nat,S3: nat,T: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ S3 @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_1354_pair__leqI1,axiom,
! [A3: nat,B3: nat,S3: nat,T: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_1355_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B5 )
= ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1356_subset__code_I1_J,axiom,
! [Xs: list_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B5 )
= ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( member_real @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1357_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B5 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1358_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B5 )
= ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1359_subset__code_I1_J,axiom,
! [Xs: list_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1360_subset__code_I1_J,axiom,
! [Xs: list_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B5 )
= ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( member_int @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1361_finite__list,axiom,
! [A4: set_VEBT_VEBT] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ? [Xs3: list_VEBT_VEBT] :
( ( set_VEBT_VEBT2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_1362_finite__list,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_1363_finite__list,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ? [Xs3: list_int] :
( ( set_int2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_1364_finite__list,axiom,
! [A4: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ? [Xs3: list_complex] :
( ( set_complex2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_1365_set__encode__inverse,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
= A4 ) ) ).
% set_encode_inverse
thf(fact_1366_frac__eq,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= X )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% frac_eq
thf(fact_1367_frac__eq,axiom,
! [X: rat] :
( ( ( archimedean_frac_rat @ X )
= X )
= ( ( ord_less_eq_rat @ zero_zero_rat @ X )
& ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% frac_eq
thf(fact_1368_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_1369_option_Osize__gen_I2_J,axiom,
! [X: nat > nat,X2: nat] :
( ( size_option_nat @ X @ ( some_nat @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1370_option_Osize__gen_I2_J,axiom,
! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1371_option_Osize__gen_I2_J,axiom,
! [X: num > nat,X2: num] :
( ( size_option_num @ X @ ( some_num @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1372_even__odd__cases,axiom,
! [X: nat] :
( ! [N2: nat] :
( X
!= ( plus_plus_nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X
!= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_1373_add__right__cancel,axiom,
! [B3: real,A3: real,C: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_1374_add__right__cancel,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ( plus_plus_rat @ B3 @ A3 )
= ( plus_plus_rat @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_1375_add__right__cancel,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_1376_add__right__cancel,axiom,
! [B3: int,A3: int,C: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= ( plus_plus_int @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_1377_add__left__cancel,axiom,
! [A3: real,B3: real,C: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1378_add__left__cancel,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ( plus_plus_rat @ A3 @ B3 )
= ( plus_plus_rat @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1379_add__left__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1380_add__left__cancel,axiom,
! [A3: int,B3: int,C: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1381_add__le__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_1382_add__le__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_1383_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_1384_add__le__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_1385_add__le__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_1386_add__le__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_1387_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_1388_add__le__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_1389_add__0,axiom,
! [A3: literal] :
( ( plus_plus_literal @ zero_zero_literal @ A3 )
= A3 ) ).
% add_0
thf(fact_1390_add__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add_0
thf(fact_1391_add__0,axiom,
! [A3: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A3 )
= A3 ) ).
% add_0
thf(fact_1392_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_1393_add__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add_0
thf(fact_1394_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1395_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1396_add__cancel__right__right,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ A3 @ B3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_1397_add__cancel__right__right,axiom,
! [A3: rat,B3: rat] :
( ( A3
= ( plus_plus_rat @ A3 @ B3 ) )
= ( B3 = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_1398_add__cancel__right__right,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1399_add__cancel__right__right,axiom,
! [A3: int,B3: int] :
( ( A3
= ( plus_plus_int @ A3 @ B3 ) )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1400_add__cancel__right__left,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ B3 @ A3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_1401_add__cancel__right__left,axiom,
! [A3: rat,B3: rat] :
( ( A3
= ( plus_plus_rat @ B3 @ A3 ) )
= ( B3 = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_1402_add__cancel__right__left,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1403_add__cancel__right__left,axiom,
! [A3: int,B3: int] :
( ( A3
= ( plus_plus_int @ B3 @ A3 ) )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1404_add__cancel__left__right,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_1405_add__cancel__left__right,axiom,
! [A3: rat,B3: rat] :
( ( ( plus_plus_rat @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_1406_add__cancel__left__right,axiom,
! [A3: nat,B3: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1407_add__cancel__left__right,axiom,
! [A3: int,B3: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1408_add__cancel__left__left,axiom,
! [B3: real,A3: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_1409_add__cancel__left__left,axiom,
! [B3: rat,A3: rat] :
( ( ( plus_plus_rat @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_1410_add__cancel__left__left,axiom,
! [B3: nat,A3: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1411_add__cancel__left__left,axiom,
! [B3: int,A3: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1412_double__zero__sym,axiom,
! [A3: real] :
( ( zero_zero_real
= ( plus_plus_real @ A3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_1413_double__zero__sym,axiom,
! [A3: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A3 @ A3 ) )
= ( A3 = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_1414_double__zero__sym,axiom,
! [A3: int] :
( ( zero_zero_int
= ( plus_plus_int @ A3 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1415_add_Oright__neutral,axiom,
! [A3: literal] :
( ( plus_plus_literal @ A3 @ zero_zero_literal )
= A3 ) ).
% add.right_neutral
thf(fact_1416_add_Oright__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.right_neutral
thf(fact_1417_add_Oright__neutral,axiom,
! [A3: rat] :
( ( plus_plus_rat @ A3 @ zero_zero_rat )
= A3 ) ).
% add.right_neutral
thf(fact_1418_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_1419_add_Oright__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.right_neutral
thf(fact_1420_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1421_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: rat] :
( ( minus_minus_rat @ A3 @ A3 )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1422_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1423_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1424_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_1425_diff__zero,axiom,
! [A3: rat] :
( ( minus_minus_rat @ A3 @ zero_zero_rat )
= A3 ) ).
% diff_zero
thf(fact_1426_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_1427_diff__zero,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_zero
thf(fact_1428_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1429_diff__0__right,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_0_right
thf(fact_1430_diff__0__right,axiom,
! [A3: rat] :
( ( minus_minus_rat @ A3 @ zero_zero_rat )
= A3 ) ).
% diff_0_right
thf(fact_1431_diff__0__right,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_0_right
thf(fact_1432_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_1433_diff__self,axiom,
! [A3: rat] :
( ( minus_minus_rat @ A3 @ A3 )
= zero_zero_rat ) ).
% diff_self
thf(fact_1434_diff__self,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% diff_self
thf(fact_1435_add__less__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_1436_add__less__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
= ( ord_less_rat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_1437_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_1438_add__less__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_1439_add__less__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_1440_add__less__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
= ( ord_less_rat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_1441_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_1442_add__less__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_1443_add__diff__cancel__right_H,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1444_add__diff__cancel__right_H,axiom,
! [A3: rat,B3: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1445_add__diff__cancel__right_H,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1446_add__diff__cancel__right_H,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1447_add__diff__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
= ( minus_minus_real @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1448_add__diff__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
= ( minus_minus_rat @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1449_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1450_add__diff__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
= ( minus_minus_int @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1451_add__diff__cancel__left_H,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1452_add__diff__cancel__left_H,axiom,
! [A3: rat,B3: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1453_add__diff__cancel__left_H,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1454_add__diff__cancel__left_H,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1455_add__diff__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
= ( minus_minus_real @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1456_add__diff__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
= ( minus_minus_rat @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1457_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1458_add__diff__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
= ( minus_minus_int @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1459_diff__add__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ B3 )
= A3 ) ).
% diff_add_cancel
thf(fact_1460_diff__add__cancel,axiom,
! [A3: rat,B3: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A3 @ B3 ) @ B3 )
= A3 ) ).
% diff_add_cancel
thf(fact_1461_diff__add__cancel,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ ( minus_minus_int @ A3 @ B3 ) @ B3 )
= A3 ) ).
% diff_add_cancel
thf(fact_1462_add__diff__cancel,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel
thf(fact_1463_add__diff__cancel,axiom,
! [A3: rat,B3: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel
thf(fact_1464_add__diff__cancel,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel
thf(fact_1465_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_1466_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_1467_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1468_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_1469_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_1470_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_1471_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1472_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1473_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_1474_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1475_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1476_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1477_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ A3 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1478_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1479_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1480_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ A3 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1481_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1482_le__add__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1483_le__add__same__cancel2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ ( plus_plus_rat @ B3 @ A3 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1484_le__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1485_le__add__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
= ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1486_le__add__same__cancel1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1487_le__add__same__cancel1,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1488_le__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1489_le__add__same__cancel1,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1490_add__le__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_1491_add__le__same__cancel2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_1492_add__le__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1493_add__le__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1494_add__le__same__cancel1,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_1495_add__le__same__cancel1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_1496_add__le__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1497_add__le__same__cancel1,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1498_diff__ge__0__iff__ge,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1499_diff__ge__0__iff__ge,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A3 @ B3 ) )
= ( ord_less_eq_rat @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1500_diff__ge__0__iff__ge,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1501_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1502_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ A3 ) )
= ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1503_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1504_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1505_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A3 @ A3 ) @ zero_zero_rat )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1506_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1507_less__add__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_1508_less__add__same__cancel2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ ( plus_plus_rat @ B3 @ A3 ) )
= ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_1509_less__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_1510_less__add__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_1511_less__add__same__cancel1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_1512_less__add__same__cancel1,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
= ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_1513_less__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_1514_less__add__same__cancel1,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_1515_add__less__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1516_add__less__same__cancel2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ B3 )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_1517_add__less__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1518_add__less__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1519_add__less__same__cancel1,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1520_add__less__same__cancel1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B3 @ A3 ) @ B3 )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_1521_add__less__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1522_add__less__same__cancel1,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1523_diff__gt__0__iff__gt,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1524_diff__gt__0__iff__gt,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A3 @ B3 ) )
= ( ord_less_rat @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1525_diff__gt__0__iff__gt,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
= ( ord_less_int @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1526_le__add__diff__inverse,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1527_le__add__diff__inverse,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1528_le__add__diff__inverse,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1529_le__add__diff__inverse,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1530_le__add__diff__inverse2,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1531_le__add__diff__inverse2,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( plus_plus_rat @ ( minus_minus_rat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1532_le__add__diff__inverse2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1533_le__add__diff__inverse2,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1534_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_1535_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1536_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_1537_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1538_diff__add__zero,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1539_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_1540_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_1541_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_1542_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_1543_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1544_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1545_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1546_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1547_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_1548_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_1549_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1550_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1551_diff__less__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( ord_less_real @ A3 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% diff_less_eq
thf(fact_1552_diff__less__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
= ( ord_less_rat @ A3 @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% diff_less_eq
thf(fact_1553_diff__less__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( ord_less_int @ A3 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% diff_less_eq
thf(fact_1554_less__diff__eq,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ A3 @ ( minus_minus_real @ C @ B3 ) )
= ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).
% less_diff_eq
thf(fact_1555_less__diff__eq,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ A3 @ ( minus_minus_rat @ C @ B3 ) )
= ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C ) ) ).
% less_diff_eq
thf(fact_1556_less__diff__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_int @ A3 @ ( minus_minus_int @ C @ B3 ) )
= ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).
% less_diff_eq
thf(fact_1557_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: real,B3: real] :
( ~ ( ord_less_real @ A3 @ B3 )
=> ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1558_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: rat,B3: rat] :
( ~ ( ord_less_rat @ A3 @ B3 )
=> ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1559_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ~ ( ord_less_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1560_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: int,B3: int] :
( ~ ( ord_less_int @ A3 @ B3 )
=> ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1561_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1562_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1563_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1564_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_1565_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_1566_diff__diff__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( minus_minus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1567_diff__diff__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
= ( minus_minus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1568_diff__diff__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1569_diff__diff__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( minus_minus_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1570_diff__right__commute,axiom,
! [A3: real,C: real,B3: real] :
( ( minus_minus_real @ ( minus_minus_real @ A3 @ C ) @ B3 )
= ( minus_minus_real @ ( minus_minus_real @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_1571_diff__right__commute,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A3 @ C ) @ B3 )
= ( minus_minus_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_1572_diff__right__commute,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_1573_diff__right__commute,axiom,
! [A3: int,C: int,B3: int] :
( ( minus_minus_int @ ( minus_minus_int @ A3 @ C ) @ B3 )
= ( minus_minus_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_1574_add__implies__diff,axiom,
! [C: real,B3: real,A3: real] :
( ( ( plus_plus_real @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_real @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1575_add__implies__diff,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ( plus_plus_rat @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_rat @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1576_add__implies__diff,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1577_add__implies__diff,axiom,
! [C: int,B3: int,A3: int] :
( ( ( plus_plus_int @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_int @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1578_add__right__imp__eq,axiom,
! [B3: real,A3: real,C: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_1579_add__right__imp__eq,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ( plus_plus_rat @ B3 @ A3 )
= ( plus_plus_rat @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_1580_add__right__imp__eq,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_1581_add__right__imp__eq,axiom,
! [B3: int,A3: int,C: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= ( plus_plus_int @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_1582_diff__add__eq__diff__diff__swap,axiom,
! [A3: real,B3: real,C: real] :
( ( minus_minus_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1583_diff__add__eq__diff__diff__swap,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( minus_minus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1584_diff__add__eq__diff__diff__swap,axiom,
! [A3: int,B3: int,C: int] :
( ( minus_minus_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1585_add__left__imp__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_1586_add__left__imp__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ( plus_plus_rat @ A3 @ B3 )
= ( plus_plus_rat @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_1587_add__left__imp__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_1588_add__left__imp__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_1589_add_Oleft__commute,axiom,
! [B3: real,A3: real,C: real] :
( ( plus_plus_real @ B3 @ ( plus_plus_real @ A3 @ C ) )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_1590_add_Oleft__commute,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( plus_plus_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) )
= ( plus_plus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_1591_add_Oleft__commute,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_1592_add_Oleft__commute,axiom,
! [B3: int,A3: int,C: int] :
( ( plus_plus_int @ B3 @ ( plus_plus_int @ A3 @ C ) )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_1593_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A: real,B: real] : ( plus_plus_real @ B @ A ) ) ) ).
% add.commute
thf(fact_1594_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A: rat,B: rat] : ( plus_plus_rat @ B @ A ) ) ) ).
% add.commute
thf(fact_1595_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ) ).
% add.commute
thf(fact_1596_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A: int,B: int] : ( plus_plus_int @ B @ A ) ) ) ).
% add.commute
thf(fact_1597_add_Oright__cancel,axiom,
! [B3: real,A3: real,C: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C @ A3 ) )
= ( B3 = C ) ) ).
% add.right_cancel
thf(fact_1598_add_Oright__cancel,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ( plus_plus_rat @ B3 @ A3 )
= ( plus_plus_rat @ C @ A3 ) )
= ( B3 = C ) ) ).
% add.right_cancel
thf(fact_1599_add_Oright__cancel,axiom,
! [B3: int,A3: int,C: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= ( plus_plus_int @ C @ A3 ) )
= ( B3 = C ) ) ).
% add.right_cancel
thf(fact_1600_diff__eq__diff__eq,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D ) )
=> ( ( A3 = B3 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1601_diff__eq__diff__eq,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A3 @ B3 )
= ( minus_minus_rat @ C @ D ) )
=> ( ( A3 = B3 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1602_diff__eq__diff__eq,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A3 = B3 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1603_add_Oleft__cancel,axiom,
! [A3: real,B3: real,C: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C ) )
= ( B3 = C ) ) ).
% add.left_cancel
thf(fact_1604_add_Oleft__cancel,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ( plus_plus_rat @ A3 @ B3 )
= ( plus_plus_rat @ A3 @ C ) )
= ( B3 = C ) ) ).
% add.left_cancel
thf(fact_1605_add_Oleft__cancel,axiom,
! [A3: int,B3: int,C: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ A3 @ C ) )
= ( B3 = C ) ) ).
% add.left_cancel
thf(fact_1606_diff__add__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_1607_diff__add__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
= ( minus_minus_rat @ ( plus_plus_rat @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_1608_diff__add__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_1609_add_Oassoc,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_1610_add_Oassoc,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
= ( plus_plus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_1611_add_Oassoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_1612_add_Oassoc,axiom,
! [A3: int,B3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_1613_diff__diff__eq2,axiom,
! [A3: real,B3: real,C: real] :
( ( minus_minus_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_1614_diff__diff__eq2,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( minus_minus_rat @ A3 @ ( minus_minus_rat @ B3 @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A3 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_1615_diff__diff__eq2,axiom,
! [A3: int,B3: int,C: int] :
( ( minus_minus_int @ A3 @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_1616_add__diff__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_1617_add__diff__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( plus_plus_rat @ A3 @ ( minus_minus_rat @ B3 @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_1618_add__diff__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( plus_plus_int @ A3 @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_1619_eq__diff__eq,axiom,
! [A3: real,C: real,B3: real] :
( ( A3
= ( minus_minus_real @ C @ B3 ) )
= ( ( plus_plus_real @ A3 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_1620_eq__diff__eq,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( A3
= ( minus_minus_rat @ C @ B3 ) )
= ( ( plus_plus_rat @ A3 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_1621_eq__diff__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( A3
= ( minus_minus_int @ C @ B3 ) )
= ( ( plus_plus_int @ A3 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_1622_diff__eq__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= C )
= ( A3
= ( plus_plus_real @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_1623_diff__eq__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ( minus_minus_rat @ A3 @ B3 )
= C )
= ( A3
= ( plus_plus_rat @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_1624_diff__eq__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= C )
= ( A3
= ( plus_plus_int @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_1625_group__cancel_Osub1,axiom,
! [A4: real,K: real,A3: real,B3: real] :
( ( A4
= ( plus_plus_real @ K @ A3 ) )
=> ( ( minus_minus_real @ A4 @ B3 )
= ( plus_plus_real @ K @ ( minus_minus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_1626_group__cancel_Osub1,axiom,
! [A4: rat,K: rat,A3: rat,B3: rat] :
( ( A4
= ( plus_plus_rat @ K @ A3 ) )
=> ( ( minus_minus_rat @ A4 @ B3 )
= ( plus_plus_rat @ K @ ( minus_minus_rat @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_1627_group__cancel_Osub1,axiom,
! [A4: int,K: int,A3: int,B3: int] :
( ( A4
= ( plus_plus_int @ K @ A3 ) )
=> ( ( minus_minus_int @ A4 @ B3 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_1628_group__cancel_Oadd2,axiom,
! [B5: real,K: real,B3: real,A3: real] :
( ( B5
= ( plus_plus_real @ K @ B3 ) )
=> ( ( plus_plus_real @ A3 @ B5 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_1629_group__cancel_Oadd2,axiom,
! [B5: rat,K: rat,B3: rat,A3: rat] :
( ( B5
= ( plus_plus_rat @ K @ B3 ) )
=> ( ( plus_plus_rat @ A3 @ B5 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_1630_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B3: nat,A3: nat] :
( ( B5
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A3 @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_1631_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B3: int,A3: int] :
( ( B5
= ( plus_plus_int @ K @ B3 ) )
=> ( ( plus_plus_int @ A3 @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_1632_group__cancel_Oadd1,axiom,
! [A4: real,K: real,A3: real,B3: real] :
( ( A4
= ( plus_plus_real @ K @ A3 ) )
=> ( ( plus_plus_real @ A4 @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_1633_group__cancel_Oadd1,axiom,
! [A4: rat,K: rat,A3: rat,B3: rat] :
( ( A4
= ( plus_plus_rat @ K @ A3 ) )
=> ( ( plus_plus_rat @ A4 @ B3 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_1634_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A3: nat,B3: nat] :
( ( A4
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A4 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_1635_group__cancel_Oadd1,axiom,
! [A4: int,K: int,A3: int,B3: int] :
( ( A4
= ( plus_plus_int @ K @ A3 ) )
=> ( ( plus_plus_int @ A4 @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_1636_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1637_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1638_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1639_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1640_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1641_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
= ( plus_plus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1642_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1643_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: int,B3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1644_is__num__normalize_I1_J,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1645_is__num__normalize_I1_J,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
= ( plus_plus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1646_is__num__normalize_I1_J,axiom,
! [A3: int,B3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1647_diff__le__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( ord_less_eq_real @ A3 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% diff_le_eq
thf(fact_1648_diff__le__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
= ( ord_less_eq_rat @ A3 @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% diff_le_eq
thf(fact_1649_diff__le__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( ord_less_eq_int @ A3 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% diff_le_eq
thf(fact_1650_le__diff__eq,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( minus_minus_real @ C @ B3 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).
% le_diff_eq
thf(fact_1651_le__diff__eq,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ ( minus_minus_rat @ C @ B3 ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C ) ) ).
% le_diff_eq
thf(fact_1652_le__diff__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_eq_int @ A3 @ ( minus_minus_int @ C @ B3 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).
% le_diff_eq
thf(fact_1653_diff__add,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% diff_add
thf(fact_1654_le__add__diff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_1655_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1656_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1657_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1658_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1659_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1660_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1661_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B3 @ A3 ) )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1662_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( minus_minus_nat @ B3 @ A3 )
= C )
= ( B3
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1663_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ 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_1664_add__le__add__imp__diff__le,axiom,
! [I: rat,K: rat,N: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ 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_1665_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ 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_1666_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ 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_1667_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1668_add__le__imp__le__diff,axiom,
! [I: rat,K: rat,N: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
=> ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1669_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1670_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1671_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_1672_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_1673_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1674_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1675_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1676_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1677_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1678_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1679_frac__1__eq,axiom,
! [X: real] :
( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( archim2898591450579166408c_real @ X ) ) ).
% frac_1_eq
thf(fact_1680_frac__1__eq,axiom,
! [X: rat] :
( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( archimedean_frac_rat @ X ) ) ).
% frac_1_eq
thf(fact_1681_frac__add,axiom,
! [X: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
= ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% frac_add
thf(fact_1682_frac__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% frac_add
thf(fact_1683_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B3 )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1684_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ( ( ord_less_nat @ A3 @ B3 )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1685_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1686_diff__eq__diff__less__eq,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A3 @ B3 )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1687_diff__eq__diff__less__eq,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A3 @ B3 )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_eq_rat @ A3 @ B3 )
= ( ord_less_eq_rat @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1688_diff__eq__diff__less__eq,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A3 @ B3 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1689_diff__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% diff_right_mono
thf(fact_1690_diff__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).
% diff_right_mono
thf(fact_1691_diff__right__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% diff_right_mono
thf(fact_1692_diff__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% diff_left_mono
thf(fact_1693_diff__left__mono,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A3 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).
% diff_left_mono
thf(fact_1694_diff__left__mono,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% diff_left_mono
thf(fact_1695_diff__mono,axiom,
! [A3: real,B3: real,D: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% diff_mono
thf(fact_1696_diff__mono,axiom,
! [A3: rat,B3: rat,D: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ D @ C )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% diff_mono
thf(fact_1697_diff__mono,axiom,
! [A3: int,B3: int,D: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% diff_mono
thf(fact_1698_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: real,Z4: real] : Y6 = Z4 )
= ( ^ [A: real,B: real] :
( ( minus_minus_real @ A @ B )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1699_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
= ( ^ [A: rat,B: rat] :
( ( minus_minus_rat @ A @ B )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1700_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
= ( ^ [A: int,B: int] :
( ( minus_minus_int @ A @ B )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1701_diff__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1702_diff__strict__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1703_diff__strict__right__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1704_diff__strict__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_1705_diff__strict__left__mono,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ord_less_rat @ ( minus_minus_rat @ C @ A3 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_1706_diff__strict__left__mono,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_1707_diff__eq__diff__less,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A3 @ B3 )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1708_diff__eq__diff__less,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A3 @ B3 )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_rat @ A3 @ B3 )
= ( ord_less_rat @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1709_diff__eq__diff__less,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A3 @ B3 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1710_diff__strict__mono,axiom,
! [A3: real,B3: real,D: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1711_diff__strict__mono,axiom,
! [A3: rat,B3: rat,D: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ D @ C )
=> ( ord_less_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1712_diff__strict__mono,axiom,
! [A3: int,B3: int,D: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1713_add__le__imp__le__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_1714_add__le__imp__le__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_1715_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_1716_add__le__imp__le__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_1717_add__le__imp__le__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_1718_add__le__imp__le__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_1719_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_1720_add__le__imp__le__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_1721_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C4: nat] :
( B
= ( plus_plus_nat @ A @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1722_add__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_1723_add__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_1724_add__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_1725_add__right__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_1726_less__eqE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( B3
!= ( plus_plus_nat @ A3 @ C3 ) ) ) ).
% less_eqE
thf(fact_1727_add__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_1728_add__left__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_1729_add__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_1730_add__left__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_1731_add__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_1732_add__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_1733_add__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_1734_add__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_1735_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1736_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1737_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1738_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1739_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1740_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1741_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1742_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1743_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1744_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1745_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1746_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1747_verit__sum__simplify,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% verit_sum_simplify
thf(fact_1748_verit__sum__simplify,axiom,
! [A3: rat] :
( ( plus_plus_rat @ A3 @ zero_zero_rat )
= A3 ) ).
% verit_sum_simplify
thf(fact_1749_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_1750_verit__sum__simplify,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% verit_sum_simplify
thf(fact_1751_add_Ogroup__left__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_1752_add_Ogroup__left__neutral,axiom,
! [A3: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_1753_add_Ogroup__left__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_1754_add_Ocomm__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.comm_neutral
thf(fact_1755_add_Ocomm__neutral,axiom,
! [A3: rat] :
( ( plus_plus_rat @ A3 @ zero_zero_rat )
= A3 ) ).
% add.comm_neutral
thf(fact_1756_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_1757_add_Ocomm__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.comm_neutral
thf(fact_1758_comm__monoid__add__class_Oadd__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_1759_comm__monoid__add__class_Oadd__0,axiom,
! [A3: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_1760_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_1761_comm__monoid__add__class_Oadd__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_1762_add__less__imp__less__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_1763_add__less__imp__less__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
=> ( ord_less_rat @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_1764_add__less__imp__less__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_1765_add__less__imp__less__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
=> ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_1766_add__less__imp__less__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_1767_add__less__imp__less__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
=> ( ord_less_rat @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_1768_add__less__imp__less__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_1769_add__less__imp__less__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
=> ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_1770_add__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1771_add__strict__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1772_add__strict__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1773_add__strict__right__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1774_add__strict__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_1775_add__strict__left__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_1776_add__strict__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_1777_add__strict__left__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_1778_add__strict__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1779_add__strict__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1780_add__strict__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1781_add__strict__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1782_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1783_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1784_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1785_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1786_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1787_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1788_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1789_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1790_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1791_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1792_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1793_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1794_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1795_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1796_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_1797_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1798_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_1799_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A3: nat] :
( ( A4
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1800_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1801_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_1802_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_1803_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_1804_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_1805_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1806_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1807_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1808_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1809_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1810_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_1811_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1812_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1813_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1814_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1815_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1816_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1817_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1818_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1819_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_1820_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1821_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1822_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_1823_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_1824_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1825_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1826_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1827_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1828_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1829_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1830_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1831_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_1832_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1833_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_1834_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1835_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A: real,B: real] : ( ord_less_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1836_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A: rat,B: rat] : ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_1837_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A: int,B: int] : ( ord_less_int @ ( minus_minus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1838_finite__set__decode,axiom,
! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_1839_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1840_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1841_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1842_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1843_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1844_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1845_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1846_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1847_add__nonpos__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1848_add__nonpos__nonpos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1849_add__nonpos__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1850_add__nonpos__nonpos,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1851_add__nonneg__nonneg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1852_add__nonneg__nonneg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1853_add__nonneg__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1854_add__nonneg__nonneg,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1855_add__increasing2,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1856_add__increasing2,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1857_add__increasing2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1858_add__increasing2,axiom,
! [C: int,B3: int,A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B3 @ A3 )
=> ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1859_add__decreasing2,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1860_add__decreasing2,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1861_add__decreasing2,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1862_add__decreasing2,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1863_add__increasing,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1864_add__increasing,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1865_add__increasing,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1866_add__increasing,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1867_add__decreasing,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1868_add__decreasing,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1869_add__decreasing,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1870_add__decreasing,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1871_add__less__le__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1872_add__less__le__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1873_add__less__le__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1874_add__less__le__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1875_add__le__less__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1876_add__le__less__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1877_add__le__less__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1878_add__le__less__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1879_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1880_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1881_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1882_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1883_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1884_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1885_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1886_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1887_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1888_add__less__zeroD,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X @ zero_zero_rat )
| ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_1889_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1890_pos__add__strict,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1891_pos__add__strict,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1892_pos__add__strict,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1893_pos__add__strict,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1894_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( ( B3
= ( plus_plus_nat @ A3 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1895_add__pos__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1896_add__pos__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1897_add__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1898_add__pos__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1899_add__neg__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1900_add__neg__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_rat @ B3 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_1901_add__neg__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1902_add__neg__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1903_add__mono1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1904_add__mono1,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( plus_plus_rat @ B3 @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_1905_add__mono1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1906_add__mono1,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1907_less__add__one,axiom,
! [A3: real] : ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ one_one_real ) ) ).
% less_add_one
thf(fact_1908_less__add__one,axiom,
! [A3: rat] : ( ord_less_rat @ A3 @ ( plus_plus_rat @ A3 @ one_one_rat ) ) ).
% less_add_one
thf(fact_1909_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1910_less__add__one,axiom,
! [A3: int] : ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ one_one_int ) ) ).
% less_add_one
thf(fact_1911_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_1912_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1913_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_1914_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_1915_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_1916_diff__less__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1917_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_1918_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_1919_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_1920_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1921_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1922_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1923_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_1924_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1925_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_1926_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1927_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1928_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1929_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1930_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_1931_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_1932_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_1933_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1934_frac__ge__0,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% frac_ge_0
thf(fact_1935_frac__ge__0,axiom,
! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% frac_ge_0
thf(fact_1936_frac__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% frac_lt_1
thf(fact_1937_frac__lt__1,axiom,
! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% frac_lt_1
thf(fact_1938_add__neg__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1939_add__neg__nonpos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_1940_add__neg__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1941_add__neg__nonpos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1942_add__nonneg__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1943_add__nonneg__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1944_add__nonneg__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1945_add__nonneg__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1946_add__nonpos__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1947_add__nonpos__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_rat @ B3 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_1948_add__nonpos__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1949_add__nonpos__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1950_add__pos__nonneg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1951_add__pos__nonneg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1952_add__pos__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1953_add__pos__nonneg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1954_add__strict__increasing,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1955_add__strict__increasing,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1956_add__strict__increasing,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1957_add__strict__increasing,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1958_add__strict__increasing2,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1959_add__strict__increasing2,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1960_add__strict__increasing2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1961_add__strict__increasing2,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1962_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1963_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_1964_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1965_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1966_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1967_finite__maxlen,axiom,
! [M7: set_list_VEBT_VEBT] :
( ( finite3004134309566078307T_VEBT @ M7 )
=> ? [N2: nat] :
! [X5: list_VEBT_VEBT] :
( ( member2936631157270082147T_VEBT @ X5 @ M7 )
=> ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1968_finite__maxlen,axiom,
! [M7: set_list_o] :
( ( finite_finite_list_o @ M7 )
=> ? [N2: nat] :
! [X5: list_o] :
( ( member_list_o @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1969_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N2: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1970_length__induct,axiom,
! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ! [Xs3: list_VEBT_VEBT] :
( ! [Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
=> ( P @ Ys ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1971_length__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ! [Xs3: list_o] :
( ! [Ys: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
=> ( P @ Ys ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1972_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1973_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_1974_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_1975_add__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( plus_plus_nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( some_nat @ Z ) ) ) ).
% add_shift
thf(fact_1976_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% add_def
thf(fact_1977_double__eq__0__iff,axiom,
! [A3: real] :
( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_1978_double__eq__0__iff,axiom,
! [A3: rat] :
( ( ( plus_plus_rat @ A3 @ A3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_1979_double__eq__0__iff,axiom,
! [A3: int] :
( ( ( plus_plus_int @ A3 @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1980_discrete,axiom,
( ord_less_nat
= ( ^ [A: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_1981_discrete,axiom,
( ord_less_int
= ( ^ [A: int] : ( ord_less_eq_int @ ( plus_plus_int @ A @ one_one_int ) ) ) ) ).
% discrete
thf(fact_1982_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1983_field__le__epsilon,axiom,
! [X: rat,Y: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1984_diff__shunt__var,axiom,
! [X: set_real,Y: set_real] :
( ( ( minus_minus_set_real @ X @ Y )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1985_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1986_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1987_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_1988_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_1989_inthall,axiom,
! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1990_inthall,axiom,
! [Xs: list_real,P: real > $o,N: nat] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1991_inthall,axiom,
! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1992_inthall,axiom,
! [Xs: list_int,P: int > $o,N: nat] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1993_inthall,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1994_inthall,axiom,
! [Xs: list_o,P: $o > $o,N: nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1995_inthall,axiom,
! [Xs: list_nat,P: nat > $o,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1996_Diff__cancel,axiom,
! [A4: set_real] :
( ( minus_minus_set_real @ A4 @ A4 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_1997_Diff__cancel,axiom,
! [A4: set_int] :
( ( minus_minus_set_int @ A4 @ A4 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_1998_Diff__cancel,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ A4 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1999_empty__Diff,axiom,
! [A4: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A4 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_2000_empty__Diff,axiom,
! [A4: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A4 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_2001_empty__Diff,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_2002_Diff__empty,axiom,
! [A4: set_real] :
( ( minus_minus_set_real @ A4 @ bot_bot_set_real )
= A4 ) ).
% Diff_empty
thf(fact_2003_Diff__empty,axiom,
! [A4: set_int] :
( ( minus_minus_set_int @ A4 @ bot_bot_set_int )
= A4 ) ).
% Diff_empty
thf(fact_2004_Diff__empty,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Diff_empty
thf(fact_2005_finite__Diff2,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( finite_finite_int @ A4 ) ) ) ).
% finite_Diff2
thf(fact_2006_finite__Diff2,axiom,
! [B5: set_complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( finite3207457112153483333omplex @ A4 ) ) ) ).
% finite_Diff2
thf(fact_2007_finite__Diff2,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_2008_finite__Diff,axiom,
! [A4: set_int,B5: set_int] :
( ( finite_finite_int @ A4 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_2009_finite__Diff,axiom,
! [A4: set_complex,B5: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_2010_finite__Diff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_2011_Diff__eq__empty__iff,axiom,
! [A4: set_real,B5: set_real] :
( ( ( minus_minus_set_real @ A4 @ B5 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_2012_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_2013_Diff__eq__empty__iff,axiom,
! [A4: set_int,B5: set_int] :
( ( ( minus_minus_set_int @ A4 @ B5 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_2014_Diff__infinite__finite,axiom,
! [T2: set_int,S2: set_int] :
( ( finite_finite_int @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_2015_Diff__infinite__finite,axiom,
! [T2: set_complex,S2: set_complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_2016_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_2017_double__diff,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_2018_double__diff,axiom,
! [A4: set_int,B5: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C2 )
=> ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_2019_Diff__subset,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_2020_Diff__subset,axiom,
! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_2021_Diff__mono,axiom,
! [A4: set_nat,C2: set_nat,D6: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ D6 @ B5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ C2 @ D6 ) ) ) ) ).
% Diff_mono
thf(fact_2022_Diff__mono,axiom,
! [A4: set_int,C2: set_int,D6: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ C2 )
=> ( ( ord_less_eq_set_int @ D6 @ B5 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ C2 @ D6 ) ) ) ) ).
% Diff_mono
thf(fact_2023_psubset__imp__ex__mem,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
=> ? [B2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B2 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2024_psubset__imp__ex__mem,axiom,
! [A4: set_real,B5: set_real] :
( ( ord_less_set_real @ A4 @ B5 )
=> ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2025_psubset__imp__ex__mem,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B5 )
=> ? [B2: set_nat] : ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2026_psubset__imp__ex__mem,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2027_psubset__imp__ex__mem,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2028_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_int,Z4: list_int] : Y6 = Z4 )
= ( ^ [Xs2: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I3 )
= ( nth_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2029_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : Y6 = Z4 )
= ( ^ [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
= ( nth_VEBT_VEBT @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2030_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_o,Z4: list_o] : Y6 = Z4 )
= ( ^ [Xs2: list_o,Ys2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ Xs2 @ I3 )
= ( nth_o @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2031_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_nat,Z4: list_nat] : Y6 = Z4 )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2032_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: int] : ( P @ I3 @ X8 ) ) )
= ( ? [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2033_Skolem__list__nth,axiom,
! [K: nat,P: nat > vEBT_VEBT > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: vEBT_VEBT] : ( P @ I3 @ X8 ) ) )
= ( ? [Xs2: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2034_Skolem__list__nth,axiom,
! [K: nat,P: nat > $o > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: $o] : ( P @ I3 @ X8 ) ) )
= ( ? [Xs2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2035_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: nat] : ( P @ I3 @ X8 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2036_nth__equalityI,axiom,
! [Xs: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys3 @ I2 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_2037_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I2 )
= ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_2038_nth__equalityI,axiom,
! [Xs: list_o,Ys3: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I2 )
= ( nth_o @ Ys3 @ I2 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_2039_nth__equalityI,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_2040_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2041_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2042_all__set__conv__all__nth,axiom,
! [Xs: list_o,P: $o > $o] :
( ( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2043_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2044_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2045_all__nth__imp__all__set,axiom,
! [Xs: list_real,P: real > $o,X: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
=> ( P @ ( nth_real @ Xs @ I2 ) ) )
=> ( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2046_all__nth__imp__all__set,axiom,
! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
=> ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2047_all__nth__imp__all__set,axiom,
! [Xs: list_int,P: int > $o,X: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I2 ) ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2048_all__nth__imp__all__set,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
=> ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2049_all__nth__imp__all__set,axiom,
! [Xs: list_o,P: $o > $o,X: $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I2 ) ) )
=> ( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2050_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2051_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2052_in__set__conv__nth,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2053_in__set__conv__nth,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
& ( ( nth_set_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2054_in__set__conv__nth,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2055_in__set__conv__nth,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( ( nth_VEBT_VEBT @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2056_in__set__conv__nth,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
& ( ( nth_o @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2057_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2058_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2059_list__ball__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2060_list__ball__nth,axiom,
! [N: nat,Xs: list_o,P: $o > $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2061_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2062_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_2063_nth__mem,axiom,
! [N: nat,Xs: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% nth_mem
thf(fact_2064_nth__mem,axiom,
! [N: nat,Xs: list_set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_2065_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_2066_nth__mem,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% nth_mem
thf(fact_2067_nth__mem,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% nth_mem
thf(fact_2068_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_2069_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_2070_linordered__field__no__ub,axiom,
! [X5: rat] :
? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_2071_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% linordered_field_no_lb
thf(fact_2072_linordered__field__no__lb,axiom,
! [X5: rat] :
? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% linordered_field_no_lb
thf(fact_2073_divides__aux__eq,axiom,
! [Q4: code_integer,R2: code_integer] :
( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( R2 = zero_z3403309356797280102nteger ) ) ).
% divides_aux_eq
thf(fact_2074_divides__aux__eq,axiom,
! [Q4: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_2075_divides__aux__eq,axiom,
! [Q4: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_2076_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_int,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
= ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_2077_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
= ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_2078_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_o,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
=> ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
= ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_2079_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_2080_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
= ( P @ B2 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_2081_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_2082_add__0__iff,axiom,
! [B3: real,A3: real] :
( ( B3
= ( plus_plus_real @ B3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_2083_add__0__iff,axiom,
! [B3: rat,A3: rat] :
( ( B3
= ( plus_plus_rat @ B3 @ A3 ) )
= ( A3 = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_2084_add__0__iff,axiom,
! [B3: nat,A3: nat] :
( ( B3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_2085_add__0__iff,axiom,
! [B3: int,A3: int] :
( ( B3
= ( plus_plus_int @ B3 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_2086_find__Some__iff2,axiom,
! [X: int,P: int > $o,Xs: list_int] :
( ( ( some_int @ X )
= ( find_int @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( P @ ( nth_int @ Xs @ I3 ) )
& ( X
= ( nth_int @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_2087_find__Some__iff2,axiom,
! [X: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( ( some_P7363390416028606310at_nat @ X )
= ( find_P8199882355184865565at_nat @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ( X
= ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_2088_find__Some__iff2,axiom,
! [X: num,P: num > $o,Xs: list_num] :
( ( ( some_num @ X )
= ( find_num @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_num @ Xs ) )
& ( P @ ( nth_num @ Xs @ I3 ) )
& ( X
= ( nth_num @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_2089_find__Some__iff2,axiom,
! [X: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ( ( some_VEBT_VEBT @ X )
= ( find_VEBT_VEBT @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ( X
= ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_2090_find__Some__iff2,axiom,
! [X: $o,P: $o > $o,Xs: list_o] :
( ( ( some_o @ X )
= ( find_o @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
& ( P @ ( nth_o @ Xs @ I3 ) )
& ( X
= ( nth_o @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_2091_find__Some__iff2,axiom,
! [X: nat,P: nat > $o,Xs: list_nat] :
( ( ( some_nat @ X )
= ( find_nat @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I3 ) )
& ( X
= ( nth_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_2092_find__Some__iff,axiom,
! [P: int > $o,Xs: list_int,X: int] :
( ( ( find_int @ P @ Xs )
= ( some_int @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( P @ ( nth_int @ Xs @ I3 ) )
& ( X
= ( nth_int @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_2093_find__Some__iff,axiom,
! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ( find_P8199882355184865565at_nat @ P @ Xs )
= ( some_P7363390416028606310at_nat @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ( X
= ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_2094_find__Some__iff,axiom,
! [P: num > $o,Xs: list_num,X: num] :
( ( ( find_num @ P @ Xs )
= ( some_num @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_num @ Xs ) )
& ( P @ ( nth_num @ Xs @ I3 ) )
& ( X
= ( nth_num @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_2095_find__Some__iff,axiom,
! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ( find_VEBT_VEBT @ P @ Xs )
= ( some_VEBT_VEBT @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ( X
= ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_2096_find__Some__iff,axiom,
! [P: $o > $o,Xs: list_o,X: $o] :
( ( ( find_o @ P @ Xs )
= ( some_o @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
& ( P @ ( nth_o @ Xs @ I3 ) )
& ( X
= ( nth_o @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_2097_find__Some__iff,axiom,
! [P: nat > $o,Xs: list_nat,X: nat] :
( ( ( find_nat @ P @ Xs )
= ( some_nat @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I3 ) )
& ( X
= ( nth_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_2098_frac__unique__iff,axiom,
! [X: real,A3: real] :
( ( ( archim2898591450579166408c_real @ X )
= A3 )
= ( ( member_real @ ( minus_minus_real @ X @ A3 ) @ ring_1_Ints_real )
& ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_real @ A3 @ one_one_real ) ) ) ).
% frac_unique_iff
thf(fact_2099_frac__unique__iff,axiom,
! [X: rat,A3: rat] :
( ( ( archimedean_frac_rat @ X )
= A3 )
= ( ( member_rat @ ( minus_minus_rat @ X @ A3 ) @ ring_1_Ints_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ A3 @ one_one_rat ) ) ) ).
% frac_unique_iff
thf(fact_2100_nth__Cons__pos,axiom,
! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
= ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_2101_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_2102_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_2103_rotate1__length01,axiom,
! [Xs: list_VEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
=> ( ( rotate1_VEBT_VEBT @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_2104_rotate1__length01,axiom,
! [Xs: list_o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
=> ( ( rotate1_o @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_2105_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_2106_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_2107_nth__Cons__Suc,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( suc @ N ) )
= ( nth_VEBT_VEBT @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_2108_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_2109_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_2110_nth__Cons__0,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_2111_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_2112_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_2113_frac__eq__0__iff,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= zero_zero_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% frac_eq_0_iff
thf(fact_2114_frac__eq__0__iff,axiom,
! [X: rat] :
( ( ( archimedean_frac_rat @ X )
= zero_zero_rat )
= ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% frac_eq_0_iff
thf(fact_2115_frac__gt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
= ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% frac_gt_0_iff
thf(fact_2116_frac__gt__0__iff,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
= ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% frac_gt_0_iff
thf(fact_2117_enumerate__simps_I2_J,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( enumerate_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_2118_enumerate__simps_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_2119_find_Osimps_I2_J,axiom,
! [P: int > $o,X: int,Xs: list_int] :
( ( ( P @ X )
=> ( ( find_int @ P @ ( cons_int @ X @ Xs ) )
= ( some_int @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_int @ P @ ( cons_int @ X @ Xs ) )
= ( find_int @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_2120_find_Osimps_I2_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( ( P @ X )
=> ( ( find_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( some_nat @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( find_nat @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_2121_find_Osimps_I2_J,axiom,
! [P: product_prod_nat_nat > $o,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( P @ X )
=> ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( some_P7363390416028606310at_nat @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( find_P8199882355184865565at_nat @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_2122_find_Osimps_I2_J,axiom,
! [P: num > $o,X: num,Xs: list_num] :
( ( ( P @ X )
=> ( ( find_num @ P @ ( cons_num @ X @ Xs ) )
= ( some_num @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_num @ P @ ( cons_num @ X @ Xs ) )
= ( find_num @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_2123_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_2124_Ints__0,axiom,
member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% Ints_0
thf(fact_2125_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_2126_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
% Ints_1
thf(fact_2127_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_2128_Ints__1,axiom,
member_rat @ one_one_rat @ ring_1_Ints_rat ).
% Ints_1
thf(fact_2129_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_2130_fact__in__Ints,axiom,
! [N: nat] : ( member_int @ ( semiri1406184849735516958ct_int @ N ) @ ring_1_Ints_int ) ).
% fact_in_Ints
thf(fact_2131_fact__in__Ints,axiom,
! [N: nat] : ( member_real @ ( semiri2265585572941072030t_real @ N ) @ ring_1_Ints_real ) ).
% fact_in_Ints
thf(fact_2132_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y3: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_2133_Suc__length__conv,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ( suc @ N )
= ( size_s6755466524823107622T_VEBT @ Xs ) )
= ( ? [Y3: vEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( Xs
= ( cons_VEBT_VEBT @ Y3 @ Ys2 ) )
& ( ( size_s6755466524823107622T_VEBT @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_2134_Suc__length__conv,axiom,
! [N: nat,Xs: list_o] :
( ( ( suc @ N )
= ( size_size_list_o @ Xs ) )
= ( ? [Y3: $o,Ys2: list_o] :
( ( Xs
= ( cons_o @ Y3 @ Ys2 ) )
& ( ( size_size_list_o @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_2135_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_2136_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y3: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_2137_length__Suc__conv,axiom,
! [Xs: list_VEBT_VEBT,N: nat] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( suc @ N ) )
= ( ? [Y3: vEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( Xs
= ( cons_VEBT_VEBT @ Y3 @ Ys2 ) )
& ( ( size_s6755466524823107622T_VEBT @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_2138_length__Suc__conv,axiom,
! [Xs: list_o,N: nat] :
( ( ( size_size_list_o @ Xs )
= ( suc @ N ) )
= ( ? [Y3: $o,Ys2: list_o] :
( ( Xs
= ( cons_o @ Y3 @ Ys2 ) )
& ( ( size_size_list_o @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_2139_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_2140_set__subset__Cons,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_2141_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_2142_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_2143_impossible__Cons,axiom,
! [Xs: list_int,Ys3: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys3 ) )
=> ( Xs
!= ( cons_int @ X @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_2144_impossible__Cons,axiom,
! [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) )
=> ( Xs
!= ( cons_VEBT_VEBT @ X @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_2145_impossible__Cons,axiom,
! [Xs: list_o,Ys3: list_o,X: $o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys3 ) )
=> ( Xs
!= ( cons_o @ X @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_2146_impossible__Cons,axiom,
! [Xs: list_nat,Ys3: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) )
=> ( Xs
!= ( cons_nat @ X @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_2147_Ints__double__eq__0__iff,axiom,
! [A3: real] :
( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_2148_Ints__double__eq__0__iff,axiom,
! [A3: rat] :
( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ( ( plus_plus_rat @ A3 @ A3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_2149_Ints__double__eq__0__iff,axiom,
! [A3: int] :
( ( member_int @ A3 @ ring_1_Ints_int )
=> ( ( ( plus_plus_int @ A3 @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_2150_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X3: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_2151_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
= ( ? [X3: vEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( Xs
= ( cons_VEBT_VEBT @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_2152_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) )
= ( ? [X3: $o,Ys2: list_o] :
( ( Xs
= ( cons_o @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_2153_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_2154_Ints__odd__nonzero,axiom,
! [A3: complex] :
( ( member_complex @ A3 @ ring_1_Ints_complex )
=> ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A3 ) @ A3 )
!= zero_zero_complex ) ) ).
% Ints_odd_nonzero
thf(fact_2155_Ints__odd__nonzero,axiom,
! [A3: real] :
( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A3 ) @ A3 )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_2156_Ints__odd__nonzero,axiom,
! [A3: rat] :
( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A3 ) @ A3 )
!= zero_zero_rat ) ) ).
% Ints_odd_nonzero
thf(fact_2157_Ints__odd__nonzero,axiom,
! [A3: int] :
( ( member_int @ A3 @ ring_1_Ints_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A3 ) @ A3 )
!= zero_zero_int ) ) ).
% Ints_odd_nonzero
thf(fact_2158_list_Osize_I4_J,axiom,
! [X21: int,X22: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_2159_list_Osize_I4_J,axiom,
! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_2160_list_Osize_I4_J,axiom,
! [X21: $o,X22: list_o] :
( ( size_size_list_o @ ( cons_o @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_o @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_2161_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_2162_Ints__odd__less__0,axiom,
! [A3: real] :
( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A3 ) @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ) ).
% Ints_odd_less_0
thf(fact_2163_Ints__odd__less__0,axiom,
! [A3: rat] :
( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A3 ) @ A3 ) @ zero_zero_rat )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ).
% Ints_odd_less_0
thf(fact_2164_Ints__odd__less__0,axiom,
! [A3: int] :
( ( member_int @ A3 @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A3 ) @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% Ints_odd_less_0
thf(fact_2165_nth__Cons_H,axiom,
! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( ( N = zero_zero_nat )
=> ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
= ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_2166_nth__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_2167_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_2168_nth__equal__first__eq,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2169_nth__equal__first__eq,axiom,
! [X: real,Xs: list_real,N: nat] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( ( nth_real @ ( cons_real @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2170_nth__equal__first__eq,axiom,
! [X: set_nat,Xs: list_set_nat,N: nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( ( ( nth_set_nat @ ( cons_set_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2171_nth__equal__first__eq,axiom,
! [X: int,Xs: list_int,N: nat] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2172_nth__equal__first__eq,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2173_nth__equal__first__eq,axiom,
! [X: $o,Xs: list_o,N: nat] :
( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2174_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_2175_nth__non__equal__first__eq,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
( ( X != Y )
=> ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_2176_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_2177_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_2178_length__Cons,axiom,
! [X: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_2179_length__Cons,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) )
= ( suc @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% length_Cons
thf(fact_2180_length__Cons,axiom,
! [X: $o,Xs: list_o] :
( ( size_size_list_o @ ( cons_o @ X @ Xs ) )
= ( suc @ ( size_size_list_o @ Xs ) ) ) ).
% length_Cons
thf(fact_2181_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_2182_nth__zip,axiom,
! [I: nat,Xs: list_Code_integer,Ys3: list_Code_integer] :
( ( ord_less_nat @ I @ ( size_s3445333598471063425nteger @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3445333598471063425nteger @ Ys3 ) )
=> ( ( nth_Pr2304437835452373666nteger @ ( zip_Co3543743374963494515nteger @ Xs @ Ys3 ) @ I )
= ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs @ I ) @ ( nth_Code_integer @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2183_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys3: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys3 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys3 ) @ I )
= ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2184_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys3: list_VEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys3 ) )
=> ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys3 ) @ I )
= ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2185_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys3: list_o] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys3 ) )
=> ( ( nth_Pr7514405829937366042_int_o @ ( zip_int_o @ Xs @ Ys3 ) @ I )
= ( product_Pair_int_o @ ( nth_int @ Xs @ I ) @ ( nth_o @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2186_nth__zip,axiom,
! [I: nat,Xs: list_Code_integer,Ys3: list_o] :
( ( ord_less_nat @ I @ ( size_s3445333598471063425nteger @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys3 ) )
=> ( ( nth_Pr8522763379788166057eger_o @ ( zip_Code_integer_o @ Xs @ Ys3 ) @ I )
= ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ I ) @ ( nth_o @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2187_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys3: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys3 ) )
=> ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys3 ) @ I )
= ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2188_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys3: list_int] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys3 ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys3 ) @ I )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2189_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys3 ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys3 ) @ I )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2190_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys3: list_o] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys3 ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys3 ) @ I )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2191_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys3: list_nat] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys3 ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys3 ) @ I )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys3 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2192_dbl__dec__def,axiom,
( neg_nu6511756317524482435omplex
= ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% dbl_dec_def
thf(fact_2193_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_2194_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_2195_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_2196_count__list_Osimps_I2_J,axiom,
! [X: int,Y: int,Xs: list_int] :
( ( ( X = Y )
=> ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_int @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
= ( count_list_int @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_2197_count__list_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( count_list_nat @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_2198_count__notin,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( count_4203492906077236349at_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_2199_count__notin,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_2200_count__notin,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_2201_count__notin,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( count_list_int @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_2202_count__notin,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ( count_list_VEBT_VEBT @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_2203_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_2204_Cons__lenlex__iff,axiom,
! [M: code_integer,Ms: list_Code_integer,N: code_integer,Ns: list_Code_integer,R2: set_Pr4811707699266497531nteger] :
( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ ( cons_Code_integer @ M @ Ms ) @ ( cons_Code_integer @ N @ Ns ) ) @ ( lenlex_Code_integer @ R2 ) )
= ( ( ord_less_nat @ ( size_s3445333598471063425nteger @ Ms ) @ ( size_s3445333598471063425nteger @ Ns ) )
| ( ( ( size_s3445333598471063425nteger @ Ms )
= ( size_s3445333598471063425nteger @ Ns ) )
& ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Ms @ Ns ) @ ( lenlex_Code_integer @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2205_Cons__lenlex__iff,axiom,
! [M: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
| ( ( ( size_s5460976970255530739at_nat @ Ms )
= ( size_s5460976970255530739at_nat @ Ns ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2206_Cons__lenlex__iff,axiom,
! [M: int,Ms: list_int,N: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
| ( ( ( size_size_list_int @ Ms )
= ( size_size_list_int @ Ns ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2207_Cons__lenlex__iff,axiom,
! [M: vEBT_VEBT,Ms: list_VEBT_VEBT,N: vEBT_VEBT,Ns: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ ( cons_VEBT_VEBT @ M @ Ms ) @ ( cons_VEBT_VEBT @ N @ Ns ) ) @ ( lenlex_VEBT_VEBT @ R2 ) )
= ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ms ) @ ( size_s6755466524823107622T_VEBT @ Ns ) )
| ( ( ( size_s6755466524823107622T_VEBT @ Ms )
= ( size_s6755466524823107622T_VEBT @ Ns ) )
& ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Ms @ Ns ) @ ( lenlex_VEBT_VEBT @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2208_Cons__lenlex__iff,axiom,
! [M: $o,Ms: list_o,N: $o,Ns: list_o,R2: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ M @ Ms ) @ ( cons_o @ N @ Ns ) ) @ ( lenlex_o @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_o @ Ms ) @ ( size_size_list_o @ Ns ) )
| ( ( ( size_size_list_o @ Ms )
= ( size_size_list_o @ Ns ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ms @ Ns ) @ ( lenlex_o @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2209_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2210_in__measures_I2_J,axiom,
! [X: code_integer,Y: code_integer,F: code_integer > nat,Fs: list_C4705013386053401436er_nat] :
( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ ( cons_C1897838848541180310er_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_2211_in__measures_I2_J,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_2212_in__measures_I2_J,axiom,
! [X: nat,Y: nat,F: nat > nat,Fs: list_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_2213_in__measures_I2_J,axiom,
! [X: int,Y: int,F: int > nat,Fs: list_int_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_2214_intind,axiom,
! [I: nat,N: nat,P: int > $o,X: int] :
( ( ord_less_nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_2215_intind,axiom,
! [I: nat,N: nat,P: nat > $o,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_2216_intind,axiom,
! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_2217_frac__neg,axiom,
! [X: real] :
( ( ( member_real @ X @ ring_1_Ints_real )
=> ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
= zero_zero_real ) )
& ( ~ ( member_real @ X @ ring_1_Ints_real )
=> ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
= ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% frac_neg
thf(fact_2218_frac__neg,axiom,
! [X: rat] :
( ( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
= zero_zero_rat ) )
& ( ~ ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
= ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% frac_neg
thf(fact_2219_nth__Cons__numeral,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,V: num] :
( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_2220_nth__Cons__numeral,axiom,
! [X: int,Xs: list_int,V: num] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_2221_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_2222_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_2223_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_2224_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_2225_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_2226_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera6620942414471956472nteger @ M )
= ( numera6620942414471956472nteger @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_2227_add_Oinverse__inverse,axiom,
! [A3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_2228_add_Oinverse__inverse,axiom,
! [A3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_2229_add_Oinverse__inverse,axiom,
! [A3: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_2230_add_Oinverse__inverse,axiom,
! [A3: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_2231_neg__equal__iff__equal,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B3 ) )
= ( A3 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_2232_neg__equal__iff__equal,axiom,
! [A3: int,B3: int] :
( ( ( uminus_uminus_int @ A3 )
= ( uminus_uminus_int @ B3 ) )
= ( A3 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_2233_neg__equal__iff__equal,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( uminus1351360451143612070nteger @ A3 )
= ( uminus1351360451143612070nteger @ B3 ) )
= ( A3 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_2234_neg__equal__iff__equal,axiom,
! [A3: rat,B3: rat] :
( ( ( uminus_uminus_rat @ A3 )
= ( uminus_uminus_rat @ B3 ) )
= ( A3 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_2235_Compl__subset__Compl__iff,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B5 ) )
= ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_2236_Compl__anti__mono,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_2237_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_2238_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_2239_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_2240_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_2241_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_2242_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_2243_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_2244_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_2245_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_2246_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_2247_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_2248_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_2249_compl__le__compl__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
= ( ord_less_eq_set_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_2250_neg__le__iff__le,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_2251_neg__le__iff__le,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A3 ) )
= ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_2252_neg__le__iff__le,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_2253_neg__le__iff__le,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_2254_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_2255_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_2256_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_2257_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_2258_neg__0__equal__iff__equal,axiom,
! [A3: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A3 ) )
= ( zero_zero_real = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_2259_neg__0__equal__iff__equal,axiom,
! [A3: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A3 ) )
= ( zero_zero_int = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_2260_neg__0__equal__iff__equal,axiom,
! [A3: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A3 ) )
= ( zero_z3403309356797280102nteger = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_2261_neg__0__equal__iff__equal,axiom,
! [A3: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A3 ) )
= ( zero_zero_rat = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_2262_neg__equal__0__iff__equal,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_2263_neg__equal__0__iff__equal,axiom,
! [A3: int] :
( ( ( uminus_uminus_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_2264_neg__equal__0__iff__equal,axiom,
! [A3: code_integer] :
( ( ( uminus1351360451143612070nteger @ A3 )
= zero_z3403309356797280102nteger )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_2265_neg__equal__0__iff__equal,axiom,
! [A3: rat] :
( ( ( uminus_uminus_rat @ A3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
thf(fact_2266_equal__neg__zero,axiom,
! [A3: real] :
( ( A3
= ( uminus_uminus_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_2267_equal__neg__zero,axiom,
! [A3: int] :
( ( A3
= ( uminus_uminus_int @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_2268_equal__neg__zero,axiom,
! [A3: code_integer] :
( ( A3
= ( uminus1351360451143612070nteger @ A3 ) )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_2269_equal__neg__zero,axiom,
! [A3: rat] :
( ( A3
= ( uminus_uminus_rat @ A3 ) )
= ( A3 = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_2270_neg__equal__zero,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= A3 )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_2271_neg__equal__zero,axiom,
! [A3: int] :
( ( ( uminus_uminus_int @ A3 )
= A3 )
= ( A3 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_2272_neg__equal__zero,axiom,
! [A3: code_integer] :
( ( ( uminus1351360451143612070nteger @ A3 )
= A3 )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_2273_neg__equal__zero,axiom,
! [A3: rat] :
( ( ( uminus_uminus_rat @ A3 )
= A3 )
= ( A3 = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_2274_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_2275_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_2276_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_2277_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_2278_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_2279_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_2280_add__numeral__left,axiom,
! [V: num,W2: num,Z: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_2281_add__numeral__left,axiom,
! [V: num,W2: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_2282_add__numeral__left,axiom,
! [V: num,W2: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_2283_add__numeral__left,axiom,
! [V: num,W2: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_2284_add__numeral__left,axiom,
! [V: num,W2: num,Z: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_2285_add__numeral__left,axiom,
! [V: num,W2: num,Z: code_integer] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_2286_neg__less__iff__less,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_2287_neg__less__iff__less,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_2288_neg__less__iff__less,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A3 ) )
= ( ord_le6747313008572928689nteger @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_2289_neg__less__iff__less,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) )
= ( ord_less_rat @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_2290_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_2291_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_2292_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_2293_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_2294_add__minus__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ A3 @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_2295_add__minus__cancel,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ A3 @ ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_2296_add__minus__cancel,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ A3 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_2297_add__minus__cancel,axiom,
! [A3: rat,B3: rat] :
( ( plus_plus_rat @ A3 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_2298_minus__add__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( plus_plus_real @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_2299_minus__add__cancel,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ ( plus_plus_int @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_2300_minus__add__cancel,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_2301_minus__add__cancel,axiom,
! [A3: rat,B3: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ ( plus_plus_rat @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_2302_minus__add__distrib,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_2303_minus__add__distrib,axiom,
! [A3: int,B3: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_2304_minus__add__distrib,axiom,
! [A3: code_integer,B3: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( uminus1351360451143612070nteger @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_2305_minus__add__distrib,axiom,
! [A3: rat,B3: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A3 @ B3 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ ( uminus_uminus_rat @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_2306_minus__diff__eq,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( minus_minus_real @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_2307_minus__diff__eq,axiom,
! [A3: int,B3: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A3 @ B3 ) )
= ( minus_minus_int @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_2308_minus__diff__eq,axiom,
! [A3: code_integer,B3: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) )
= ( minus_8373710615458151222nteger @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_2309_minus__diff__eq,axiom,
! [A3: rat,B3: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A3 @ B3 ) )
= ( minus_minus_rat @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_2310_replicate__eq__replicate,axiom,
! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( ( replicate_VEBT_VEBT @ M @ X )
= ( replicate_VEBT_VEBT @ N @ Y ) )
= ( ( M = N )
& ( ( M != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_2311_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6511756317524482435omplex @ one_one_complex )
= one_one_complex ) ).
% dbl_dec_simps(3)
thf(fact_2312_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_2313_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
= one_one_rat ) ).
% dbl_dec_simps(3)
thf(fact_2314_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_2315_neg__0__le__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_2316_neg__0__le__iff__le,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A3 ) )
= ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger ) ) ).
% neg_0_le_iff_le
thf(fact_2317_neg__0__le__iff__le,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A3 ) )
= ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_2318_neg__0__le__iff__le,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_2319_neg__le__0__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_2320_neg__le__0__iff__le,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_2321_neg__le__0__iff__le,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_2322_neg__le__0__iff__le,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_2323_less__eq__neg__nonpos,axiom,
! [A3: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_2324_less__eq__neg__nonpos,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ A3 @ ( uminus1351360451143612070nteger @ A3 ) )
= ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_2325_less__eq__neg__nonpos,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ A3 ) )
= ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_2326_less__eq__neg__nonpos,axiom,
! [A3: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_2327_neg__less__eq__nonneg,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_2328_neg__less__eq__nonneg,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ A3 )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_2329_neg__less__eq__nonneg,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
= ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_2330_neg__less__eq__nonneg,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_2331_less__neg__neg,axiom,
! [A3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_2332_less__neg__neg,axiom,
! [A3: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_2333_less__neg__neg,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ A3 @ ( uminus1351360451143612070nteger @ A3 ) )
= ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_2334_less__neg__neg,axiom,
! [A3: rat] :
( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ A3 ) )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_2335_neg__less__pos,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_pos
thf(fact_2336_neg__less__pos,axiom,
! [A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% neg_less_pos
thf(fact_2337_neg__less__pos,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A3 ) @ A3 )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).
% neg_less_pos
thf(fact_2338_neg__less__pos,axiom,
! [A3: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
= ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).
% neg_less_pos
thf(fact_2339_neg__0__less__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_2340_neg__0__less__iff__less,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_2341_neg__0__less__iff__less,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A3 ) )
= ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) ) ).
% neg_0_less_iff_less
thf(fact_2342_neg__0__less__iff__less,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A3 ) )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_2343_neg__less__0__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_2344_neg__less__0__iff__less,axiom,
! [A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_2345_neg__less__0__iff__less,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A3 ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_2346_neg__less__0__iff__less,axiom,
! [A3: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_2347_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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_add_Oright__inverse,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_2356_add_Oright__inverse,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_2357_add_Oright__inverse,axiom,
! [A3: code_integer] :
( ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ A3 ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_2358_add_Oright__inverse,axiom,
! [A3: rat] :
( ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ A3 ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_2359_ab__left__minus,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_2360_ab__left__minus,axiom,
! [A3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_2361_ab__left__minus,axiom,
! [A3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A3 ) @ A3 )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_2362_ab__left__minus,axiom,
! [A3: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_2363_diff__0,axiom,
! [A3: real] :
( ( minus_minus_real @ zero_zero_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ).
% diff_0
thf(fact_2364_diff__0,axiom,
! [A3: int] :
( ( minus_minus_int @ zero_zero_int @ A3 )
= ( uminus_uminus_int @ A3 ) ) ).
% diff_0
thf(fact_2365_diff__0,axiom,
! [A3: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A3 )
= ( uminus1351360451143612070nteger @ A3 ) ) ).
% diff_0
thf(fact_2366_diff__0,axiom,
! [A3: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A3 )
= ( uminus_uminus_rat @ A3 ) ) ).
% diff_0
thf(fact_2367_verit__minus__simplify_I3_J,axiom,
! [B3: real] :
( ( minus_minus_real @ zero_zero_real @ B3 )
= ( uminus_uminus_real @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_2368_verit__minus__simplify_I3_J,axiom,
! [B3: int] :
( ( minus_minus_int @ zero_zero_int @ B3 )
= ( uminus_uminus_int @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_2369_verit__minus__simplify_I3_J,axiom,
! [B3: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B3 )
= ( uminus1351360451143612070nteger @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_2370_verit__minus__simplify_I3_J,axiom,
! [B3: rat] :
( ( minus_minus_rat @ zero_zero_rat @ B3 )
= ( uminus_uminus_rat @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_2371_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_2372_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_2373_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_2374_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_2375_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_2376_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_2377_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_2378_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_2379_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_2380_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_2381_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_2382_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_2383_diff__minus__eq__add,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( plus_plus_real @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_2384_diff__minus__eq__add,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( plus_plus_int @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_2385_diff__minus__eq__add,axiom,
! [A3: code_integer,B3: code_integer] :
( ( minus_8373710615458151222nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( plus_p5714425477246183910nteger @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_2386_diff__minus__eq__add,axiom,
! [A3: rat,B3: rat] :
( ( minus_minus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
= ( plus_plus_rat @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_2387_uminus__add__conv__diff,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( minus_minus_real @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_2388_uminus__add__conv__diff,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( minus_minus_int @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_2389_uminus__add__conv__diff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( minus_8373710615458151222nteger @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_2390_uminus__add__conv__diff,axiom,
! [A3: rat,B3: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
= ( minus_minus_rat @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_2391_euclidean__size__numeral,axiom,
! [K: num] :
( ( euclid6377331345833325938nteger @ ( numera6620942414471956472nteger @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% euclidean_size_numeral
thf(fact_2392_euclidean__size__numeral,axiom,
! [K: num] :
( ( euclid4774559944035922753ze_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% euclidean_size_numeral
thf(fact_2393_euclidean__size__numeral,axiom,
! [K: num] :
( ( euclid4777050414544973029ze_nat @ ( numeral_numeral_nat @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% euclidean_size_numeral
thf(fact_2394_Ball__set__replicate,axiom,
! [N: nat,A3: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A3 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A3 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_2395_Ball__set__replicate,axiom,
! [N: nat,A3: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A3 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A3 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_2396_Bex__set__replicate,axiom,
! [N: nat,A3: nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A3 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A3 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_2397_Bex__set__replicate,axiom,
! [N: nat,A3: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ? [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A3 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A3 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_2398_in__set__replicate,axiom,
! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_2399_in__set__replicate,axiom,
! [X: real,N: nat,Y: real] :
( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_2400_in__set__replicate,axiom,
! [X: set_nat,N: nat,Y: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_2401_in__set__replicate,axiom,
! [X: int,N: nat,Y: int] :
( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_2402_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_2403_in__set__replicate,axiom,
! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_2404_nth__replicate,axiom,
! [I: nat,N: nat,X: int] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_2405_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_2406_nth__replicate,axiom,
! [I: nat,N: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_2407_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_2408_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_2409_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_2410_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_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_2420_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_2421_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_2422_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_2423_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_2424_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_2425_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_2426_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_2427_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_2428_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_2429_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_2430_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_2431_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_dec_simps(2)
thf(fact_2432_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_2433_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_2434_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_2435_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_2436_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_2437_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_2438_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_2439_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_2440_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_2441_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_2442_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
!= ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_2443_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_2444_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_2445_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_2446_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6620942414471956472nteger @ M )
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_2447_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_2448_equation__minus__iff,axiom,
! [A3: real,B3: real] :
( ( A3
= ( uminus_uminus_real @ B3 ) )
= ( B3
= ( uminus_uminus_real @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_2449_equation__minus__iff,axiom,
! [A3: int,B3: int] :
( ( A3
= ( uminus_uminus_int @ B3 ) )
= ( B3
= ( uminus_uminus_int @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_2450_equation__minus__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3
= ( uminus1351360451143612070nteger @ B3 ) )
= ( B3
= ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_2451_equation__minus__iff,axiom,
! [A3: rat,B3: rat] :
( ( A3
= ( uminus_uminus_rat @ B3 ) )
= ( B3
= ( uminus_uminus_rat @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_2452_minus__equation__iff,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= B3 )
= ( ( uminus_uminus_real @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_2453_minus__equation__iff,axiom,
! [A3: int,B3: int] :
( ( ( uminus_uminus_int @ A3 )
= B3 )
= ( ( uminus_uminus_int @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_2454_minus__equation__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( uminus1351360451143612070nteger @ A3 )
= B3 )
= ( ( uminus1351360451143612070nteger @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_2455_minus__equation__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( uminus_uminus_rat @ A3 )
= B3 )
= ( ( uminus_uminus_rat @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_2456_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_2457_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_2458_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_2459_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_2460_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_2461_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_2462_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_2463_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_2464_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_2465_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_2466_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_2467_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_2468_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_2469_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_2470_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_2471_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_2472_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_2473_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ N )
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% numeral_neq_neg_one
thf(fact_2474_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_2475_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_2476_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ N )
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% numeral_neq_neg_one
thf(fact_2477_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_rat @ N )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% numeral_neq_neg_one
thf(fact_2478_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_2479_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_2480_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_2481_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_2482_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_2483_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_2484_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_2485_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_2486_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_2487_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_2488_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_2489_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_2490_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_2491_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_2492_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_2493_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_2494_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_2495_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_2496_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_2497_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_2498_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_2499_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_2500_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_2501_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_2502_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_2503_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_2504_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_2505_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_2506_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_2507_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_le_numeral
thf(fact_2508_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_2509_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_2510_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_2511_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_2512_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_2513_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_2514_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_2515_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_2516_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_2517_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_2518_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_2519_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_2520_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_2521_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_2522_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_2523_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_2524_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_less_one
thf(fact_2525_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_2526_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_2527_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_2528_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_less_numeral
thf(fact_2529_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_2530_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_2531_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_2532_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_less_neg_one
thf(fact_2533_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_2534_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_2535_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_2536_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_2537_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_2538_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_2539_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_2540_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_2541_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_2542_compl__le__swap2,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_2543_compl__le__swap1,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
=> ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_2544_compl__mono,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% compl_mono
thf(fact_2545_le__minus__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% le_minus_iff
thf(fact_2546_le__minus__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( ord_le3102999989581377725nteger @ B3 @ ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% le_minus_iff
thf(fact_2547_le__minus__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
= ( ord_less_eq_rat @ B3 @ ( uminus_uminus_rat @ A3 ) ) ) ).
% le_minus_iff
thf(fact_2548_le__minus__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_minus_iff
thf(fact_2549_minus__le__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% minus_le_iff
thf(fact_2550_minus__le__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A3 ) ) ).
% minus_le_iff
thf(fact_2551_minus__le__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ A3 ) ) ).
% minus_le_iff
thf(fact_2552_minus__le__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A3 ) ) ).
% minus_le_iff
thf(fact_2553_le__imp__neg__le,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_2554_le__imp__neg__le,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ A3 @ B3 )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_2555_le__imp__neg__le,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_2556_le__imp__neg__le,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_2557_less__minus__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% less_minus_iff
thf(fact_2558_less__minus__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_int @ B3 @ ( uminus_uminus_int @ A3 ) ) ) ).
% less_minus_iff
thf(fact_2559_less__minus__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le6747313008572928689nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( ord_le6747313008572928689nteger @ B3 @ ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% less_minus_iff
thf(fact_2560_less__minus__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
= ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ A3 ) ) ) ).
% less_minus_iff
thf(fact_2561_minus__less__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_2562_minus__less__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_2563_minus__less__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_2564_minus__less__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
= ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_2565_verit__negate__coefficient_I2_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2566_verit__negate__coefficient_I2_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2567_verit__negate__coefficient_I2_J,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le6747313008572928689nteger @ A3 @ B3 )
=> ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2568_verit__negate__coefficient_I2_J,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2569_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_2570_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_2571_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_2572_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_2573_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_2574_zero__neq__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( numera6620942414471956472nteger @ N ) ) ).
% zero_neq_numeral
thf(fact_2575_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_2576_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_2577_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_2578_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_2579_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_2580_is__num__normalize_I8_J,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% is_num_normalize(8)
thf(fact_2581_is__num__normalize_I8_J,axiom,
! [A3: int,B3: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% is_num_normalize(8)
thf(fact_2582_is__num__normalize_I8_J,axiom,
! [A3: code_integer,B3: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% is_num_normalize(8)
thf(fact_2583_is__num__normalize_I8_J,axiom,
! [A3: rat,B3: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A3 @ B3 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).
% is_num_normalize(8)
thf(fact_2584_group__cancel_Oneg1,axiom,
! [A4: real,K: real,A3: real] :
( ( A4
= ( plus_plus_real @ K @ A3 ) )
=> ( ( uminus_uminus_real @ A4 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_2585_group__cancel_Oneg1,axiom,
! [A4: int,K: int,A3: int] :
( ( A4
= ( plus_plus_int @ K @ A3 ) )
=> ( ( uminus_uminus_int @ A4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_2586_group__cancel_Oneg1,axiom,
! [A4: code_integer,K: code_integer,A3: code_integer] :
( ( A4
= ( plus_p5714425477246183910nteger @ K @ A3 ) )
=> ( ( uminus1351360451143612070nteger @ A4 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_2587_group__cancel_Oneg1,axiom,
! [A4: rat,K: rat,A3: rat] :
( ( A4
= ( plus_plus_rat @ K @ A3 ) )
=> ( ( uminus_uminus_rat @ A4 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_2588_add_Oinverse__distrib__swap,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2589_add_Oinverse__distrib__swap,axiom,
! [A3: int,B3: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2590_add_Oinverse__distrib__swap,axiom,
! [A3: code_integer,B3: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2591_add_Oinverse__distrib__swap,axiom,
! [A3: rat,B3: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A3 @ B3 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2592_minus__diff__commute,axiom,
! [B3: real,A3: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B3 ) @ A3 )
= ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_2593_minus__diff__commute,axiom,
! [B3: int,A3: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B3 ) @ A3 )
= ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_2594_minus__diff__commute,axiom,
! [B3: code_integer,A3: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A3 )
= ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_2595_minus__diff__commute,axiom,
! [B3: rat,A3: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B3 ) @ A3 )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_2596_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_2597_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_2598_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_2599_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_2600_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_2601_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_2602_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_2603_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_2604_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_2605_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_2606_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_2607_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_2608_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_2609_neg__eq__iff__add__eq__0,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= B3 )
= ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2610_neg__eq__iff__add__eq__0,axiom,
! [A3: int,B3: int] :
( ( ( uminus_uminus_int @ A3 )
= B3 )
= ( ( plus_plus_int @ A3 @ B3 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2611_neg__eq__iff__add__eq__0,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( uminus1351360451143612070nteger @ A3 )
= B3 )
= ( ( plus_p5714425477246183910nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2612_neg__eq__iff__add__eq__0,axiom,
! [A3: rat,B3: rat] :
( ( ( uminus_uminus_rat @ A3 )
= B3 )
= ( ( plus_plus_rat @ A3 @ B3 )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2613_eq__neg__iff__add__eq__0,axiom,
! [A3: real,B3: real] :
( ( A3
= ( uminus_uminus_real @ B3 ) )
= ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2614_eq__neg__iff__add__eq__0,axiom,
! [A3: int,B3: int] :
( ( A3
= ( uminus_uminus_int @ B3 ) )
= ( ( plus_plus_int @ A3 @ B3 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2615_eq__neg__iff__add__eq__0,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3
= ( uminus1351360451143612070nteger @ B3 ) )
= ( ( plus_p5714425477246183910nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2616_eq__neg__iff__add__eq__0,axiom,
! [A3: rat,B3: rat] :
( ( A3
= ( uminus_uminus_rat @ B3 ) )
= ( ( plus_plus_rat @ A3 @ B3 )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2617_add_Oinverse__unique,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A3 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_2618_add_Oinverse__unique,axiom,
! [A3: int,B3: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A3 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_2619_add_Oinverse__unique,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A3 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_2620_add_Oinverse__unique,axiom,
! [A3: rat,B3: rat] :
( ( ( plus_plus_rat @ A3 @ B3 )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A3 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_2621_ab__group__add__class_Oab__left__minus,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_2622_ab__group__add__class_Oab__left__minus,axiom,
! [A3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_2623_ab__group__add__class_Oab__left__minus,axiom,
! [A3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A3 ) @ A3 )
= zero_z3403309356797280102nteger ) ).
% ab_group_add_class.ab_left_minus
thf(fact_2624_ab__group__add__class_Oab__left__minus,axiom,
! [A3: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_2625_add__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= zero_zero_real )
= ( B3
= ( uminus_uminus_real @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_2626_add__eq__0__iff,axiom,
! [A3: int,B3: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= zero_zero_int )
= ( B3
= ( uminus_uminus_int @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_2627_add__eq__0__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger )
= ( B3
= ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_2628_add__eq__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( plus_plus_rat @ A3 @ B3 )
= zero_zero_rat )
= ( B3
= ( uminus_uminus_rat @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_2629_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_2630_zero__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_2631_zero__le__numeral,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).
% zero_le_numeral
thf(fact_2632_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_le_numeral
thf(fact_2633_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_2634_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_2635_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_2636_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_2637_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).
% not_numeral_le_zero
thf(fact_2638_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_2639_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_2640_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_2641_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_2642_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_2643_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_2644_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_2645_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_2646_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_2647_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_2648_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_2649_zero__less__numeral,axiom,
! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_less_numeral
thf(fact_2650_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_2651_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_2652_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_2653_zero__less__numeral,axiom,
! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_less_numeral
thf(fact_2654_zero__less__numeral,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).
% zero_less_numeral
thf(fact_2655_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_2656_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_2657_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_2658_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_2659_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_less_zero
thf(fact_2660_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).
% not_numeral_less_zero
thf(fact_2661_replicate__Suc,axiom,
! [N: nat,X: int] :
( ( replicate_int @ ( suc @ N ) @ X )
= ( cons_int @ X @ ( replicate_int @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_2662_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_2663_replicate__Suc,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( replicate_VEBT_VEBT @ ( suc @ N ) @ X )
= ( cons_VEBT_VEBT @ X @ ( replicate_VEBT_VEBT @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_2664_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_2665_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_2666_one__le__numeral,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).
% one_le_numeral
thf(fact_2667_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_2668_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_2669_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_2670_group__cancel_Osub2,axiom,
! [B5: real,K: real,B3: real,A3: real] :
( ( B5
= ( plus_plus_real @ K @ B3 ) )
=> ( ( minus_minus_real @ A3 @ B5 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_2671_group__cancel_Osub2,axiom,
! [B5: int,K: int,B3: int,A3: int] :
( ( B5
= ( plus_plus_int @ K @ B3 ) )
=> ( ( minus_minus_int @ A3 @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_2672_group__cancel_Osub2,axiom,
! [B5: code_integer,K: code_integer,B3: code_integer,A3: code_integer] :
( ( B5
= ( plus_p5714425477246183910nteger @ K @ B3 ) )
=> ( ( minus_8373710615458151222nteger @ A3 @ B5 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_2673_group__cancel_Osub2,axiom,
! [B5: rat,K: rat,B3: rat,A3: rat] :
( ( B5
= ( plus_plus_rat @ K @ B3 ) )
=> ( ( minus_minus_rat @ A3 @ B5 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_2674_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A: real,B: real] : ( plus_plus_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2675_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A: int,B: int] : ( plus_plus_int @ A @ ( uminus_uminus_int @ B ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2676_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A: code_integer,B: code_integer] : ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2677_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A: rat,B: rat] : ( plus_plus_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2678_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A: real,B: real] : ( plus_plus_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2679_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A: int,B: int] : ( plus_plus_int @ A @ ( uminus_uminus_int @ B ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2680_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A: code_integer,B: code_integer] : ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2681_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A: rat,B: rat] : ( plus_plus_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2682_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_2683_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_2684_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_2685_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_2686_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% not_numeral_less_one
thf(fact_2687_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).
% not_numeral_less_one
thf(fact_2688_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_2689_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_2690_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_2691_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_2692_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_2693_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_2694_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).
% one_plus_numeral_commute
thf(fact_2695_subset__Compl__self__eq,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ A4 ) )
= ( A4 = bot_bot_set_real ) ) ).
% subset_Compl_self_eq
thf(fact_2696_subset__Compl__self__eq,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_2697_subset__Compl__self__eq,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_2698_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_2699_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_2700_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_2701_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_2702_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_2703_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_2704_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_2705_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_2706_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_2707_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_2708_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_2709_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_2710_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_2711_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_2712_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_2713_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_2714_lenlex__length,axiom,
! [Ms: list_VEBT_VEBT,Ns: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Ms @ Ns ) @ ( lenlex_VEBT_VEBT @ R2 ) )
=> ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Ms ) @ ( size_s6755466524823107622T_VEBT @ Ns ) ) ) ).
% lenlex_length
thf(fact_2715_lenlex__length,axiom,
! [Ms: list_o,Ns: list_o,R2: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ms @ Ns ) @ ( lenlex_o @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Ms ) @ ( size_size_list_o @ Ns ) ) ) ).
% lenlex_length
thf(fact_2716_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_2717_measures__less,axiom,
! [F: code_integer > nat,X: code_integer,Y: code_integer,Fs: list_C4705013386053401436er_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ ( cons_C1897838848541180310er_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_2718_measures__less,axiom,
! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_2719_measures__less,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_2720_measures__less,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_2721_measures__lesseq,axiom,
! [F: code_integer > nat,X: code_integer,Y: code_integer,Fs: list_C4705013386053401436er_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ Fs ) )
=> ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ ( cons_C1897838848541180310er_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_2722_measures__lesseq,axiom,
! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_2723_measures__lesseq,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_2724_measures__lesseq,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_2725_count__le__length,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% count_le_length
thf(fact_2726_count__le__length,axiom,
! [Xs: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X ) @ ( size_size_list_o @ Xs ) ) ).
% count_le_length
thf(fact_2727_count__le__length,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_2728_Cons__replicate__eq,axiom,
! [X: int,Xs: list_int,N: nat,Y: int] :
( ( ( cons_int @ X @ Xs )
= ( replicate_int @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_int @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_2729_Cons__replicate__eq,axiom,
! [X: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_2730_Cons__replicate__eq,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( ( cons_VEBT_VEBT @ X @ Xs )
= ( replicate_VEBT_VEBT @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_VEBT_VEBT @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_2731_pred__max,axiom,
! [Deg: nat,Ma2: nat,X: nat,Mi2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_nat @ Ma2 @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some_nat @ Ma2 ) ) ) ) ).
% pred_max
thf(fact_2732_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2733_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2734_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2735_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2736_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2737_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2738_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
= ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2739_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2740_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2741_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2742_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2743_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2744_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2745_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2746_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2747_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2748_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2749_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2750_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2751_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2752_listrel__iff__nth,axiom,
! [Xs: list_Code_integer,Ys3: list_Code_integer,R2: set_Pr4811707699266497531nteger] :
( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs @ Ys3 ) @ ( listre5734910445319291053nteger @ R2 ) )
= ( ( ( size_s3445333598471063425nteger @ Xs )
= ( size_s3445333598471063425nteger @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3445333598471063425nteger @ Xs ) )
=> ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs @ N3 ) @ ( nth_Code_integer @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2753_listrel__iff__nth,axiom,
! [Xs: list_int,Ys3: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys3 ) @ ( listrel_int_int @ R2 ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N3 ) @ ( nth_int @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2754_listrel__iff__nth,axiom,
! [Xs: list_int,Ys3: list_VEBT_VEBT,R2: set_Pr8044002425091019955T_VEBT] :
( ( member4376149543098372618T_VEBT @ ( produc6743464080745587621T_VEBT @ Xs @ Ys3 ) @ ( listre8491537028387690453T_VEBT @ R2 ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
=> ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ N3 ) @ ( nth_VEBT_VEBT @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2755_listrel__iff__nth,axiom,
! [Xs: list_int,Ys3: list_o,R2: set_Pr903927857289325719_int_o] :
( ( member9156582987741540206list_o @ ( produc3167582181186427401list_o @ Xs @ Ys3 ) @ ( listrel_int_o @ R2 ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_o @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
=> ( member4489920277610959864_int_o @ ( product_Pair_int_o @ ( nth_int @ Xs @ N3 ) @ ( nth_o @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2756_listrel__iff__nth,axiom,
! [Xs: list_Code_integer,Ys3: list_o,R2: set_Pr448751882837621926eger_o] :
( ( member7510714728986300413list_o @ ( produc2864564883805000344list_o @ Xs @ Ys3 ) @ ( listre7327554457731897160eger_o @ R2 ) )
= ( ( ( size_s3445333598471063425nteger @ Xs )
= ( size_size_list_o @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3445333598471063425nteger @ Xs ) )
=> ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ N3 ) @ ( nth_o @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2757_listrel__iff__nth,axiom,
! [Xs: list_int,Ys3: list_nat,R2: set_Pr3448869479623346877nt_nat] :
( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ Ys3 ) @ ( listrel_int_nat @ R2 ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
=> ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ ( nth_int @ Xs @ N3 ) @ ( nth_nat @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2758_listrel__iff__nth,axiom,
! [Xs: list_VEBT_VEBT,Ys3: list_int,R2: set_Pr5066593544530342725BT_int] :
( ( member3703241499402361532st_int @ ( produc1392282695434103839st_int @ Xs @ Ys3 ) @ ( listre5898179758603845167BT_int @ R2 ) )
= ( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_int @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ N3 ) @ ( nth_int @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2759_listrel__iff__nth,axiom,
! [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys3 ) @ ( listre1230615542750757617T_VEBT @ R2 ) )
= ( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N3 ) @ ( nth_VEBT_VEBT @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2760_listrel__iff__nth,axiom,
! [Xs: list_VEBT_VEBT,Ys3: list_o,R2: set_Pr3175402225741728619VEBT_o] :
( ( member3126162362653435956list_o @ ( produc2717590391345394939list_o @ Xs @ Ys3 ) @ ( listrel_VEBT_VEBT_o @ R2 ) )
= ( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_o @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ N3 ) @ ( nth_o @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2761_listrel__iff__nth,axiom,
! [Xs: list_VEBT_VEBT,Ys3: list_nat,R2: set_Pr7556676689462069481BT_nat] :
( ( member6193324644334088288st_nat @ ( produc5570133714943300547st_nat @ Xs @ Ys3 ) @ ( listre5900670229112895443BT_nat @ R2 ) )
= ( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ N3 ) @ ( nth_nat @ Ys3 @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_2762_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_2763_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_2764_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_2765_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_2766_neg__numeral__le__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_2767_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_2768_ceiling__less__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2769_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2770__092_060open_0622_A_092_060le_062_Ax_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ xa ).
% \<open>2 \<le> x\<close>
thf(fact_2771_insert__simp__mima,axiom,
! [X: nat,Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi2 )
| ( X = Ma2 ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_2772_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_2773_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_2774_ceiling__zero,axiom,
( ( archim2889992004027027881ng_rat @ zero_zero_rat )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_2775_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_2776_ceiling__one,axiom,
( ( archim2889992004027027881ng_rat @ one_one_rat )
= one_one_int ) ).
% ceiling_one
thf(fact_2777_ceiling__one,axiom,
( ( archim7802044766580827645g_real @ one_one_real )
= one_one_int ) ).
% ceiling_one
thf(fact_2778_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_2779_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% dbl_simps(2)
thf(fact_2780_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_2781_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2782_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2783_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2784_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2785_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2786_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2787_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_Code_integer
= ( numera6620942414471956472nteger @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_2788_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6690914467698888265omplex @ N )
= one_one_complex )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2789_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_rat @ N )
= one_one_rat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2790_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2791_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2792_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2793_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2794_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6620942414471956472nteger @ N )
= one_one_Code_integer )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_2795_ceiling__add__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_2796_ceiling__add__one,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_2797_ceiling__diff__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
= ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_diff_one
thf(fact_2798_ceiling__diff__one,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
= ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% ceiling_diff_one
thf(fact_2799_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_2800_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_2801_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) )
= ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_2802_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_2803_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_2804_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_2805_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_2806_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_2807_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_2808_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_2809_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_2810_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_2811_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_2812_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_2813_ceiling__le__zero,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_le_zero
thf(fact_2814_ceiling__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_less_one
thf(fact_2815_ceiling__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_less_one
thf(fact_2816_one__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% one_le_ceiling
thf(fact_2817_one__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_le_ceiling
thf(fact_2818_zero__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% zero_less_ceiling
thf(fact_2819_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_2820_ceiling__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% ceiling_le_one
thf(fact_2821_ceiling__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% ceiling_le_one
thf(fact_2822_one__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ one_one_rat @ X ) ) ).
% one_less_ceiling
thf(fact_2823_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_2824_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_2825_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_2826_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_2827_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_2828_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_2829_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_2830_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_2831_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_2832_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_2833_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2834_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2835_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2836_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2837_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2838_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2839_one__add__one,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2840_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_2841_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_2842_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_2843_fact__2,axiom,
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_2844_fact__2,axiom,
( ( semiri4449623510593786356d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_2845_fact__2,axiom,
( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_2846_fact__2,axiom,
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_2847_fact__2,axiom,
( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_2848_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_2849_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_2850_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_2851_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_2852_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_2853_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2854_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2855_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_2856_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_2857_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_2858_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_2859_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_2860_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2861_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2862_ceiling__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% ceiling_less_zero
thf(fact_2863_ceiling__less__zero,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% ceiling_less_zero
thf(fact_2864_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_2865_zero__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% zero_le_ceiling
thf(fact_2866_ceiling__le__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% ceiling_le_numeral
thf(fact_2867_ceiling__le__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% ceiling_le_numeral
thf(fact_2868_numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% numeral_less_ceiling
thf(fact_2869_numeral__less__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% numeral_less_ceiling
thf(fact_2870_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_2871_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2872_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2881_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2882_dbl__simps_I3_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_2883_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_2884_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_2885_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_2886_dbl__simps_I3_J,axiom,
( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_2887_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_2888_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_2889_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_2890_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_2891_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_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_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_2903_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_2904_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_2905_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_2906_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_2907_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_2908_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_2909_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_2910_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_2911_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_2912_dbl__simps_I4_J,axiom,
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_2913_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_2914_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_2915_dbl__simps_I4_J,axiom,
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_2916_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_2917_ceiling__less__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% ceiling_less_numeral
thf(fact_2918_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% ceiling_less_numeral
thf(fact_2919_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_2920_numeral__le__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_2921_ceiling__le__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2922_ceiling__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2923_neg__numeral__less__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_2924_neg__numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_2925_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_2926_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_2927_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_2928_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_2929_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_2930_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_2931_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_2932_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_2933_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_2934_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_2935_imp__le__cong,axiom,
! [X: int,X9: int,P: $o,P4: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_2936_conj__le__cong,axiom,
! [X: int,X9: int,P: $o,P4: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_2937_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_2938_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_2939_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq_int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq_int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_2940_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_2941_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_2942_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_2943_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_2944_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_2945_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_2946_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_2947_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_2948_ceiling__mono,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% ceiling_mono
thf(fact_2949_ceiling__less__cancel,axiom,
! [X: rat,Y: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
=> ( ord_less_rat @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_2950_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_2951_ceiling__add__le,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% ceiling_add_le
thf(fact_2952_ceiling__add__le,axiom,
! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% ceiling_add_le
thf(fact_2953_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_2954_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_2955_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_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_2963_numeral__Bit0,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_Bit0
thf(fact_2964_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_2965_dbl__def,axiom,
( neg_numeral_dbl_rat
= ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_2966_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_2967_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_2968_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_2969_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_2970_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_2971_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_2972_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_2973_numeral__One,axiom,
( ( numera6620942414471956472nteger @ one )
= one_one_Code_integer ) ).
% numeral_One
thf(fact_2974_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_2975_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_2976_uminus__numeral__One,axiom,
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% uminus_numeral_One
thf(fact_2977_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_2978_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_2979_uminus__numeral__One,axiom,
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% uminus_numeral_One
thf(fact_2980_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_2981_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2982_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_2983_numeral__inc,axiom,
! [X: num] :
( ( numera6690914467698888265omplex @ ( inc @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% numeral_inc
thf(fact_2984_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_rat @ ( inc @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% numeral_inc
thf(fact_2985_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_2986_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_real @ ( inc @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% numeral_inc
thf(fact_2987_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_2988_numeral__inc,axiom,
! [X: num] :
( ( numera1916890842035813515d_enat @ ( inc @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_inc
thf(fact_2989_numeral__inc,axiom,
! [X: num] :
( ( numera6620942414471956472nteger @ ( inc @ X ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).
% numeral_inc
thf(fact_2990_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_2991_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_2992_mi__ma__2__deg,axiom,
! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
& ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_2993_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_2994_post__member__pre__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_2995_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_2996_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_2997_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_2998_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_2999_power__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( power_power_nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( some_nat @ Z ) ) ) ).
% power_shift
thf(fact_3000_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% local.power_def
thf(fact_3001_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_3002_set__n__deg__not__0,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_3003_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_3004_valid__insert__both__member__options__pres,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_3005_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_3006_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_3007_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_3008_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_3009_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_3010_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_3011_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( 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 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_3012_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( 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 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_3013_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_3014_sum__power2__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_3015_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_3016_zero__less__power2,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A3 != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_3017_zero__less__power2,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A3 != zero_zero_rat ) ) ).
% zero_less_power2
thf(fact_3018_zero__less__power2,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A3 != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_3019_power2__less__eq__zero__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_3020_power2__less__eq__zero__iff,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% power2_less_eq_zero_iff
thf(fact_3021_power2__less__eq__zero__iff,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_3022_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_3023_power2__eq__iff__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_3024_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_3025_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_3026_power__decreasing__iff,axiom,
! [B3: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_real @ B3 @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3027_power__decreasing__iff,axiom,
! [B3: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_rat @ B3 @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3028_power__decreasing__iff,axiom,
! [B3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3029_power__decreasing__iff,axiom,
! [B3: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ B3 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3030_zero__eq__power2,axiom,
! [A3: rat] :
( ( ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% zero_eq_power2
thf(fact_3031_zero__eq__power2,axiom,
! [A3: nat] :
( ( ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_3032_zero__eq__power2,axiom,
! [A3: real] :
( ( ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_3033_zero__eq__power2,axiom,
! [A3: int] :
( ( ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_3034_zero__eq__power2,axiom,
! [A3: complex] :
( ( ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A3 = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_3035_power__mono__iff,axiom,
! [A3: real,B3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B3 @ N ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_3036_power__mono__iff,axiom,
! [A3: rat,B3: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ B3 @ N ) )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_3037_power__mono__iff,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_3038_power__mono__iff,axiom,
! [A3: int,B3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_3039_power__increasing__iff,axiom,
! [B3: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B3 @ X ) @ ( power_power_real @ B3 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_3040_power__increasing__iff,axiom,
! [B3: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B3 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ X ) @ ( power_power_rat @ B3 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_3041_power__increasing__iff,axiom,
! [B3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_3042_power__increasing__iff,axiom,
! [B3: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B3 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B3 @ X ) @ ( power_power_int @ B3 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_3043_power__strict__decreasing__iff,axiom,
! [B3: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_real @ B3 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3044_power__strict__decreasing__iff,axiom,
! [B3: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_rat @ B3 @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3045_power__strict__decreasing__iff,axiom,
! [B3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3046_power__strict__decreasing__iff,axiom,
! [B3: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ B3 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3047_power__one,axiom,
! [N: nat] :
( ( power_power_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_one
thf(fact_3048_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_3049_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_3050_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_3051_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_3052_power__one__right,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ one_one_nat )
= A3 ) ).
% power_one_right
thf(fact_3053_power__one__right,axiom,
! [A3: real] :
( ( power_power_real @ A3 @ one_one_nat )
= A3 ) ).
% power_one_right
thf(fact_3054_power__one__right,axiom,
! [A3: int] :
( ( power_power_int @ A3 @ one_one_nat )
= A3 ) ).
% power_one_right
thf(fact_3055_power__one__right,axiom,
! [A3: complex] :
( ( power_power_complex @ A3 @ one_one_nat )
= A3 ) ).
% power_one_right
thf(fact_3056_power__inject__exp,axiom,
! [A3: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ( power_power_real @ A3 @ M )
= ( power_power_real @ A3 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3057_power__inject__exp,axiom,
! [A3: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ( ( power_power_rat @ A3 @ M )
= ( power_power_rat @ A3 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3058_power__inject__exp,axiom,
! [A3: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ( power_power_nat @ A3 @ M )
= ( power_power_nat @ A3 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3059_power__inject__exp,axiom,
! [A3: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ( power_power_int @ A3 @ M )
= ( power_power_int @ A3 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3060_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_3061_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_3062_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_3063_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_3064_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_3065_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_3066_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_3067_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_3068_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_3069_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_3070_power__Suc0__right,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_3071_power__Suc0__right,axiom,
! [A3: real] :
( ( power_power_real @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_3072_power__Suc0__right,axiom,
! [A3: int] :
( ( power_power_int @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_3073_power__Suc0__right,axiom,
! [A3: complex] :
( ( power_power_complex @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_3074_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_3075_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_3076_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_3077_power__strict__increasing__iff,axiom,
! [B3: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ ( power_power_real @ B3 @ X ) @ ( power_power_real @ B3 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_3078_power__strict__increasing__iff,axiom,
! [B3: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B3 )
=> ( ( ord_less_rat @ ( power_power_rat @ B3 @ X ) @ ( power_power_rat @ B3 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_3079_power__strict__increasing__iff,axiom,
! [B3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_nat @ ( power_power_nat @ B3 @ X ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_3080_power__strict__increasing__iff,axiom,
! [B3: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B3 )
=> ( ( ord_less_int @ ( power_power_int @ B3 @ X ) @ ( power_power_int @ B3 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_3081_power__eq__0__iff,axiom,
! [A3: rat,N: nat] :
( ( ( power_power_rat @ A3 @ N )
= zero_zero_rat )
= ( ( A3 = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3082_power__eq__0__iff,axiom,
! [A3: nat,N: nat] :
( ( ( power_power_nat @ A3 @ N )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3083_power__eq__0__iff,axiom,
! [A3: real,N: nat] :
( ( ( power_power_real @ A3 @ N )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3084_power__eq__0__iff,axiom,
! [A3: int,N: nat] :
( ( ( power_power_int @ A3 @ N )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3085_power__eq__0__iff,axiom,
! [A3: complex,N: nat] :
( ( ( power_power_complex @ A3 @ N )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3086_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_3087_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_3088_power__not__zero,axiom,
! [A3: rat,N: nat] :
( ( A3 != zero_zero_rat )
=> ( ( power_power_rat @ A3 @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_3089_power__not__zero,axiom,
! [A3: nat,N: nat] :
( ( A3 != zero_zero_nat )
=> ( ( power_power_nat @ A3 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_3090_power__not__zero,axiom,
! [A3: real,N: nat] :
( ( A3 != zero_zero_real )
=> ( ( power_power_real @ A3 @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_3091_power__not__zero,axiom,
! [A3: int,N: nat] :
( ( A3 != zero_zero_int )
=> ( ( power_power_int @ A3 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_3092_power__not__zero,axiom,
! [A3: complex,N: nat] :
( ( A3 != zero_zero_complex )
=> ( ( power_power_complex @ A3 @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_3093_zero__le__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_3094_zero__le__power,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_3095_zero__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_3096_zero__le__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_3097_power__mono,axiom,
! [A3: real,B3: real,N: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ).
% power_mono
thf(fact_3098_power__mono,axiom,
! [A3: rat,B3: rat,N: nat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ).
% power_mono
thf(fact_3099_power__mono,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ).
% power_mono
thf(fact_3100_power__mono,axiom,
! [A3: int,B3: int,N: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ).
% power_mono
thf(fact_3101_zero__less__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_3102_zero__less__power,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_3103_zero__less__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_3104_zero__less__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_3105_one__le__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A3 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A3 @ N ) ) ) ).
% one_le_power
thf(fact_3106_one__le__power,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A3 )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A3 @ N ) ) ) ).
% one_le_power
thf(fact_3107_one__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% one_le_power
thf(fact_3108_one__le__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A3 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A3 @ N ) ) ) ).
% one_le_power
thf(fact_3109_power__0,axiom,
! [A3: rat] :
( ( power_power_rat @ A3 @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_3110_power__0,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_3111_power__0,axiom,
! [A3: real] :
( ( power_power_real @ A3 @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_3112_power__0,axiom,
! [A3: int] :
( ( power_power_int @ A3 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_3113_power__0,axiom,
! [A3: complex] :
( ( power_power_complex @ A3 @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_3114_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_3115_power__less__imp__less__base,axiom,
! [A3: real,N: nat,B3: real] :
( ( ord_less_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B3 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_3116_power__less__imp__less__base,axiom,
! [A3: rat,N: nat,B3: rat] :
( ( ord_less_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ B3 @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_rat @ A3 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_3117_power__less__imp__less__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_3118_power__less__imp__less__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_3119_power__le__one,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_3120_power__le__one,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ A3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_3121_power__le__one,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_3122_power__le__one,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_3123_power__le__imp__le__base,axiom,
! [A3: real,N: nat,B3: real] :
( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( suc @ N ) ) @ ( power_power_real @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_3124_power__le__imp__le__base,axiom,
! [A3: rat,N: nat,B3: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( suc @ N ) ) @ ( power_power_rat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_3125_power__le__imp__le__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ ( power_power_nat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_3126_power__le__imp__le__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ ( power_power_int @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_3127_power__inject__base,axiom,
! [A3: real,N: nat,B3: real] :
( ( ( power_power_real @ A3 @ ( suc @ N ) )
= ( power_power_real @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_3128_power__inject__base,axiom,
! [A3: rat,N: nat,B3: rat] :
( ( ( power_power_rat @ A3 @ ( suc @ N ) )
= ( power_power_rat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_3129_power__inject__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( power_power_nat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_3130_power__inject__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ( power_power_int @ A3 @ ( suc @ N ) )
= ( power_power_int @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_3131_power__gt1,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A3 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3132_power__gt1,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A3 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3133_power__gt1,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3134_power__gt1,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3135_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_3136_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_3137_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_3138_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_3139_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_3140_power__strict__increasing,axiom,
! [N: nat,N6: nat,A3: real] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ one_one_real @ A3 )
=> ( ord_less_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ A3 @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_3141_power__strict__increasing,axiom,
! [N: nat,N6: nat,A3: rat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ord_less_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ A3 @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_3142_power__strict__increasing,axiom,
! [N: nat,N6: nat,A3: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ A3 @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_3143_power__strict__increasing,axiom,
! [N: nat,N6: nat,A3: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ A3 @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_3144_power__less__imp__less__exp,axiom,
! [A3: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3145_power__less__imp__less__exp,axiom,
! [A3: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ( ord_less_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3146_power__less__imp__less__exp,axiom,
! [A3: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3147_power__less__imp__less__exp,axiom,
! [A3: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3148_power__increasing,axiom,
! [N: nat,N6: nat,A3: real] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ A3 )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ A3 @ N6 ) ) ) ) ).
% power_increasing
thf(fact_3149_power__increasing,axiom,
! [N: nat,N6: nat,A3: rat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A3 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ A3 @ N6 ) ) ) ) ).
% power_increasing
thf(fact_3150_power__increasing,axiom,
! [N: nat,N6: nat,A3: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ A3 @ N6 ) ) ) ) ).
% power_increasing
thf(fact_3151_power__increasing,axiom,
! [N: nat,N6: nat,A3: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ one_one_int @ A3 )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ A3 @ N6 ) ) ) ) ).
% power_increasing
thf(fact_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_3159_power__Suc__le__self,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_3160_power__Suc__le__self,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ A3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_3161_power__Suc__le__self,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_3162_power__Suc__le__self,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_3163_power__Suc__less__one,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ A3 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A3 @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_3164_power__Suc__less__one,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ A3 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A3 @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_3165_power__Suc__less__one,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_3166_power__Suc__less__one,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_3167_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A3: real] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ A3 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A3 @ N6 ) @ ( power_power_real @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3168_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A3: rat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ A3 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A3 @ N6 ) @ ( power_power_rat @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3169_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A3: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N6 ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3170_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A3: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ N6 ) @ ( power_power_int @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3171_power__decreasing,axiom,
! [N: nat,N6: nat,A3: real] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N6 ) @ ( power_power_real @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3172_power__decreasing,axiom,
! [N: nat,N6: nat,A3: rat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ A3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N6 ) @ ( power_power_rat @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3173_power__decreasing,axiom,
! [N: nat,N6: nat,A3: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N6 ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3174_power__decreasing,axiom,
! [N: nat,N6: nat,A3: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N6 ) @ ( power_power_int @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3175_power__le__imp__le__exp,axiom,
! [A3: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3176_power__le__imp__le__exp,axiom,
! [A3: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3177_power__le__imp__le__exp,axiom,
! [A3: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3178_power__le__imp__le__exp,axiom,
! [A3: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3179_power__eq__iff__eq__base,axiom,
! [N: nat,A3: real,B3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ( power_power_real @ A3 @ N )
= ( power_power_real @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3180_power__eq__iff__eq__base,axiom,
! [N: nat,A3: rat,B3: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ( ( power_power_rat @ A3 @ N )
= ( power_power_rat @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3181_power__eq__iff__eq__base,axiom,
! [N: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ( power_power_nat @ A3 @ N )
= ( power_power_nat @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3182_power__eq__iff__eq__base,axiom,
! [N: nat,A3: int,B3: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ( power_power_int @ A3 @ N )
= ( power_power_int @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3183_power__eq__imp__eq__base,axiom,
! [A3: real,N: nat,B3: real] :
( ( ( power_power_real @ A3 @ N )
= ( power_power_real @ B3 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3184_power__eq__imp__eq__base,axiom,
! [A3: rat,N: nat,B3: rat] :
( ( ( power_power_rat @ A3 @ N )
= ( power_power_rat @ B3 @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3185_power__eq__imp__eq__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ( power_power_nat @ A3 @ N )
= ( power_power_nat @ B3 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3186_power__eq__imp__eq__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ( power_power_int @ A3 @ N )
= ( power_power_int @ B3 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3187_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_3188_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_3189_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_3190_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_3191_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_3192_self__le__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A3 @ ( power_power_real @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_3193_self__le__power,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A3 @ ( power_power_rat @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_3194_self__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_3195_self__le__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_3196_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_3197_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_3198_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_3199_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_3200_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex ) ).
% one_power2
thf(fact_3201_one__less__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_3202_one__less__power,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_3203_one__less__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_3204_one__less__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_3205_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_3206_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_3207_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_3208_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_3209_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_3210_power2__le__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_3211_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_3212_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_3213_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3214_power2__eq__imp__eq,axiom,
! [X: rat,Y: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3215_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3216_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3217_zero__le__power2,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_3218_zero__le__power2,axiom,
! [A3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_3219_zero__le__power2,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_3220_power__strict__mono,axiom,
! [A3: real,B3: real,N: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_3221_power__strict__mono,axiom,
! [A3: rat,B3: rat,N: nat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_3222_power__strict__mono,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_3223_power__strict__mono,axiom,
! [A3: int,B3: int,N: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_3224_power2__less__0,axiom,
! [A3: real] :
~ ( ord_less_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_3225_power2__less__0,axiom,
! [A3: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% power2_less_0
thf(fact_3226_power2__less__0,axiom,
! [A3: int] :
~ ( ord_less_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_3227_power2__eq__1__iff,axiom,
! [A3: complex] :
( ( ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex )
= ( ( A3 = one_one_complex )
| ( A3
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3228_power2__eq__1__iff,axiom,
! [A3: real] :
( ( ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( A3 = one_one_real )
| ( A3
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3229_power2__eq__1__iff,axiom,
! [A3: int] :
( ( ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A3 = one_one_int )
| ( A3
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3230_power2__eq__1__iff,axiom,
! [A3: code_integer] :
( ( ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( A3 = one_one_Code_integer )
| ( A3
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3231_power2__eq__1__iff,axiom,
! [A3: rat] :
( ( ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( A3 = one_one_rat )
| ( A3
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3232_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_3233_power2__less__imp__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_3234_power2__less__imp__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_3235_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_3236_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_3237_sum__power2__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_3238_sum__power2__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_3239_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_3240_sum__power2__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_3241_sum__power2__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_3242_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_3243_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_3244_not__sum__power2__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% not_sum_power2_lt_zero
thf(fact_3245_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_3246_sum__power2__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_3247_sum__power2__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_3248_sum__power2__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_3249_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_3250_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_3251_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_3252_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_3253_ex__power__ivl2,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ ( power_power_nat @ B3 @ N2 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_3254_ex__power__ivl1,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N2 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_3255_mintlistlength,axiom,
! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( Mi2 != Ma2 )
=> ( ( ord_less_nat @ Mi2 @ Ma2 )
& ? [M3: nat] :
( ( ( some_nat @ M3 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_3256_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_3257_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_3258_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_3259_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
!= zero_z3403309356797280102nteger )
=> ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_z3403309356797280102nteger ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3260_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_3261_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_3262_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_z3403309356797280102nteger )
=> ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
!= zero_z3403309356797280102nteger ) ) ).
% exp_add_not_zero_imp_left
thf(fact_3263_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_3264_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_3265_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_z3403309356797280102nteger )
=> ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
!= zero_z3403309356797280102nteger ) ) ).
% exp_add_not_zero_imp_right
thf(fact_3266_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_3267_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_3268_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_3269_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_3270_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_3271_pow__sum,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ B3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ).
% pow_sum
thf(fact_3272_power__minus__is__div,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A3 @ B3 ) )
= ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).
% power_minus_is_div
thf(fact_3273_bits__div__by__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_3274_bits__div__by__0,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_3275_bits__div__by__0,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ A3 @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% bits_div_by_0
thf(fact_3276_bits__div__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_3277_bits__div__0,axiom,
! [A3: int] :
( ( divide_divide_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_3278_bits__div__0,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A3 )
= zero_z3403309356797280102nteger ) ).
% bits_div_0
thf(fact_3279_division__ring__divide__zero,axiom,
! [A3: rat] :
( ( divide_divide_rat @ A3 @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_3280_division__ring__divide__zero,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_3281_divide__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ( divide_divide_rat @ A3 @ C )
= ( divide_divide_rat @ B3 @ C ) )
= ( ( C = zero_zero_rat )
| ( A3 = B3 ) ) ) ).
% divide_cancel_right
thf(fact_3282_divide__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ( divide_divide_real @ A3 @ C )
= ( divide_divide_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% divide_cancel_right
thf(fact_3283_divide__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ( divide_divide_rat @ C @ A3 )
= ( divide_divide_rat @ C @ B3 ) )
= ( ( C = zero_zero_rat )
| ( A3 = B3 ) ) ) ).
% divide_cancel_left
thf(fact_3284_divide__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ( divide_divide_real @ C @ A3 )
= ( divide_divide_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% divide_cancel_left
thf(fact_3285_divide__eq__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( divide_divide_rat @ A3 @ B3 )
= zero_zero_rat )
= ( ( A3 = zero_zero_rat )
| ( B3 = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_3286_divide__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( divide_divide_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_3287_div__by__0,axiom,
! [A3: rat] :
( ( divide_divide_rat @ A3 @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_3288_div__by__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_3289_div__by__0,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_3290_div__by__0,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_3291_div__by__0,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ A3 @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% div_by_0
thf(fact_3292_div__0,axiom,
! [A3: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A3 )
= zero_zero_rat ) ).
% div_0
thf(fact_3293_div__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% div_0
thf(fact_3294_div__0,axiom,
! [A3: int] :
( ( divide_divide_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% div_0
thf(fact_3295_div__0,axiom,
! [A3: real] :
( ( divide_divide_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% div_0
thf(fact_3296_div__0,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A3 )
= zero_z3403309356797280102nteger ) ).
% div_0
thf(fact_3297_bits__div__by__1,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ one_one_nat )
= A3 ) ).
% bits_div_by_1
thf(fact_3298_bits__div__by__1,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ one_one_int )
= A3 ) ).
% bits_div_by_1
thf(fact_3299_bits__div__by__1,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ A3 @ one_one_Code_integer )
= A3 ) ).
% bits_div_by_1
thf(fact_3300_div__by__1,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ A3 @ one_one_complex )
= A3 ) ).
% div_by_1
thf(fact_3301_div__by__1,axiom,
! [A3: rat] :
( ( divide_divide_rat @ A3 @ one_one_rat )
= A3 ) ).
% div_by_1
thf(fact_3302_div__by__1,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ one_one_nat )
= A3 ) ).
% div_by_1
thf(fact_3303_div__by__1,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ one_one_int )
= A3 ) ).
% div_by_1
thf(fact_3304_div__by__1,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ one_one_real )
= A3 ) ).
% div_by_1
thf(fact_3305_div__by__1,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ A3 @ one_one_Code_integer )
= A3 ) ).
% div_by_1
thf(fact_3306_div__minus__minus,axiom,
! [A3: int,B3: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) )
= ( divide_divide_int @ A3 @ B3 ) ) ).
% div_minus_minus
thf(fact_3307_div__minus__minus,axiom,
! [A3: code_integer,B3: code_integer] :
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( uminus1351360451143612070nteger @ B3 ) )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ).
% div_minus_minus
thf(fact_3308_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3309_atLeastAtMost__iff,axiom,
! [I: set_int,L: set_int,U: set_int] :
( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
= ( ( ord_less_eq_set_int @ L @ I )
& ( ord_less_eq_set_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3310_atLeastAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I )
& ( ord_less_eq_rat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3311_atLeastAtMost__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_eq_num @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3312_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3313_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3314_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_3315_Icc__eq__Icc,axiom,
! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
( ( ( set_or370866239135849197et_int @ L @ H )
= ( set_or370866239135849197et_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_int @ L @ H )
& ~ ( ord_less_eq_set_int @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3316_Icc__eq__Icc,axiom,
! [L: rat,H: rat,L2: rat,H2: rat] :
( ( ( set_or633870826150836451st_rat @ L @ H )
= ( set_or633870826150836451st_rat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_rat @ L @ H )
& ~ ( ord_less_eq_rat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3317_Icc__eq__Icc,axiom,
! [L: num,H: num,L2: num,H2: num] :
( ( ( set_or7049704709247886629st_num @ L @ H )
= ( set_or7049704709247886629st_num @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_num @ L @ H )
& ~ ( ord_less_eq_num @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3318_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3319_Icc__eq__Icc,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or1266510415728281911st_int @ L @ H )
= ( set_or1266510415728281911st_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_int @ L @ H )
& ~ ( ord_less_eq_int @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3320_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3321_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_3322_divide__eq__1__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( divide1717551699836669952omplex @ A3 @ B3 )
= one_one_complex )
= ( ( B3 != zero_zero_complex )
& ( A3 = B3 ) ) ) ).
% divide_eq_1_iff
thf(fact_3323_divide__eq__1__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( divide_divide_rat @ A3 @ B3 )
= one_one_rat )
= ( ( B3 != zero_zero_rat )
& ( A3 = B3 ) ) ) ).
% divide_eq_1_iff
thf(fact_3324_divide__eq__1__iff,axiom,
! [A3: real,B3: real] :
( ( ( divide_divide_real @ A3 @ B3 )
= one_one_real )
= ( ( B3 != zero_zero_real )
& ( A3 = B3 ) ) ) ).
% divide_eq_1_iff
thf(fact_3325_one__eq__divide__iff,axiom,
! [A3: complex,B3: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A3 @ B3 ) )
= ( ( B3 != zero_zero_complex )
& ( A3 = B3 ) ) ) ).
% one_eq_divide_iff
thf(fact_3326_one__eq__divide__iff,axiom,
! [A3: rat,B3: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A3 @ B3 ) )
= ( ( B3 != zero_zero_rat )
& ( A3 = B3 ) ) ) ).
% one_eq_divide_iff
thf(fact_3327_one__eq__divide__iff,axiom,
! [A3: real,B3: real] :
( ( one_one_real
= ( divide_divide_real @ A3 @ B3 ) )
= ( ( B3 != zero_zero_real )
& ( A3 = B3 ) ) ) ).
% one_eq_divide_iff
thf(fact_3328_divide__self,axiom,
! [A3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ A3 )
= one_one_complex ) ) ).
% divide_self
thf(fact_3329_divide__self,axiom,
! [A3: rat] :
( ( A3 != zero_zero_rat )
=> ( ( divide_divide_rat @ A3 @ A3 )
= one_one_rat ) ) ).
% divide_self
thf(fact_3330_divide__self,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ).
% divide_self
thf(fact_3331_divide__self__if,axiom,
! [A3: complex] :
( ( ( A3 = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ A3 )
= zero_zero_complex ) )
& ( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ A3 )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_3332_divide__self__if,axiom,
! [A3: rat] :
( ( ( A3 = zero_zero_rat )
=> ( ( divide_divide_rat @ A3 @ A3 )
= zero_zero_rat ) )
& ( ( A3 != zero_zero_rat )
=> ( ( divide_divide_rat @ A3 @ A3 )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_3333_divide__self__if,axiom,
! [A3: real] :
( ( ( A3 = zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= zero_zero_real ) )
& ( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_3334_divide__eq__eq__1,axiom,
! [B3: rat,A3: rat] :
( ( ( divide_divide_rat @ B3 @ A3 )
= one_one_rat )
= ( ( A3 != zero_zero_rat )
& ( A3 = B3 ) ) ) ).
% divide_eq_eq_1
thf(fact_3335_divide__eq__eq__1,axiom,
! [B3: real,A3: real] :
( ( ( divide_divide_real @ B3 @ A3 )
= one_one_real )
= ( ( A3 != zero_zero_real )
& ( A3 = B3 ) ) ) ).
% divide_eq_eq_1
thf(fact_3336_eq__divide__eq__1,axiom,
! [B3: rat,A3: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B3 @ A3 ) )
= ( ( A3 != zero_zero_rat )
& ( A3 = B3 ) ) ) ).
% eq_divide_eq_1
thf(fact_3337_eq__divide__eq__1,axiom,
! [B3: real,A3: real] :
( ( one_one_real
= ( divide_divide_real @ B3 @ A3 ) )
= ( ( A3 != zero_zero_real )
& ( A3 = B3 ) ) ) ).
% eq_divide_eq_1
thf(fact_3338_one__divide__eq__0__iff,axiom,
! [A3: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_3339_one__divide__eq__0__iff,axiom,
! [A3: real] :
( ( ( divide_divide_real @ one_one_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_3340_zero__eq__1__divide__iff,axiom,
! [A3: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A3 ) )
= ( A3 = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_3341_zero__eq__1__divide__iff,axiom,
! [A3: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_3342_div__self,axiom,
! [A3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ A3 )
= one_one_complex ) ) ).
% div_self
thf(fact_3343_div__self,axiom,
! [A3: rat] :
( ( A3 != zero_zero_rat )
=> ( ( divide_divide_rat @ A3 @ A3 )
= one_one_rat ) ) ).
% div_self
thf(fact_3344_div__self,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ A3 @ A3 )
= one_one_nat ) ) ).
% div_self
thf(fact_3345_div__self,axiom,
! [A3: int] :
( ( A3 != zero_zero_int )
=> ( ( divide_divide_int @ A3 @ A3 )
= one_one_int ) ) ).
% div_self
thf(fact_3346_div__self,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ).
% div_self
thf(fact_3347_div__self,axiom,
! [A3: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ A3 @ A3 )
= one_one_Code_integer ) ) ).
% div_self
thf(fact_3348_div__minus1__right,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A3 ) ) ).
% div_minus1_right
thf(fact_3349_div__minus1__right,axiom,
! [A3: code_integer] :
( ( divide6298287555418463151nteger @ A3 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ A3 ) ) ).
% div_minus1_right
thf(fact_3350_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_3351_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_3352_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_3353_atLeastatMost__empty__iff2,axiom,
! [A3: set_int,B3: set_int] :
( ( bot_bot_set_set_int
= ( set_or370866239135849197et_int @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3354_atLeastatMost__empty__iff2,axiom,
! [A3: rat,B3: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3355_atLeastatMost__empty__iff2,axiom,
! [A3: num,B3: num] :
( ( bot_bot_set_num
= ( set_or7049704709247886629st_num @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3356_atLeastatMost__empty__iff2,axiom,
! [A3: nat,B3: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3357_atLeastatMost__empty__iff2,axiom,
! [A3: int,B3: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3358_atLeastatMost__empty__iff2,axiom,
! [A3: real,B3: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3359_atLeastatMost__empty__iff,axiom,
! [A3: set_int,B3: set_int] :
( ( ( set_or370866239135849197et_int @ A3 @ B3 )
= bot_bot_set_set_int )
= ( ~ ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3360_atLeastatMost__empty__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( set_or633870826150836451st_rat @ A3 @ B3 )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3361_atLeastatMost__empty__iff,axiom,
! [A3: num,B3: num] :
( ( ( set_or7049704709247886629st_num @ A3 @ B3 )
= bot_bot_set_num )
= ( ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3362_atLeastatMost__empty__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( set_or1269000886237332187st_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3363_atLeastatMost__empty__iff,axiom,
! [A3: int,B3: int] :
( ( ( set_or1266510415728281911st_int @ A3 @ B3 )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3364_atLeastatMost__empty__iff,axiom,
! [A3: real,B3: real] :
( ( ( set_or1222579329274155063t_real @ A3 @ B3 )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3365_atLeastatMost__subset__iff,axiom,
! [A3: set_int,B3: set_int,C: set_int,D: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A3 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
= ( ~ ( ord_less_eq_set_int @ A3 @ B3 )
| ( ( ord_less_eq_set_int @ C @ A3 )
& ( ord_less_eq_set_int @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3366_atLeastatMost__subset__iff,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ~ ( ord_less_eq_rat @ A3 @ B3 )
| ( ( ord_less_eq_rat @ C @ A3 )
& ( ord_less_eq_rat @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3367_atLeastatMost__subset__iff,axiom,
! [A3: num,B3: num,C: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A3 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ~ ( ord_less_eq_num @ A3 @ B3 )
| ( ( ord_less_eq_num @ C @ A3 )
& ( ord_less_eq_num @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3368_atLeastatMost__subset__iff,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( ( ord_less_eq_nat @ C @ A3 )
& ( ord_less_eq_nat @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3369_atLeastatMost__subset__iff,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A3 @ B3 )
| ( ( ord_less_eq_int @ C @ A3 )
& ( ord_less_eq_int @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3370_atLeastatMost__subset__iff,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( ( ord_less_eq_real @ C @ A3 )
& ( ord_less_eq_real @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3371_atLeastatMost__empty,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( set_or633870826150836451st_rat @ A3 @ B3 )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_3372_atLeastatMost__empty,axiom,
! [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
=> ( ( set_or7049704709247886629st_num @ A3 @ B3 )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_3373_atLeastatMost__empty,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( set_or1269000886237332187st_nat @ A3 @ B3 )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_3374_atLeastatMost__empty,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( set_or1266510415728281911st_int @ A3 @ B3 )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_3375_atLeastatMost__empty,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( set_or1222579329274155063t_real @ A3 @ B3 )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_3376_infinite__Icc__iff,axiom,
! [A3: rat,B3: rat] :
( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) ) )
= ( ord_less_rat @ A3 @ B3 ) ) ).
% infinite_Icc_iff
thf(fact_3377_infinite__Icc__iff,axiom,
! [A3: real,B3: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% infinite_Icc_iff
thf(fact_3378_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_3379_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_3380_divide__le__0__1__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_3381_divide__le__0__1__iff,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_3382_zero__le__divide__1__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_divide_1_iff
thf(fact_3383_zero__le__divide__1__iff,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).
% zero_le_divide_1_iff
thf(fact_3384_divide__less__0__1__iff,axiom,
! [A3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) @ zero_zero_rat )
= ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_3385_divide__less__0__1__iff,axiom,
! [A3: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_3386_divide__less__eq__1__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
= ( ord_less_rat @ A3 @ B3 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_3387_divide__less__eq__1__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_3388_divide__less__eq__1__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
= ( ord_less_rat @ B3 @ A3 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_3389_divide__less__eq__1__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_3390_less__divide__eq__1__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
= ( ord_less_rat @ B3 @ A3 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_3391_less__divide__eq__1__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_3392_less__divide__eq__1__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
= ( ord_less_rat @ A3 @ B3 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_3393_less__divide__eq__1__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_3394_zero__less__divide__1__iff,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) )
= ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).
% zero_less_divide_1_iff
thf(fact_3395_zero__less__divide__1__iff,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A3 ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% zero_less_divide_1_iff
thf(fact_3396_divide__le__eq__1__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% divide_le_eq_1_neg
thf(fact_3397_divide__le__eq__1__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% divide_le_eq_1_neg
thf(fact_3398_divide__le__eq__1__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_3399_divide__le__eq__1__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
= ( ord_less_eq_rat @ B3 @ A3 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_3400_le__divide__eq__1__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_3401_le__divide__eq__1__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
= ( ord_less_eq_rat @ B3 @ A3 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_3402_le__divide__eq__1__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% le_divide_eq_1_pos
thf(fact_3403_le__divide__eq__1__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% le_divide_eq_1_pos
thf(fact_3404_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_3405_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_3406_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_3407_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_3408_bits__1__div__2,axiom,
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% bits_1_div_2
thf(fact_3409_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_3410_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_3411_one__div__two__eq__zero,axiom,
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% one_div_two_eq_zero
thf(fact_3412_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_3413_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_3414_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_3415_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_3416_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_3417_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_3418_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_3419_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_3420_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_3421_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_3422_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z5: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z5 ) )
=> ? [Y4: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y4 ) )
& ! [Z5: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z5 ) )
=> ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_3423_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_3424_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_3425_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_3426_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_3427_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_3428_div__minus__right,axiom,
! [A3: int,B3: int] :
( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).
% div_minus_right
thf(fact_3429_div__minus__right,axiom,
! [A3: code_integer,B3: code_integer] :
( ( divide6298287555418463151nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).
% div_minus_right
thf(fact_3430_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_3431_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_3432_infinite__Icc,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) ) ) ).
% infinite_Icc
thf(fact_3433_infinite__Icc,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) ) ).
% infinite_Icc
thf(fact_3434_divide__le__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% divide_le_0_iff
thf(fact_3435_divide__le__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ B3 ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% divide_le_0_iff
thf(fact_3436_divide__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% divide_right_mono
thf(fact_3437_divide__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% divide_right_mono
thf(fact_3438_zero__le__divide__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_3439_zero__le__divide__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ B3 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
| ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_3440_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_3441_divide__nonneg__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_3442_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_3443_divide__nonneg__nonpos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_3444_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_3445_divide__nonpos__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_3446_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_3447_divide__nonpos__nonpos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_3448_divide__right__mono__neg,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A3 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_3449_divide__right__mono__neg,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( divide_divide_rat @ A3 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_3450_divide__neg__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_3451_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_3452_divide__neg__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_3453_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_3454_divide__pos__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_3455_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_3456_divide__pos__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_3457_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_3458_divide__less__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A3 @ B3 ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ B3 @ zero_zero_rat ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% divide_less_0_iff
thf(fact_3459_divide__less__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% divide_less_0_iff
thf(fact_3460_divide__less__cancel,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ B3 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ A3 ) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
thf(fact_3461_divide__less__cancel,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_3462_zero__less__divide__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ B3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ zero_zero_rat @ B3 ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% zero_less_divide_iff
thf(fact_3463_zero__less__divide__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_3464_divide__strict__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_3465_divide__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_3466_divide__strict__right__mono__neg,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_3467_divide__strict__right__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_3468_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_3469_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_3470_right__inverse__eq,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A3 @ B3 )
= one_one_complex )
= ( A3 = B3 ) ) ) ).
% right_inverse_eq
thf(fact_3471_right__inverse__eq,axiom,
! [B3: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A3 @ B3 )
= one_one_rat )
= ( A3 = B3 ) ) ) ).
% right_inverse_eq
thf(fact_3472_right__inverse__eq,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( ( divide_divide_real @ A3 @ B3 )
= one_one_real )
= ( A3 = B3 ) ) ) ).
% right_inverse_eq
thf(fact_3473_nonzero__minus__divide__right,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
= ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_3474_nonzero__minus__divide__right,axiom,
! [B3: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) )
= ( divide_divide_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_3475_nonzero__minus__divide__divide,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_3476_nonzero__minus__divide__divide,axiom,
! [B3: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A3 ) @ ( uminus_uminus_rat @ B3 ) )
= ( divide_divide_rat @ A3 @ B3 ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_3477_divide__numeral__1,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ ( numeral_numeral_real @ one ) )
= A3 ) ).
% divide_numeral_1
thf(fact_3478_power__one__over,axiom,
! [A3: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A3 @ N ) ) ) ).
% power_one_over
thf(fact_3479_power__one__over,axiom,
! [A3: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A3 ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A3 @ N ) ) ) ).
% power_one_over
thf(fact_3480_power__one__over,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A3 ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A3 @ N ) ) ) ).
% power_one_over
thf(fact_3481_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_3482_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_3483_atLeastatMost__psubset__iff,axiom,
! [A3: set_int,B3: set_int,C: set_int,D: set_int] :
( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A3 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_int @ A3 @ B3 )
| ( ( ord_less_eq_set_int @ C @ A3 )
& ( ord_less_eq_set_int @ B3 @ D )
& ( ( ord_less_set_int @ C @ A3 )
| ( ord_less_set_int @ B3 @ D ) ) ) )
& ( ord_less_eq_set_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3484_atLeastatMost__psubset__iff,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ~ ( ord_less_eq_rat @ A3 @ B3 )
| ( ( ord_less_eq_rat @ C @ A3 )
& ( ord_less_eq_rat @ B3 @ D )
& ( ( ord_less_rat @ C @ A3 )
| ( ord_less_rat @ B3 @ D ) ) ) )
& ( ord_less_eq_rat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3485_atLeastatMost__psubset__iff,axiom,
! [A3: num,B3: num,C: num,D: num] :
( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A3 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ( ~ ( ord_less_eq_num @ A3 @ B3 )
| ( ( ord_less_eq_num @ C @ A3 )
& ( ord_less_eq_num @ B3 @ D )
& ( ( ord_less_num @ C @ A3 )
| ( ord_less_num @ B3 @ D ) ) ) )
& ( ord_less_eq_num @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3486_atLeastatMost__psubset__iff,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( ( ord_less_eq_nat @ C @ A3 )
& ( ord_less_eq_nat @ B3 @ D )
& ( ( ord_less_nat @ C @ A3 )
| ( ord_less_nat @ B3 @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3487_atLeastatMost__psubset__iff,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A3 @ B3 )
| ( ( ord_less_eq_int @ C @ A3 )
& ( ord_less_eq_int @ B3 @ D )
& ( ( ord_less_int @ C @ A3 )
| ( ord_less_int @ B3 @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3488_atLeastatMost__psubset__iff,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( ( ord_less_eq_real @ C @ A3 )
& ( ord_less_eq_real @ B3 @ D )
& ( ( ord_less_real @ C @ A3 )
| ( ord_less_real @ B3 @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3489_frac__le,axiom,
! [Y: real,X: real,W2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_3490_frac__le,axiom,
! [Y: rat,X: rat,W2: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_3491_frac__less,axiom,
! [X: real,Y: real,W2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_3492_frac__less,axiom,
! [X: rat,Y: rat,W2: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_3493_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_3494_frac__less2,axiom,
! [X: rat,Y: rat,W2: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_rat @ W2 @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_3495_divide__le__cancel,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% divide_le_cancel
thf(fact_3496_divide__le__cancel,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ B3 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% divide_le_cancel
thf(fact_3497_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_3498_divide__nonneg__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_3499_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_3500_divide__nonneg__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_3501_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_3502_divide__nonpos__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_3503_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_3504_divide__nonpos__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_3505_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
=> ( ( divide6298287555418463151nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3506_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ B3 )
=> ( ( divide_divide_nat @ A3 @ B3 )
= zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3507_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ B3 )
=> ( ( divide_divide_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3508_div__positive,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_positive
thf(fact_3509_div__positive,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_positive
thf(fact_3510_div__positive,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A3 )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_positive
thf(fact_3511_divide__less__eq__1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ B3 @ A3 ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_rat @ A3 @ B3 ) )
| ( A3 = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_3512_divide__less__eq__1,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B3 @ A3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ A3 @ B3 ) )
| ( A3 = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_3513_less__divide__eq__1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ A3 @ B3 ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_rat @ B3 @ A3 ) ) ) ) ).
% less_divide_eq_1
thf(fact_3514_less__divide__eq__1,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% less_divide_eq_1
thf(fact_3515_div__add__self1,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_3516_div__add__self1,axiom,
! [B3: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
= ( plus_plus_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_3517_div__add__self1,axiom,
! [B3: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B3 @ A3 ) @ B3 )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ one_one_Code_integer ) ) ) ).
% div_add_self1
thf(fact_3518_div__add__self2,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_3519_div__add__self2,axiom,
! [B3: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= ( plus_plus_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_3520_div__add__self2,axiom,
! [B3: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ B3 )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ one_one_Code_integer ) ) ) ).
% div_add_self2
thf(fact_3521_gt__half__sum,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B3 ) ) ).
% gt_half_sum
thf(fact_3522_gt__half__sum,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B3 ) ) ).
% gt_half_sum
thf(fact_3523_less__half__sum,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ord_less_rat @ A3 @ ( divide_divide_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_3524_less__half__sum,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ A3 @ ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_3525_divide__eq__minus__1__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( divide1717551699836669952omplex @ A3 @ B3 )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B3 != zero_zero_complex )
& ( A3
= ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_3526_divide__eq__minus__1__iff,axiom,
! [A3: real,B3: real] :
( ( ( divide_divide_real @ A3 @ B3 )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B3 != zero_zero_real )
& ( A3
= ( uminus_uminus_real @ B3 ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_3527_divide__eq__minus__1__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( divide_divide_rat @ A3 @ B3 )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ( B3 != zero_zero_rat )
& ( A3
= ( uminus_uminus_rat @ B3 ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_3528_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_3529_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_3530_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_3531_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_3532_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_3533_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_3534_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_3535_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_3536_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_3537_divide__le__eq__1,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ A3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ A3 @ B3 ) )
| ( A3 = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_3538_divide__le__eq__1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ B3 @ A3 ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ A3 @ B3 ) )
| ( A3 = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_3539_le__divide__eq__1,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% le_divide_eq_1
thf(fact_3540_le__divide__eq__1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ A3 @ B3 ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% le_divide_eq_1
thf(fact_3541_power__diff,axiom,
! [A3: rat,N: nat,M: nat] :
( ( A3 != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_rat @ A3 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N ) ) ) ) ) ).
% power_diff
thf(fact_3542_power__diff,axiom,
! [A3: complex,N: nat,M: nat] :
( ( A3 != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_complex @ A3 @ ( minus_minus_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A3 @ M ) @ ( power_power_complex @ A3 @ N ) ) ) ) ) ).
% power_diff
thf(fact_3543_power__diff,axiom,
! [A3: nat,N: nat,M: nat] :
( ( A3 != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A3 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_diff
thf(fact_3544_power__diff,axiom,
! [A3: int,N: nat,M: nat] :
( ( A3 != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A3 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N ) ) ) ) ) ).
% power_diff
thf(fact_3545_power__diff,axiom,
! [A3: real,N: nat,M: nat] :
( ( A3 != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_real @ A3 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N ) ) ) ) ) ).
% power_diff
thf(fact_3546_power__diff,axiom,
! [A3: code_integer,N: nat,M: nat] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ M @ N ) )
= ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N ) ) ) ) ) ).
% power_diff
thf(fact_3547_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_3548_div__if,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M6 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_3549_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_3550_half__gt__zero,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_3551_half__gt__zero,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_3552_half__gt__zero__iff,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).
% half_gt_zero_iff
thf(fact_3553_half__gt__zero__iff,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% half_gt_zero_iff
thf(fact_3554_field__less__half__sum,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_3555_field__less__half__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_3556_power__diff__power__eq,axiom,
! [A3: nat,N: nat,M: nat] :
( ( A3 != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N ) )
= ( power_power_nat @ A3 @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A3 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3557_power__diff__power__eq,axiom,
! [A3: int,N: nat,M: nat] :
( ( A3 != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N ) )
= ( power_power_int @ A3 @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A3 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3558_power__diff__power__eq,axiom,
! [A3: code_integer,N: nat,M: nat] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N ) )
= ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3559_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_3560_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_3561_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_3562_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_3563_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_3564_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma2: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma2 )
=> ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( 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 @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= none_nat ) ) ) ) ).
% pred_list_to_short
thf(fact_3565_high__bound__aux,axiom,
! [Ma2: nat,N: nat,M: nat] :
( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_3566_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X3: nat,N3: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% high_def
thf(fact_3567_even__succ__div__exp,axiom,
! [A3: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3568_even__succ__div__exp,axiom,
! [A3: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3569_even__succ__div__exp,axiom,
! [A3: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3570_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_3571_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_3572_dvd__0__right,axiom,
! [A3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ zero_z3403309356797280102nteger ) ).
% dvd_0_right
thf(fact_3573_dvd__0__right,axiom,
! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).
% dvd_0_right
thf(fact_3574_dvd__0__right,axiom,
! [A3: rat] : ( dvd_dvd_rat @ A3 @ zero_zero_rat ) ).
% dvd_0_right
thf(fact_3575_dvd__0__right,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_3576_dvd__0__right,axiom,
! [A3: int] : ( dvd_dvd_int @ A3 @ zero_zero_int ) ).
% dvd_0_right
thf(fact_3577_dvd__0__left__iff,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A3 )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left_iff
thf(fact_3578_dvd__0__left__iff,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
= ( A3 = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_3579_dvd__0__left__iff,axiom,
! [A3: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A3 )
= ( A3 = zero_zero_rat ) ) ).
% dvd_0_left_iff
thf(fact_3580_dvd__0__left__iff,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_3581_dvd__0__left__iff,axiom,
! [A3: int] :
( ( dvd_dvd_int @ zero_zero_int @ A3 )
= ( A3 = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_3582_dvd__add__triv__right__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ A3 ) )
= ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).
% dvd_add_triv_right_iff
thf(fact_3583_dvd__add__triv__right__iff,axiom,
! [A3: real,B3: real] :
( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( dvd_dvd_real @ A3 @ B3 ) ) ).
% dvd_add_triv_right_iff
thf(fact_3584_dvd__add__triv__right__iff,axiom,
! [A3: rat,B3: rat] :
( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ A3 ) )
= ( dvd_dvd_rat @ A3 @ B3 ) ) ).
% dvd_add_triv_right_iff
thf(fact_3585_dvd__add__triv__right__iff,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ).
% dvd_add_triv_right_iff
thf(fact_3586_dvd__add__triv__right__iff,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ).
% dvd_add_triv_right_iff
thf(fact_3587_dvd__add__triv__left__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
= ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).
% dvd_add_triv_left_iff
thf(fact_3588_dvd__add__triv__left__iff,axiom,
! [A3: real,B3: real] :
( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( dvd_dvd_real @ A3 @ B3 ) ) ).
% dvd_add_triv_left_iff
thf(fact_3589_dvd__add__triv__left__iff,axiom,
! [A3: rat,B3: rat] :
( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
= ( dvd_dvd_rat @ A3 @ B3 ) ) ).
% dvd_add_triv_left_iff
thf(fact_3590_dvd__add__triv__left__iff,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ).
% dvd_add_triv_left_iff
thf(fact_3591_dvd__add__triv__left__iff,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ).
% dvd_add_triv_left_iff
thf(fact_3592_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_3593_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_3594_div__dvd__div,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ A3 @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B3 @ A3 ) @ ( divide_divide_nat @ C @ A3 ) )
= ( dvd_dvd_nat @ B3 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_3595_div__dvd__div,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ A3 @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B3 @ A3 ) @ ( divide_divide_int @ C @ A3 ) )
= ( dvd_dvd_int @ B3 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_3596_div__dvd__div,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ A3 @ C )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B3 @ A3 ) @ ( divide6298287555418463151nteger @ C @ A3 ) )
= ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_3597_minus__dvd__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
= ( dvd_dvd_real @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3598_minus__dvd__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( dvd_dvd_int @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3599_minus__dvd__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
= ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3600_minus__dvd__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3601_dvd__minus__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
= ( dvd_dvd_real @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3602_dvd__minus__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( dvd_dvd_int @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3603_dvd__minus__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
= ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3604_dvd__minus__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3605_unit__div__1__div__1,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) )
= A3 ) ) ).
% unit_div_1_div_1
thf(fact_3606_unit__div__1__div__1,axiom,
! [A3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A3 ) )
= A3 ) ) ).
% unit_div_1_div_1
thf(fact_3607_unit__div__1__div__1,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 ) )
= A3 ) ) ).
% unit_div_1_div_1
thf(fact_3608_unit__div__1__unit,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_3609_unit__div__1__unit,axiom,
! [A3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A3 ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_3610_unit__div__1__unit,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 ) @ one_one_Code_integer ) ) ).
% unit_div_1_unit
thf(fact_3611_unit__div,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_3612_unit__div,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_3613_unit__div,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ one_one_Code_integer ) ) ) ).
% unit_div
thf(fact_3614_div__add,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( dvd_dvd_nat @ C @ A3 )
=> ( ( dvd_dvd_nat @ C @ B3 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) ) ) ) ).
% div_add
thf(fact_3615_div__add,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ C @ A3 )
=> ( ( dvd_dvd_int @ C @ B3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ) ).
% div_add
thf(fact_3616_div__add,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A3 )
=> ( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ) ).
% div_add
thf(fact_3617_div__diff,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ C @ A3 )
=> ( ( dvd_dvd_int @ C @ B3 )
=> ( ( divide_divide_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ) ).
% div_diff
thf(fact_3618_div__diff,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A3 )
=> ( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C )
= ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ) ).
% div_diff
thf(fact_3619_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_3620_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_3621_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_3622_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_3623_pow__divides__pow__iff,axiom,
! [N: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ) ).
% pow_divides_pow_iff
thf(fact_3624_pow__divides__pow__iff,axiom,
! [N: nat,A3: int,B3: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ) ).
% pow_divides_pow_iff
thf(fact_3625_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_3626_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_3627_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_3628_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_3629_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_3630_zero__le__power__eq__numeral,axiom,
! [A3: real,W2: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3631_zero__le__power__eq__numeral,axiom,
! [A3: rat,W2: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3632_zero__le__power__eq__numeral,axiom,
! [A3: int,W2: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3633_power__less__zero__eq,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A3 @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A3 @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_3634_power__less__zero__eq,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ ( power_power_rat @ A3 @ N ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_3635_power__less__zero__eq,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A3 @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_3636_power__less__zero__eq__numeral,axiom,
! [A3: real,W2: num] :
( ( ord_less_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ A3 @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3637_power__less__zero__eq__numeral,axiom,
! [A3: rat,W2: num] :
( ( ord_less_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3638_power__less__zero__eq__numeral,axiom,
! [A3: int,W2: num] :
( ( ord_less_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3639_even__plus__one__iff,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ).
% even_plus_one_iff
thf(fact_3640_even__plus__one__iff,axiom,
! [A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A3 @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ).
% even_plus_one_iff
thf(fact_3641_even__plus__one__iff,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ).
% even_plus_one_iff
thf(fact_3642_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_3643_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_3644_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_3645_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_3646_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_3647_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_3648_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_3649_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_3650_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_3651_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_3652_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_3653_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_3654_zero__less__power__eq__numeral,axiom,
! [A3: real,W2: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A3 != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3655_zero__less__power__eq__numeral,axiom,
! [A3: rat,W2: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A3 != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3656_zero__less__power__eq__numeral,axiom,
! [A3: int,W2: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A3 != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ zero_zero_int @ A3 ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3657_odd__succ__div__two,axiom,
! [A3: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_3658_odd__succ__div__two,axiom,
! [A3: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_3659_odd__succ__div__two,axiom,
! [A3: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% odd_succ_div_two
thf(fact_3660_even__succ__div__two,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3661_even__succ__div__two,axiom,
! [A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3662_even__succ__div__two,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3663_even__succ__div__2,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3664_even__succ__div__2,axiom,
! [A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3665_even__succ__div__2,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A3 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3666_even__power,axiom,
! [A3: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A3 @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3667_even__power,axiom,
! [A3: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A3 @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3668_even__power,axiom,
! [A3: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A3 @ N ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3669_power__le__zero__eq__numeral,axiom,
! [A3: real,W2: num] :
( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ A3 @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A3 = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3670_power__le__zero__eq__numeral,axiom,
! [A3: rat,W2: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ A3 @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A3 = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3671_power__le__zero__eq__numeral,axiom,
! [A3: int,W2: num] :
( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ A3 @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A3 = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3672_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_3673_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_3674_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_3675_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_3676_dvd__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ B3 @ C )
=> ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_trans
thf(fact_3677_dvd__trans,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ B3 @ C )
=> ( dvd_dvd_int @ A3 @ C ) ) ) ).
% dvd_trans
thf(fact_3678_dvd__trans,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ B3 @ C )
=> ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% dvd_trans
thf(fact_3679_dvd__refl,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% dvd_refl
thf(fact_3680_dvd__refl,axiom,
! [A3: int] : ( dvd_dvd_int @ A3 @ A3 ) ).
% dvd_refl
thf(fact_3681_dvd__refl,axiom,
! [A3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ A3 ) ).
% dvd_refl
thf(fact_3682_dvd__0__left,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A3 )
=> ( A3 = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left
thf(fact_3683_dvd__0__left,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
=> ( A3 = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_3684_dvd__0__left,axiom,
! [A3: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A3 )
=> ( A3 = zero_zero_rat ) ) ).
% dvd_0_left
thf(fact_3685_dvd__0__left,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_3686_dvd__0__left,axiom,
! [A3: int] :
( ( dvd_dvd_int @ zero_zero_int @ A3 )
=> ( A3 = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_3687_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A: real,B: real] :
( ( A = zero_zero_real )
=> ( B = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_3688_dvd__field__iff,axiom,
( dvd_dvd_rat
= ( ^ [A: rat,B: rat] :
( ( A = zero_zero_rat )
=> ( B = zero_zero_rat ) ) ) ) ).
% dvd_field_iff
thf(fact_3689_dvd__unit__imp__unit,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer ) ) ) ).
% dvd_unit_imp_unit
thf(fact_3690_dvd__unit__imp__unit,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_3691_dvd__unit__imp__unit,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( dvd_dvd_int @ A3 @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_3692_unit__imp__dvd,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).
% unit_imp_dvd
thf(fact_3693_unit__imp__dvd,axiom,
! [B3: nat,A3: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( dvd_dvd_nat @ B3 @ A3 ) ) ).
% unit_imp_dvd
thf(fact_3694_unit__imp__dvd,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( dvd_dvd_int @ B3 @ A3 ) ) ).
% unit_imp_dvd
thf(fact_3695_one__dvd,axiom,
! [A3: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A3 ) ).
% one_dvd
thf(fact_3696_one__dvd,axiom,
! [A3: complex] : ( dvd_dvd_complex @ one_one_complex @ A3 ) ).
% one_dvd
thf(fact_3697_one__dvd,axiom,
! [A3: real] : ( dvd_dvd_real @ one_one_real @ A3 ) ).
% one_dvd
thf(fact_3698_one__dvd,axiom,
! [A3: rat] : ( dvd_dvd_rat @ one_one_rat @ A3 ) ).
% one_dvd
thf(fact_3699_one__dvd,axiom,
! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).
% one_dvd
thf(fact_3700_one__dvd,axiom,
! [A3: int] : ( dvd_dvd_int @ one_one_int @ A3 ) ).
% one_dvd
thf(fact_3701_dvd__add__right__iff,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ C ) )
= ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_3702_dvd__add__right__iff,axiom,
! [A3: real,B3: real,C: real] :
( ( dvd_dvd_real @ A3 @ B3 )
=> ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
= ( dvd_dvd_real @ A3 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_3703_dvd__add__right__iff,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( dvd_dvd_rat @ A3 @ B3 )
=> ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
= ( dvd_dvd_rat @ A3 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_3704_dvd__add__right__iff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_3705_dvd__add__right__iff,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
= ( dvd_dvd_int @ A3 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_3706_dvd__add__left__iff,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ C )
=> ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ C ) )
= ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ).
% dvd_add_left_iff
thf(fact_3707_dvd__add__left__iff,axiom,
! [A3: real,C: real,B3: real] :
( ( dvd_dvd_real @ A3 @ C )
=> ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
= ( dvd_dvd_real @ A3 @ B3 ) ) ) ).
% dvd_add_left_iff
thf(fact_3708_dvd__add__left__iff,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( dvd_dvd_rat @ A3 @ C )
=> ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
= ( dvd_dvd_rat @ A3 @ B3 ) ) ) ).
% dvd_add_left_iff
thf(fact_3709_dvd__add__left__iff,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ C )
=> ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ) ).
% dvd_add_left_iff
thf(fact_3710_dvd__add__left__iff,axiom,
! [A3: int,C: int,B3: int] :
( ( dvd_dvd_int @ A3 @ C )
=> ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ) ).
% dvd_add_left_iff
thf(fact_3711_dvd__add,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ A3 @ C )
=> ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ C ) ) ) ) ).
% dvd_add
thf(fact_3712_dvd__add,axiom,
! [A3: real,B3: real,C: real] :
( ( dvd_dvd_real @ A3 @ B3 )
=> ( ( dvd_dvd_real @ A3 @ C )
=> ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ) ).
% dvd_add
thf(fact_3713_dvd__add,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( dvd_dvd_rat @ A3 @ B3 )
=> ( ( dvd_dvd_rat @ A3 @ C )
=> ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ) ).
% dvd_add
thf(fact_3714_dvd__add,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ A3 @ C )
=> ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ) ).
% dvd_add
thf(fact_3715_dvd__add,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ A3 @ C )
=> ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ) ).
% dvd_add
thf(fact_3716_dvd__diff__commute,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ ( minus_8373710615458151222nteger @ C @ B3 ) )
= ( dvd_dvd_Code_integer @ A3 @ ( minus_8373710615458151222nteger @ B3 @ C ) ) ) ).
% dvd_diff_commute
thf(fact_3717_dvd__diff__commute,axiom,
! [A3: int,C: int,B3: int] :
( ( dvd_dvd_int @ A3 @ ( minus_minus_int @ C @ B3 ) )
= ( dvd_dvd_int @ A3 @ ( minus_minus_int @ B3 @ C ) ) ) ).
% dvd_diff_commute
thf(fact_3718_dvd__diff,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( dvd_dvd_Code_integer @ X @ Y )
=> ( ( dvd_dvd_Code_integer @ X @ Z )
=> ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3719_dvd__diff,axiom,
! [X: real,Y: real,Z: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3720_dvd__diff,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( dvd_dvd_rat @ X @ Y )
=> ( ( dvd_dvd_rat @ X @ Z )
=> ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3721_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3722_div__div__div__same,axiom,
! [D: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ D @ B3 )
=> ( ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ D ) @ ( divide_divide_nat @ B3 @ D ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_div_div_same
thf(fact_3723_div__div__div__same,axiom,
! [D: int,B3: int,A3: int] :
( ( dvd_dvd_int @ D @ B3 )
=> ( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( divide_divide_int @ ( divide_divide_int @ A3 @ D ) @ ( divide_divide_int @ B3 @ D ) )
= ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_div_div_same
thf(fact_3724_div__div__div__same,axiom,
! [D: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ D @ B3 )
=> ( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ D ) @ ( divide6298287555418463151nteger @ B3 @ D ) )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_div_div_same
thf(fact_3725_dvd__div__eq__cancel,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ( divide_divide_nat @ A3 @ C )
= ( divide_divide_nat @ B3 @ C ) )
=> ( ( dvd_dvd_nat @ C @ A3 )
=> ( ( dvd_dvd_nat @ C @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3726_dvd__div__eq__cancel,axiom,
! [A3: int,C: int,B3: int] :
( ( ( divide_divide_int @ A3 @ C )
= ( divide_divide_int @ B3 @ C ) )
=> ( ( dvd_dvd_int @ C @ A3 )
=> ( ( dvd_dvd_int @ C @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3727_dvd__div__eq__cancel,axiom,
! [A3: real,C: real,B3: real] :
( ( ( divide_divide_real @ A3 @ C )
= ( divide_divide_real @ B3 @ C ) )
=> ( ( dvd_dvd_real @ C @ A3 )
=> ( ( dvd_dvd_real @ C @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3728_dvd__div__eq__cancel,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( ( divide6298287555418463151nteger @ A3 @ C )
= ( divide6298287555418463151nteger @ B3 @ C ) )
=> ( ( dvd_dvd_Code_integer @ C @ A3 )
=> ( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3729_dvd__div__eq__iff,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( dvd_dvd_nat @ C @ A3 )
=> ( ( dvd_dvd_nat @ C @ B3 )
=> ( ( ( divide_divide_nat @ A3 @ C )
= ( divide_divide_nat @ B3 @ C ) )
= ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3730_dvd__div__eq__iff,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ C @ A3 )
=> ( ( dvd_dvd_int @ C @ B3 )
=> ( ( ( divide_divide_int @ A3 @ C )
= ( divide_divide_int @ B3 @ C ) )
= ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3731_dvd__div__eq__iff,axiom,
! [C: real,A3: real,B3: real] :
( ( dvd_dvd_real @ C @ A3 )
=> ( ( dvd_dvd_real @ C @ B3 )
=> ( ( ( divide_divide_real @ A3 @ C )
= ( divide_divide_real @ B3 @ C ) )
= ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3732_dvd__div__eq__iff,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A3 )
=> ( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( ( divide6298287555418463151nteger @ A3 @ C )
= ( divide6298287555418463151nteger @ B3 @ C ) )
= ( A3 = B3 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3733_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_3734_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
& ( zero_zero_nat != A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_3735_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_3736_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
& ( A3 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_3737_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_3738_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_3739_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% not_is_unit_0
thf(fact_3740_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_3741_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_3742_pinf_I9_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ Z3 @ X5 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3743_pinf_I9_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3744_pinf_I9_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3745_pinf_I9_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3746_pinf_I9_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3747_pinf_I10_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3748_pinf_I10_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3749_pinf_I10_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3750_pinf_I10_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3751_pinf_I10_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3752_minf_I9_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ X5 @ Z3 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3753_minf_I9_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3754_minf_I9_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3755_minf_I9_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3756_minf_I9_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3757_minf_I10_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3758_minf_I10_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3759_minf_I10_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3760_minf_I10_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3761_minf_I10_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3762_dvd__div__eq__0__iff,axiom,
! [B3: rat,A3: rat] :
( ( dvd_dvd_rat @ B3 @ A3 )
=> ( ( ( divide_divide_rat @ A3 @ B3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3763_dvd__div__eq__0__iff,axiom,
! [B3: nat,A3: nat] :
( ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( ( divide_divide_nat @ A3 @ B3 )
= zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3764_dvd__div__eq__0__iff,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( ( divide_divide_int @ A3 @ B3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3765_dvd__div__eq__0__iff,axiom,
! [B3: real,A3: real] :
( ( dvd_dvd_real @ B3 @ A3 )
=> ( ( ( divide_divide_real @ A3 @ B3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3766_dvd__div__eq__0__iff,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( ( divide6298287555418463151nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger )
= ( A3 = zero_z3403309356797280102nteger ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3767_dvd__div__unit__iff,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ C @ B3 ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_3768_dvd__div__unit__iff,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( dvd_dvd_int @ A3 @ ( divide_divide_int @ C @ B3 ) )
= ( dvd_dvd_int @ A3 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_3769_dvd__div__unit__iff,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A3 @ ( divide6298287555418463151nteger @ C @ B3 ) )
= ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_3770_div__unit__dvd__iff,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_3771_div__unit__dvd__iff,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A3 @ B3 ) @ C )
= ( dvd_dvd_int @ A3 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_3772_div__unit__dvd__iff,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C )
= ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_3773_unit__div__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ B3 @ A3 )
= ( divide_divide_nat @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% unit_div_cancel
thf(fact_3774_unit__div__cancel,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( ( divide_divide_int @ B3 @ A3 )
= ( divide_divide_int @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% unit_div_cancel
thf(fact_3775_unit__div__cancel,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ B3 @ A3 )
= ( divide6298287555418463151nteger @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% unit_div_cancel
thf(fact_3776_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B3 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3777_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ B3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3778_div__plus__div__distrib__dvd__right,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3779_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( dvd_dvd_nat @ C @ A3 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3780_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ C @ A3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3781_div__plus__div__distrib__dvd__left,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A3 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3782_dvd__neg__div,axiom,
! [B3: real,A3: real] :
( ( dvd_dvd_real @ B3 @ A3 )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) ) ) ).
% dvd_neg_div
thf(fact_3783_dvd__neg__div,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% dvd_neg_div
thf(fact_3784_dvd__neg__div,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% dvd_neg_div
thf(fact_3785_dvd__neg__div,axiom,
! [B3: rat,A3: rat] :
( ( dvd_dvd_rat @ B3 @ A3 )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) ) ) ) ).
% dvd_neg_div
thf(fact_3786_dvd__div__neg,axiom,
! [B3: real,A3: real] :
( ( dvd_dvd_real @ B3 @ A3 )
=> ( ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) ) ) ).
% dvd_div_neg
thf(fact_3787_dvd__div__neg,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% dvd_div_neg
thf(fact_3788_dvd__div__neg,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( divide6298287555418463151nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% dvd_div_neg
thf(fact_3789_dvd__div__neg,axiom,
! [B3: rat,A3: rat] :
( ( dvd_dvd_rat @ B3 @ A3 )
=> ( ( divide_divide_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) ) ) ) ).
% dvd_div_neg
thf(fact_3790_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A3: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3791_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3792_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A3: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3793_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A3: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3794_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A3: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A3 @ M ) @ ( power_power_complex @ A3 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3795_power__le__dvd,axiom,
! [A3: code_integer,N: nat,B3: code_integer,M: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ M ) @ B3 ) ) ) ).
% power_le_dvd
thf(fact_3796_power__le__dvd,axiom,
! [A3: nat,N: nat,B3: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A3 @ M ) @ B3 ) ) ) ).
% power_le_dvd
thf(fact_3797_power__le__dvd,axiom,
! [A3: real,N: nat,B3: real,M: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A3 @ M ) @ B3 ) ) ) ).
% power_le_dvd
thf(fact_3798_power__le__dvd,axiom,
! [A3: int,N: nat,B3: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A3 @ M ) @ B3 ) ) ) ).
% power_le_dvd
thf(fact_3799_power__le__dvd,axiom,
! [A3: complex,N: nat,B3: complex,M: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A3 @ M ) @ B3 ) ) ) ).
% power_le_dvd
thf(fact_3800_dvd__power__le,axiom,
! [X: code_integer,Y: code_integer,N: nat,M: nat] :
( ( dvd_dvd_Code_integer @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_3801_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_3802_dvd__power__le,axiom,
! [X: real,Y: real,N: nat,M: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_3803_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_3804_dvd__power__le,axiom,
! [X: complex,Y: complex,N: nat,M: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_3805_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_3806_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_3807_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_3808_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_3809_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_3810_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_3811_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_3812_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_3813_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_3814_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_3815_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_3816_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_3817_div__neg__pos__less0,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_3818_neg__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_3819_pos__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_3820_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( ( euclid6377331345833325938nteger @ A3 )
= ( euclid6377331345833325938nteger @ B3 ) )
=> ( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_3821_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( ( euclid4774559944035922753ze_int @ A3 )
= ( euclid4774559944035922753ze_int @ B3 ) )
=> ( ( dvd_dvd_int @ B3 @ A3 )
=> ( dvd_dvd_int @ A3 @ B3 ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_3822_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( ( euclid4777050414544973029ze_nat @ A3 )
= ( euclid4777050414544973029ze_nat @ B3 ) )
=> ( ( dvd_dvd_nat @ B3 @ A3 )
=> ( dvd_dvd_nat @ A3 @ B3 ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_3823_euclidean__size__unit,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( euclid6377331345833325938nteger @ A3 )
= ( euclid6377331345833325938nteger @ one_one_Code_integer ) ) ) ).
% euclidean_size_unit
thf(fact_3824_euclidean__size__unit,axiom,
! [A3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( euclid4774559944035922753ze_int @ A3 )
= ( euclid4774559944035922753ze_int @ one_one_int ) ) ) ).
% euclidean_size_unit
thf(fact_3825_euclidean__size__unit,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( euclid4777050414544973029ze_nat @ A3 )
= ( euclid4777050414544973029ze_nat @ one_one_nat ) ) ) ).
% euclidean_size_unit
thf(fact_3826_unit__div__eq__0__iff,axiom,
! [B3: nat,A3: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A3 @ B3 )
= zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3827_unit__div__eq__0__iff,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( ( divide_divide_int @ A3 @ B3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3828_unit__div__eq__0__iff,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger )
= ( A3 = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3829_is__unit__power__iff,axiom,
! [A3: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3830_is__unit__power__iff,axiom,
! [A3: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3831_is__unit__power__iff,axiom,
! [A3: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A3 @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A3 @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3832_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_3833_unit__iff__euclidean__size,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
= ( ( ( euclid6377331345833325938nteger @ A3 )
= ( euclid6377331345833325938nteger @ one_one_Code_integer ) )
& ( A3 != zero_z3403309356797280102nteger ) ) ) ).
% unit_iff_euclidean_size
thf(fact_3834_unit__iff__euclidean__size,axiom,
! [A3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
= ( ( ( euclid4774559944035922753ze_int @ A3 )
= ( euclid4774559944035922753ze_int @ one_one_int ) )
& ( A3 != zero_zero_int ) ) ) ).
% unit_iff_euclidean_size
thf(fact_3835_unit__iff__euclidean__size,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
= ( ( ( euclid4777050414544973029ze_nat @ A3 )
= ( euclid4777050414544973029ze_nat @ one_one_nat ) )
& ( A3 != zero_zero_nat ) ) ) ).
% unit_iff_euclidean_size
thf(fact_3836_dvd__proper__imp__size__less,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ~ ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( B3 != zero_z3403309356797280102nteger )
=> ( ord_less_nat @ ( euclid6377331345833325938nteger @ A3 ) @ ( euclid6377331345833325938nteger @ B3 ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_3837_dvd__proper__imp__size__less,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ~ ( dvd_dvd_int @ B3 @ A3 )
=> ( ( B3 != zero_zero_int )
=> ( ord_less_nat @ ( euclid4774559944035922753ze_int @ A3 ) @ ( euclid4774559944035922753ze_int @ B3 ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_3838_dvd__proper__imp__size__less,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ~ ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( B3 != zero_zero_nat )
=> ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ A3 ) @ ( euclid4777050414544973029ze_nat @ B3 ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_3839_dvd__imp__size__le,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( B3 != zero_z3403309356797280102nteger )
=> ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A3 ) @ ( euclid6377331345833325938nteger @ B3 ) ) ) ) ).
% dvd_imp_size_le
thf(fact_3840_dvd__imp__size__le,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( B3 != zero_zero_int )
=> ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A3 ) @ ( euclid4774559944035922753ze_int @ B3 ) ) ) ) ).
% dvd_imp_size_le
thf(fact_3841_dvd__imp__size__le,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( B3 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A3 ) @ ( euclid4777050414544973029ze_nat @ B3 ) ) ) ) ).
% dvd_imp_size_le
thf(fact_3842_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_3843_zdiv__mono1,axiom,
! [A3: int,A7: int,B3: int] :
( ( ord_less_eq_int @ A3 @ A7 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ A7 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_3844_zdiv__mono2,axiom,
! [A3: int,B7: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ A3 @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_3845_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_3846_zdiv__mono1__neg,axiom,
! [A3: int,A7: int,B3: int] :
( ( ord_less_eq_int @ A3 @ A7 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B3 ) @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_3847_zdiv__mono2__neg,axiom,
! [A3: int,B7: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B7 ) @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_3848_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_3849_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ L @ K )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_3850_div__nonneg__neg__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_3851_div__nonpos__pos__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_3852_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_3853_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_3854_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_3855_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_3856_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_3857_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_3858_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_3859_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_3860_even__zero,axiom,
dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% even_zero
thf(fact_3861_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_3862_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_3863_odd__one,axiom,
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% odd_one
thf(fact_3864_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_3865_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_3866_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_3867_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_3868_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_3869_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_3870_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_3871_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_3872_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_3873_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_3874_verit__less__mono__div__int2,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_3875_div__eq__minus1,axiom,
! [B3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_3876_aset_I7_J,axiom,
! [D6: int,A4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D6 ) ) ) ) ) ).
% aset(7)
thf(fact_3877_aset_I5_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_3878_aset_I4_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D6 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_3879_aset_I3_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D6 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_3880_bset_I7_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D6 ) ) ) ) ) ) ).
% bset(7)
thf(fact_3881_bset_I5_J,axiom,
! [D6: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D6 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_3882_bset_I4_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D6 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_3883_bset_I3_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D6 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_3884_power__mono__odd,axiom,
! [N: nat,A3: real,B3: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3885_power__mono__odd,axiom,
! [N: nat,A3: rat,B3: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3886_power__mono__odd,axiom,
! [N: nat,A3: int,B3: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3887_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_3888_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_3889_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_3890_aset_I8_J,axiom,
! [D6: int,A4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D6 ) ) ) ) ) ).
% aset(8)
thf(fact_3891_aset_I6_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_3892_bset_I8_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D6 ) ) ) ) ) ) ).
% bset(8)
thf(fact_3893_bset_I6_J,axiom,
! [D6: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D6 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_3894_zero__le__power__eq,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ 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 @ A3 ) ) ) ) ).
% zero_le_power_eq
thf(fact_3895_zero__le__power__eq,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ 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 @ A3 ) ) ) ) ).
% zero_le_power_eq
thf(fact_3896_zero__le__power__eq,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ 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 @ A3 ) ) ) ) ).
% zero_le_power_eq
thf(fact_3897_zero__le__odd__power,axiom,
! [N: nat,A3: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ).
% zero_le_odd_power
thf(fact_3898_zero__le__odd__power,axiom,
! [N: nat,A3: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ).
% zero_le_odd_power
thf(fact_3899_zero__le__odd__power,axiom,
! [N: nat,A3: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).
% zero_le_odd_power
thf(fact_3900_zero__le__even__power,axiom,
! [N: nat,A3: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N ) ) ) ).
% zero_le_even_power
thf(fact_3901_zero__le__even__power,axiom,
! [N: nat,A3: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N ) ) ) ).
% zero_le_even_power
thf(fact_3902_zero__le__even__power,axiom,
! [N: nat,A3: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) ) ) ).
% zero_le_even_power
thf(fact_3903_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_3904_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_3905_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_3906_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_3907_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_3908_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_3909_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_3910_even__set__encode__iff,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3911_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_3912_zero__less__power__eq,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A3 != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ).
% zero_less_power_eq
thf(fact_3913_zero__less__power__eq,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A3 != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ).
% zero_less_power_eq
thf(fact_3914_zero__less__power__eq,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A3 != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A3 ) ) ) ) ).
% zero_less_power_eq
thf(fact_3915_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_3916_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_3917_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_3918_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_3919_power__le__zero__eq,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A3 @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A3 = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3920_power__le__zero__eq,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ A3 @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A3 = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3921_power__le__zero__eq,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A3 @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A3 = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3922_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_3923_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_3924_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_3925_summaxma,axiom,
! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi2 != Ma2 )
=> ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% summaxma
thf(fact_3926_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Ma2: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_3927_member__inv,axiom,
! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X = Mi2 )
| ( X = Ma2 )
| ( ( ord_less_nat @ X @ Ma2 )
& ( ord_less_nat @ Mi2 @ X )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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_3928_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi2: nat,Ma2: nat,TreeList: 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 @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 = Mi2 )
| ( X = Ma2 ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_3929_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_3930_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_3931_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A2: $o,B2: $o] :
( A1
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( A22
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: nat,Ma: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: nat,Ma: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_3932_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A: $o,B: $o] :
( A12
= ( vEBT_Leaf @ A @ B ) )
& ( A23
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
& ( A23
= ( plus_plus_nat @ N3 @ N3 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
& ( A23
= ( plus_plus_nat @ N3 @ N3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( 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 ) ) @ N3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( 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 @ N3 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_3933_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_3934_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_3935_mult__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_3936_mult__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ( times_times_rat @ A3 @ C )
= ( times_times_rat @ B3 @ C ) )
= ( ( C = zero_zero_rat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_3937_mult__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_3938_mult__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B3 @ C ) )
= ( ( C = zero_zero_int )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_3939_mult__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_3940_mult__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ( times_times_rat @ C @ A3 )
= ( times_times_rat @ C @ B3 ) )
= ( ( C = zero_zero_rat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_3941_mult__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B3 ) )
= ( ( C = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_3942_mult__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B3 ) )
= ( ( C = zero_zero_int )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_3943_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_3944_mult__eq__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( times_times_rat @ A3 @ B3 )
= zero_zero_rat )
= ( ( A3 = zero_zero_rat )
| ( B3 = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_3945_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_3946_mult__eq__0__iff,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_3947_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_3948_mult__zero__right,axiom,
! [A3: rat] :
( ( times_times_rat @ A3 @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_3949_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_3950_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_3951_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_3952_mult__zero__left,axiom,
! [A3: rat] :
( ( times_times_rat @ zero_zero_rat @ A3 )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_3953_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_3954_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_3955_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_3956_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_3957_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_3958_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_3959_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_3960_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_3961_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_3962_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_3963_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_3964_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_3965_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_3966_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_3967_mult_Oright__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.right_neutral
thf(fact_3968_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_3969_mult_Oright__neutral,axiom,
! [A3: rat] :
( ( times_times_rat @ A3 @ one_one_rat )
= A3 ) ).
% mult.right_neutral
thf(fact_3970_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_3971_mult_Oright__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.right_neutral
thf(fact_3972_mult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% mult_1
thf(fact_3973_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_3974_mult__1,axiom,
! [A3: rat] :
( ( times_times_rat @ one_one_rat @ A3 )
= A3 ) ).
% mult_1
thf(fact_3975_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_3976_mult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% mult_1
thf(fact_3977_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_3978_mult__minus__left,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_3979_mult__minus__left,axiom,
! [A3: int,B3: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( uminus_uminus_int @ ( times_times_int @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_3980_mult__minus__left,axiom,
! [A3: code_integer,B3: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_3981_mult__minus__left,axiom,
! [A3: rat,B3: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
= ( uminus_uminus_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_3982_minus__mult__minus,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( times_times_real @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_3983_minus__mult__minus,axiom,
! [A3: int,B3: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) )
= ( times_times_int @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_3984_minus__mult__minus,axiom,
! [A3: code_integer,B3: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( uminus1351360451143612070nteger @ B3 ) )
= ( times_3573771949741848930nteger @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_3985_minus__mult__minus,axiom,
! [A3: rat,B3: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A3 ) @ ( uminus_uminus_rat @ B3 ) )
= ( times_times_rat @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_3986_mult__minus__right,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_3987_mult__minus__right,axiom,
! [A3: int,B3: int] :
( ( times_times_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( uminus_uminus_int @ ( times_times_int @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_3988_mult__minus__right,axiom,
! [A3: code_integer,B3: code_integer] :
( ( times_3573771949741848930nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_3989_mult__minus__right,axiom,
! [A3: rat,B3: rat] :
( ( times_times_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
= ( uminus_uminus_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_3990_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_3991_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_3992_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_3993_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_3994_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_3995_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_3996_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H3: nat,L3: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L3 ) ) ) ).
% bit_concat_def
thf(fact_3997_mult__cancel__left1,axiom,
! [C: complex,B3: complex] :
( ( C
= ( times_times_complex @ C @ B3 ) )
= ( ( C = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_3998_mult__cancel__left1,axiom,
! [C: real,B3: real] :
( ( C
= ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_3999_mult__cancel__left1,axiom,
! [C: rat,B3: rat] :
( ( C
= ( times_times_rat @ C @ B3 ) )
= ( ( C = zero_zero_rat )
| ( B3 = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_4000_mult__cancel__left1,axiom,
! [C: int,B3: int] :
( ( C
= ( times_times_int @ C @ B3 ) )
= ( ( C = zero_zero_int )
| ( B3 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_4001_mult__cancel__left2,axiom,
! [C: complex,A3: complex] :
( ( ( times_times_complex @ C @ A3 )
= C )
= ( ( C = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_4002_mult__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ( times_times_real @ C @ A3 )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_4003_mult__cancel__left2,axiom,
! [C: rat,A3: rat] :
( ( ( times_times_rat @ C @ A3 )
= C )
= ( ( C = zero_zero_rat )
| ( A3 = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_4004_mult__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ( times_times_int @ C @ A3 )
= C )
= ( ( C = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_4005_mult__cancel__right1,axiom,
! [C: complex,B3: complex] :
( ( C
= ( times_times_complex @ B3 @ C ) )
= ( ( C = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_4006_mult__cancel__right1,axiom,
! [C: real,B3: real] :
( ( C
= ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_4007_mult__cancel__right1,axiom,
! [C: rat,B3: rat] :
( ( C
= ( times_times_rat @ B3 @ C ) )
= ( ( C = zero_zero_rat )
| ( B3 = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_4008_mult__cancel__right1,axiom,
! [C: int,B3: int] :
( ( C
= ( times_times_int @ B3 @ C ) )
= ( ( C = zero_zero_int )
| ( B3 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_4009_mult__cancel__right2,axiom,
! [A3: complex,C: complex] :
( ( ( times_times_complex @ A3 @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_4010_mult__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ( times_times_real @ A3 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_4011_mult__cancel__right2,axiom,
! [A3: rat,C: rat] :
( ( ( times_times_rat @ A3 @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A3 = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_4012_mult__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ( times_times_int @ A3 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_4013_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_4014_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_4015_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_4016_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ C @ B3 ) )
= ( divide_divide_rat @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_4017_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ C @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_4018_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
= ( divide_divide_rat @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_4019_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_4020_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ B3 @ C ) )
= ( divide_divide_rat @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_4021_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ B3 @ C ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_4022_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( divide_divide_rat @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_4023_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_4024_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( divide_divide_rat @ A3 @ B3 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_4025_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A3: real,B3: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( divide_divide_real @ A3 @ B3 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_4026_div__mult__mult1,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ).
% div_mult_mult1
thf(fact_4027_div__mult__mult1,axiom,
! [C: int,A3: int,B3: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( divide_divide_int @ A3 @ B3 ) ) ) ).
% div_mult_mult1
thf(fact_4028_div__mult__mult1,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ).
% div_mult_mult1
thf(fact_4029_div__mult__mult2,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ).
% div_mult_mult2
thf(fact_4030_div__mult__mult2,axiom,
! [C: int,A3: int,B3: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ A3 @ B3 ) ) ) ).
% div_mult_mult2
thf(fact_4031_div__mult__mult2,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ).
% div_mult_mult2
thf(fact_4032_div__mult__mult1__if,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_4033_div__mult__mult1__if,axiom,
! [C: int,A3: int,B3: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_4034_div__mult__mult1__if,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( ( C = zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) )
= zero_z3403309356797280102nteger ) )
& ( ( C != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_4035_nonzero__mult__div__cancel__left,axiom,
! [A3: rat,B3: rat] :
( ( A3 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_4036_nonzero__mult__div__cancel__left,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_4037_nonzero__mult__div__cancel__left,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_4038_nonzero__mult__div__cancel__left,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_4039_nonzero__mult__div__cancel__left,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_4040_nonzero__mult__div__cancel__right,axiom,
! [B3: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_4041_nonzero__mult__div__cancel__right,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_4042_nonzero__mult__div__cancel__right,axiom,
! [B3: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_4043_nonzero__mult__div__cancel__right,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_4044_nonzero__mult__div__cancel__right,axiom,
! [B3: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_4045_distrib__left__numeral,axiom,
! [V: num,B3: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B3 @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B3 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4046_distrib__left__numeral,axiom,
! [V: num,B3: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B3 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4047_distrib__left__numeral,axiom,
! [V: num,B3: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B3 @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B3 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4048_distrib__left__numeral,axiom,
! [V: num,B3: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B3 @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4049_distrib__left__numeral,axiom,
! [V: num,B3: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B3 @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B3 ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4050_distrib__left__numeral,axiom,
! [V: num,B3: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B3 @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4051_distrib__right__numeral,axiom,
! [A3: rat,B3: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B3 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4052_distrib__right__numeral,axiom,
! [A3: nat,B3: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B3 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4053_distrib__right__numeral,axiom,
! [A3: real,B3: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B3 @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4054_distrib__right__numeral,axiom,
! [A3: int,B3: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B3 @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4055_distrib__right__numeral,axiom,
! [A3: extended_enat,B3: extended_enat,V: num] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A3 @ B3 ) @ ( numera1916890842035813515d_enat @ V ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A3 @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B3 @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4056_distrib__right__numeral,axiom,
! [A3: code_integer,B3: code_integer,V: num] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ ( numera6620942414471956472nteger @ V ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A3 @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B3 @ ( numera6620942414471956472nteger @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4057_left__diff__distrib__numeral,axiom,
! [A3: rat,B3: rat,V: num] :
( ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ ( numeral_numeral_rat @ V ) )
= ( minus_minus_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B3 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_4058_left__diff__distrib__numeral,axiom,
! [A3: real,B3: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B3 @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_4059_left__diff__distrib__numeral,axiom,
! [A3: int,B3: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B3 @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_4060_left__diff__distrib__numeral,axiom,
! [A3: code_integer,B3: code_integer,V: num] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ ( numera6620942414471956472nteger @ V ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A3 @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B3 @ ( numera6620942414471956472nteger @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_4061_right__diff__distrib__numeral,axiom,
! [V: num,B3: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B3 @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B3 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_4062_right__diff__distrib__numeral,axiom,
! [V: num,B3: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B3 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_4063_right__diff__distrib__numeral,axiom,
! [V: num,B3: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_4064_right__diff__distrib__numeral,axiom,
! [V: num,B3: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( minus_8373710615458151222nteger @ B3 @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_4065_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_4066_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_4067_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_4068_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_4069_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_4070_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_4071_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_4072_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_4073_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_4074_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_4075_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_4076_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_4077_dvd__mult__cancel__left,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4078_dvd__mult__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4079_dvd__mult__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4080_dvd__mult__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4081_dvd__mult__cancel__right,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4082_dvd__mult__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4083_dvd__mult__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4084_dvd__mult__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A3 @ B3 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4085_dvd__times__left__cancel__iff,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ ( times_3573771949741848930nteger @ A3 @ C ) )
= ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4086_dvd__times__left__cancel__iff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) )
= ( dvd_dvd_nat @ B3 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4087_dvd__times__left__cancel__iff,axiom,
! [A3: int,B3: int,C: int] :
( ( A3 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) )
= ( dvd_dvd_int @ B3 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4088_dvd__times__right__cancel__iff,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B3 @ A3 ) @ ( times_3573771949741848930nteger @ C @ A3 ) )
= ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4089_dvd__times__right__cancel__iff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B3 @ A3 ) @ ( times_times_nat @ C @ A3 ) )
= ( dvd_dvd_nat @ B3 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4090_dvd__times__right__cancel__iff,axiom,
! [A3: int,B3: int,C: int] :
( ( A3 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B3 @ A3 ) @ ( times_times_int @ C @ A3 ) )
= ( dvd_dvd_int @ B3 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4091_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_4092_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_4093_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_4094_mult__minus1,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1
thf(fact_4095_mult__minus1,axiom,
! [Z: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1
thf(fact_4096_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_4097_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_4098_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_4099_mult__minus1__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1_right
thf(fact_4100_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_4101_unit__prod,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ one_one_Code_integer ) ) ) ).
% unit_prod
thf(fact_4102_unit__prod,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_4103_unit__prod,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_4104_dvd__add__times__triv__right__iff,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ ( times_3573771949741848930nteger @ C @ A3 ) ) )
= ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4105_dvd__add__times__triv__right__iff,axiom,
! [A3: real,B3: real,C: real] :
( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ ( times_times_real @ C @ A3 ) ) )
= ( dvd_dvd_real @ A3 @ B3 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4106_dvd__add__times__triv__right__iff,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ ( times_times_rat @ C @ A3 ) ) )
= ( dvd_dvd_rat @ A3 @ B3 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4107_dvd__add__times__triv__right__iff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ ( times_times_nat @ C @ A3 ) ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4108_dvd__add__times__triv__right__iff,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ ( times_times_int @ C @ A3 ) ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4109_dvd__add__times__triv__left__iff,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ B3 ) )
= ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4110_dvd__add__times__triv__left__iff,axiom,
! [A3: real,C: real,B3: real] :
( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ ( times_times_real @ C @ A3 ) @ B3 ) )
= ( dvd_dvd_real @ A3 @ B3 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4111_dvd__add__times__triv__left__iff,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ ( times_times_rat @ C @ A3 ) @ B3 ) )
= ( dvd_dvd_rat @ A3 @ B3 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4112_dvd__add__times__triv__left__iff,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ ( times_times_nat @ C @ A3 ) @ B3 ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4113_dvd__add__times__triv__left__iff,axiom,
! [A3: int,C: int,B3: int] :
( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ ( times_times_int @ C @ A3 ) @ B3 ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4114_dvd__mult__div__cancel,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ A3 ) )
= B3 ) ) ).
% dvd_mult_div_cancel
thf(fact_4115_dvd__mult__div__cancel,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ A3 ) )
= B3 ) ) ).
% dvd_mult_div_cancel
thf(fact_4116_dvd__mult__div__cancel,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ A3 ) )
= B3 ) ) ).
% dvd_mult_div_cancel
thf(fact_4117_dvd__div__mult__self,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% dvd_div_mult_self
thf(fact_4118_dvd__div__mult__self,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( times_times_int @ ( divide_divide_int @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% dvd_div_mult_self
thf(fact_4119_dvd__div__mult__self,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% dvd_div_mult_self
thf(fact_4120_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_4121_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_4122_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_4123_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_4124_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_4125_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_4126_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_4127_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_4128_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_4129_option_Ocollapse,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_4130_option_Ocollapse,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( ( some_num @ ( the_num @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_4131_le__divide__eq__numeral1_I1_J,axiom,
! [A3: real,B3: real,W2: num] :
( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_4132_le__divide__eq__numeral1_I1_J,axiom,
! [A3: rat,B3: rat,W2: num] :
( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_4133_divide__le__eq__numeral1_I1_J,axiom,
! [B3: real,W2: num,A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A3 )
= ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_4134_divide__le__eq__numeral1_I1_J,axiom,
! [B3: rat,W2: num,A3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A3 )
= ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_4135_divide__eq__eq__numeral1_I1_J,axiom,
! [B3: rat,W2: num,A3: rat] :
( ( ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( B3
= ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W2 ) ) ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_4136_divide__eq__eq__numeral1_I1_J,axiom,
! [B3: real,W2: num,A3: real] :
( ( ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( B3
= ( times_times_real @ A3 @ ( numeral_numeral_real @ W2 ) ) ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_4137_eq__divide__eq__numeral1_I1_J,axiom,
! [A3: rat,B3: rat,W2: num] :
( ( A3
= ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W2 ) )
= B3 ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_4138_eq__divide__eq__numeral1_I1_J,axiom,
! [A3: real,B3: real,W2: num] :
( ( A3
= ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( ( times_times_real @ A3 @ ( numeral_numeral_real @ W2 ) )
= B3 ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_4139_less__divide__eq__numeral1_I1_J,axiom,
! [A3: rat,B3: rat,W2: num] :
( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_4140_less__divide__eq__numeral1_I1_J,axiom,
! [A3: real,B3: real,W2: num] :
( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_4141_divide__less__eq__numeral1_I1_J,axiom,
! [B3: rat,W2: num,A3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A3 )
= ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_4142_divide__less__eq__numeral1_I1_J,axiom,
! [B3: real,W2: num,A3: real] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A3 )
= ( ord_less_real @ B3 @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_4143_nonzero__divide__mult__cancel__left,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ A3 @ B3 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B3 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_4144_nonzero__divide__mult__cancel__left,axiom,
! [A3: rat,B3: rat] :
( ( A3 != zero_zero_rat )
=> ( ( divide_divide_rat @ A3 @ ( times_times_rat @ A3 @ B3 ) )
= ( divide_divide_rat @ one_one_rat @ B3 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_4145_nonzero__divide__mult__cancel__left,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ ( times_times_real @ A3 @ B3 ) )
= ( divide_divide_real @ one_one_real @ B3 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_4146_nonzero__divide__mult__cancel__right,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B3 @ ( times_times_complex @ A3 @ B3 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_4147_nonzero__divide__mult__cancel__right,axiom,
! [B3: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( divide_divide_rat @ B3 @ ( times_times_rat @ A3 @ B3 ) )
= ( divide_divide_rat @ one_one_rat @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_4148_nonzero__divide__mult__cancel__right,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ B3 @ ( times_times_real @ A3 @ B3 ) )
= ( divide_divide_real @ one_one_real @ A3 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_4149_div__mult__self1,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self1
thf(fact_4150_div__mult__self1,axiom,
! [B3: int,A3: int,C: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ C @ B3 ) ) @ B3 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_mult_self1
thf(fact_4151_div__mult__self1,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) ) @ B3 )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_mult_self1
thf(fact_4152_div__mult__self2,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self2
thf(fact_4153_div__mult__self2,axiom,
! [B3: int,A3: int,C: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ B3 @ C ) ) @ B3 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_mult_self2
thf(fact_4154_div__mult__self2,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) @ B3 )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_mult_self2
thf(fact_4155_div__mult__self3,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A3 ) @ B3 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self3
thf(fact_4156_div__mult__self3,axiom,
! [B3: int,C: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A3 ) @ B3 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_mult_self3
thf(fact_4157_div__mult__self3,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B3 ) @ A3 ) @ B3 )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_mult_self3
thf(fact_4158_div__mult__self4,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A3 ) @ B3 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% div_mult_self4
thf(fact_4159_div__mult__self4,axiom,
! [B3: int,C: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A3 ) @ B3 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% div_mult_self4
thf(fact_4160_div__mult__self4,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ C ) @ A3 ) @ B3 )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% div_mult_self4
thf(fact_4161_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_4162_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_4163_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_4164_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_4165_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_4166_left__minus__one__mult__self,axiom,
! [N: nat,A3: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_4167_left__minus__one__mult__self,axiom,
! [N: nat,A3: 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 ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_4168_left__minus__one__mult__self,axiom,
! [N: nat,A3: 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 ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_4169_left__minus__one__mult__self,axiom,
! [N: nat,A3: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_4170_left__minus__one__mult__self,axiom,
! [N: nat,A3: 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 ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_4171_unit__mult__div__div,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( times_times_nat @ B3 @ ( divide_divide_nat @ one_one_nat @ A3 ) )
= ( divide_divide_nat @ B3 @ A3 ) ) ) ).
% unit_mult_div_div
thf(fact_4172_unit__mult__div__div,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( times_times_int @ B3 @ ( divide_divide_int @ one_one_int @ A3 ) )
= ( divide_divide_int @ B3 @ A3 ) ) ) ).
% unit_mult_div_div
thf(fact_4173_unit__mult__div__div,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 ) )
= ( divide6298287555418463151nteger @ B3 @ A3 ) ) ) ).
% unit_mult_div_div
thf(fact_4174_unit__div__mult__self,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% unit_div_mult_self
thf(fact_4175_unit__div__mult__self,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% unit_div_mult_self
thf(fact_4176_unit__div__mult__self,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% unit_div_mult_self
thf(fact_4177_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_4178_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_4179_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_4180_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_4181_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_4182_le__divide__eq__numeral1_I2_J,axiom,
! [A3: real,B3: real,W2: num] :
( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4183_le__divide__eq__numeral1_I2_J,axiom,
! [A3: rat,B3: rat,W2: num] :
( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4184_divide__le__eq__numeral1_I2_J,axiom,
! [B3: real,W2: num,A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A3 )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4185_divide__le__eq__numeral1_I2_J,axiom,
! [B3: rat,W2: num,A3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A3 )
= ( ord_less_eq_rat @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4186_divide__eq__eq__numeral1_I2_J,axiom,
! [B3: real,W2: num,A3: real] :
( ( ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( B3
= ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4187_divide__eq__eq__numeral1_I2_J,axiom,
! [B3: rat,W2: num,A3: rat] :
( ( ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( B3
= ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4188_eq__divide__eq__numeral1_I2_J,axiom,
! [A3: real,B3: real,W2: num] :
( ( A3
= ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= B3 ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4189_eq__divide__eq__numeral1_I2_J,axiom,
! [A3: rat,B3: rat,W2: num] :
( ( A3
= ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= B3 ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4190_less__divide__eq__numeral1_I2_J,axiom,
! [A3: real,B3: real,W2: num] :
( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_real @ B3 @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4191_less__divide__eq__numeral1_I2_J,axiom,
! [A3: rat,B3: rat,W2: num] :
( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4192_divide__less__eq__numeral1_I2_J,axiom,
! [B3: real,W2: num,A3: real] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4193_divide__less__eq__numeral1_I2_J,axiom,
! [B3: rat,W2: num,A3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A3 )
= ( ord_less_rat @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4194_nested__mint,axiom,
! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va ) ) )
=> ( ~ ( ord_less_nat @ Ma2 @ Mi2 )
=> ( ( Ma2 != Mi2 )
=> ( 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 @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% nested_mint
thf(fact_4195_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_4196_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_4197_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_4198_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_4199_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_4200_odd__two__times__div__two__succ,axiom,
! [A3: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A3 ) ) ).
% odd_two_times_div_two_succ
thf(fact_4201_odd__two__times__div__two__succ,axiom,
! [A3: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A3 ) ) ).
% odd_two_times_div_two_succ
thf(fact_4202_odd__two__times__div__two__succ,axiom,
! [A3: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
= A3 ) ) ).
% odd_two_times_div_two_succ
thf(fact_4203_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_4204_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_4205_euclidean__size__mult,axiom,
! [A3: code_integer,B3: code_integer] :
( ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
= ( times_times_nat @ ( euclid6377331345833325938nteger @ A3 ) @ ( euclid6377331345833325938nteger @ B3 ) ) ) ).
% euclidean_size_mult
thf(fact_4206_euclidean__size__mult,axiom,
! [A3: int,B3: int] :
( ( euclid4774559944035922753ze_int @ ( times_times_int @ A3 @ B3 ) )
= ( times_times_nat @ ( euclid4774559944035922753ze_int @ A3 ) @ ( euclid4774559944035922753ze_int @ B3 ) ) ) ).
% euclidean_size_mult
thf(fact_4207_euclidean__size__mult,axiom,
! [A3: nat,B3: nat] :
( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A3 @ B3 ) )
= ( times_times_nat @ ( euclid4777050414544973029ze_nat @ A3 ) @ ( euclid4777050414544973029ze_nat @ B3 ) ) ) ).
% euclidean_size_mult
thf(fact_4208_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4209_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A3 @ B3 ) @ C )
= ( times_times_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4210_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4211_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B3 ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4212_mult_Oassoc,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).
% mult.assoc
thf(fact_4213_mult_Oassoc,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A3 @ B3 ) @ C )
= ( times_times_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).
% mult.assoc
thf(fact_4214_mult_Oassoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% mult.assoc
thf(fact_4215_mult_Oassoc,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B3 ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).
% mult.assoc
thf(fact_4216_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A: real,B: real] : ( times_times_real @ B @ A ) ) ) ).
% mult.commute
thf(fact_4217_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A: rat,B: rat] : ( times_times_rat @ B @ A ) ) ) ).
% mult.commute
thf(fact_4218_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A: nat,B: nat] : ( times_times_nat @ B @ A ) ) ) ).
% mult.commute
thf(fact_4219_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A: int,B: int] : ( times_times_int @ B @ A ) ) ) ).
% mult.commute
thf(fact_4220_mult_Oleft__commute,axiom,
! [B3: real,A3: real,C: real] :
( ( times_times_real @ B3 @ ( times_times_real @ A3 @ C ) )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).
% mult.left_commute
thf(fact_4221_mult_Oleft__commute,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( times_times_rat @ B3 @ ( times_times_rat @ A3 @ C ) )
= ( times_times_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).
% mult.left_commute
thf(fact_4222_mult_Oleft__commute,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A3 @ C ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% mult.left_commute
thf(fact_4223_mult_Oleft__commute,axiom,
! [B3: int,A3: int,C: int] :
( ( times_times_int @ B3 @ ( times_times_int @ A3 @ C ) )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).
% mult.left_commute
thf(fact_4224_mult__right__cancel,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_4225_mult__right__cancel,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A3 @ C )
= ( times_times_rat @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_4226_mult__right__cancel,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_4227_mult__right__cancel,axiom,
! [C: int,A3: int,B3: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_4228_mult__left__cancel,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_4229_mult__left__cancel,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A3 )
= ( times_times_rat @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_4230_mult__left__cancel,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_4231_mult__left__cancel,axiom,
! [C: int,A3: int,B3: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_4232_no__zero__divisors,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_4233_no__zero__divisors,axiom,
! [A3: rat,B3: rat] :
( ( A3 != zero_zero_rat )
=> ( ( B3 != zero_zero_rat )
=> ( ( times_times_rat @ A3 @ B3 )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_4234_no__zero__divisors,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_4235_no__zero__divisors,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( B3 != zero_zero_int )
=> ( ( times_times_int @ A3 @ B3 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_4236_divisors__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_4237_divisors__zero,axiom,
! [A3: rat,B3: rat] :
( ( ( times_times_rat @ A3 @ B3 )
= zero_zero_rat )
=> ( ( A3 = zero_zero_rat )
| ( B3 = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_4238_divisors__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_4239_divisors__zero,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_4240_mult__not__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_4241_mult__not__zero,axiom,
! [A3: rat,B3: rat] :
( ( ( times_times_rat @ A3 @ B3 )
!= zero_zero_rat )
=> ( ( A3 != zero_zero_rat )
& ( B3 != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_4242_mult__not__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_4243_mult__not__zero,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B3 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_4244_comm__monoid__mult__class_Omult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_4245_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_4246_comm__monoid__mult__class_Omult__1,axiom,
! [A3: rat] :
( ( times_times_rat @ one_one_rat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_4247_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_4248_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_4249_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_4250_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_4251_mult_Ocomm__neutral,axiom,
! [A3: rat] :
( ( times_times_rat @ A3 @ one_one_rat )
= A3 ) ).
% mult.comm_neutral
thf(fact_4252_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_4253_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_4254_ring__class_Oring__distribs_I2_J,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_4255_ring__class_Oring__distribs_I2_J,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_4256_ring__class_Oring__distribs_I2_J,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_4257_ring__class_Oring__distribs_I1_J,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_4258_ring__class_Oring__distribs_I1_J,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_4259_ring__class_Oring__distribs_I1_J,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_4260_comm__semiring__class_Odistrib,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4261_comm__semiring__class_Odistrib,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4262_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4263_comm__semiring__class_Odistrib,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4264_distrib__left,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_4265_distrib__left,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_4266_distrib__left,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_4267_distrib__left,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_4268_distrib__right,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).
% distrib_right
thf(fact_4269_distrib__right,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).
% distrib_right
thf(fact_4270_distrib__right,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).
% distrib_right
thf(fact_4271_distrib__right,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).
% distrib_right
thf(fact_4272_combine__common__factor,axiom,
! [A3: real,E2: real,B3: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4273_combine__common__factor,axiom,
! [A3: rat,E2: rat,B3: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4274_combine__common__factor,axiom,
! [A3: nat,E2: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B3 @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4275_combine__common__factor,axiom,
! [A3: int,E2: int,B3: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4276_left__diff__distrib,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( minus_minus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).
% left_diff_distrib
thf(fact_4277_left__diff__distrib,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).
% left_diff_distrib
thf(fact_4278_left__diff__distrib,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( minus_minus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).
% left_diff_distrib
thf(fact_4279_right__diff__distrib,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).
% right_diff_distrib
thf(fact_4280_right__diff__distrib,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ A3 @ ( minus_minus_rat @ B3 @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).
% right_diff_distrib
thf(fact_4281_right__diff__distrib,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).
% right_diff_distrib
thf(fact_4282_left__diff__distrib_H,axiom,
! [B3: real,C: real,A3: real] :
( ( times_times_real @ ( minus_minus_real @ B3 @ C ) @ A3 )
= ( minus_minus_real @ ( times_times_real @ B3 @ A3 ) @ ( times_times_real @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_4283_left__diff__distrib_H,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( times_times_rat @ ( minus_minus_rat @ B3 @ C ) @ A3 )
= ( minus_minus_rat @ ( times_times_rat @ B3 @ A3 ) @ ( times_times_rat @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_4284_left__diff__distrib_H,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B3 @ C ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B3 @ A3 ) @ ( times_times_nat @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_4285_left__diff__distrib_H,axiom,
! [B3: int,C: int,A3: int] :
( ( times_times_int @ ( minus_minus_int @ B3 @ C ) @ A3 )
= ( minus_minus_int @ ( times_times_int @ B3 @ A3 ) @ ( times_times_int @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_4286_right__diff__distrib_H,axiom,
! [A3: real,B3: real,C: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_4287_right__diff__distrib_H,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( times_times_rat @ A3 @ ( minus_minus_rat @ B3 @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_4288_right__diff__distrib_H,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B3 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_4289_right__diff__distrib_H,axiom,
! [A3: int,B3: int,C: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_4290_square__eq__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ A3 )
= ( times_times_real @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_real @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_4291_square__eq__iff,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ A3 )
= ( times_times_int @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_int @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_4292_square__eq__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( times_3573771949741848930nteger @ A3 @ A3 )
= ( times_3573771949741848930nteger @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_4293_square__eq__iff,axiom,
! [A3: rat,B3: rat] :
( ( ( times_times_rat @ A3 @ A3 )
= ( times_times_rat @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_rat @ B3 ) ) ) ) ).
% square_eq_iff
thf(fact_4294_minus__mult__commute,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_4295_minus__mult__commute,axiom,
! [A3: int,B3: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( times_times_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_4296_minus__mult__commute,axiom,
! [A3: code_integer,B3: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( times_3573771949741848930nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_4297_minus__mult__commute,axiom,
! [A3: rat,B3: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
= ( times_times_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ).
% minus_mult_commute
thf(fact_4298_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_4299_dvdE,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ~ ! [K2: code_integer] :
( A3
!= ( times_3573771949741848930nteger @ B3 @ K2 ) ) ) ).
% dvdE
thf(fact_4300_dvdE,axiom,
! [B3: real,A3: real] :
( ( dvd_dvd_real @ B3 @ A3 )
=> ~ ! [K2: real] :
( A3
!= ( times_times_real @ B3 @ K2 ) ) ) ).
% dvdE
thf(fact_4301_dvdE,axiom,
! [B3: rat,A3: rat] :
( ( dvd_dvd_rat @ B3 @ A3 )
=> ~ ! [K2: rat] :
( A3
!= ( times_times_rat @ B3 @ K2 ) ) ) ).
% dvdE
thf(fact_4302_dvdE,axiom,
! [B3: nat,A3: nat] :
( ( dvd_dvd_nat @ B3 @ A3 )
=> ~ ! [K2: nat] :
( A3
!= ( times_times_nat @ B3 @ K2 ) ) ) ).
% dvdE
thf(fact_4303_dvdE,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ A3 )
=> ~ ! [K2: int] :
( A3
!= ( times_times_int @ B3 @ K2 ) ) ) ).
% dvdE
thf(fact_4304_dvdI,axiom,
! [A3: code_integer,B3: code_integer,K: code_integer] :
( ( A3
= ( times_3573771949741848930nteger @ B3 @ K ) )
=> ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).
% dvdI
thf(fact_4305_dvdI,axiom,
! [A3: real,B3: real,K: real] :
( ( A3
= ( times_times_real @ B3 @ K ) )
=> ( dvd_dvd_real @ B3 @ A3 ) ) ).
% dvdI
thf(fact_4306_dvdI,axiom,
! [A3: rat,B3: rat,K: rat] :
( ( A3
= ( times_times_rat @ B3 @ K ) )
=> ( dvd_dvd_rat @ B3 @ A3 ) ) ).
% dvdI
thf(fact_4307_dvdI,axiom,
! [A3: nat,B3: nat,K: nat] :
( ( A3
= ( times_times_nat @ B3 @ K ) )
=> ( dvd_dvd_nat @ B3 @ A3 ) ) ).
% dvdI
thf(fact_4308_dvdI,axiom,
! [A3: int,B3: int,K: int] :
( ( A3
= ( times_times_int @ B3 @ K ) )
=> ( dvd_dvd_int @ B3 @ A3 ) ) ).
% dvdI
thf(fact_4309_dvd__def,axiom,
( dvd_dvd_Code_integer
= ( ^ [B: code_integer,A: code_integer] :
? [K3: code_integer] :
( A
= ( times_3573771949741848930nteger @ B @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4310_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B: real,A: real] :
? [K3: real] :
( A
= ( times_times_real @ B @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4311_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B: rat,A: rat] :
? [K3: rat] :
( A
= ( times_times_rat @ B @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4312_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B: nat,A: nat] :
? [K3: nat] :
( A
= ( times_times_nat @ B @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4313_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B: int,A: int] :
? [K3: int] :
( A
= ( times_times_int @ B @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4314_dvd__mult,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ C )
=> ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) ) ).
% dvd_mult
thf(fact_4315_dvd__mult,axiom,
! [A3: real,C: real,B3: real] :
( ( dvd_dvd_real @ A3 @ C )
=> ( dvd_dvd_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).
% dvd_mult
thf(fact_4316_dvd__mult,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( dvd_dvd_rat @ A3 @ C )
=> ( dvd_dvd_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).
% dvd_mult
thf(fact_4317_dvd__mult,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ C )
=> ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% dvd_mult
thf(fact_4318_dvd__mult,axiom,
! [A3: int,C: int,B3: int] :
( ( dvd_dvd_int @ A3 @ C )
=> ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).
% dvd_mult
thf(fact_4319_dvd__mult2,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) ) ).
% dvd_mult2
thf(fact_4320_dvd__mult2,axiom,
! [A3: real,B3: real,C: real] :
( ( dvd_dvd_real @ A3 @ B3 )
=> ( dvd_dvd_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).
% dvd_mult2
thf(fact_4321_dvd__mult2,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( dvd_dvd_rat @ A3 @ B3 )
=> ( dvd_dvd_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).
% dvd_mult2
thf(fact_4322_dvd__mult2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% dvd_mult2
thf(fact_4323_dvd__mult2,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).
% dvd_mult2
thf(fact_4324_dvd__mult__left,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
=> ( dvd_dvd_Code_integer @ A3 @ C ) ) ).
% dvd_mult_left
thf(fact_4325_dvd__mult__left,axiom,
! [A3: real,B3: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ B3 ) @ C )
=> ( dvd_dvd_real @ A3 @ C ) ) ).
% dvd_mult_left
thf(fact_4326_dvd__mult__left,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A3 @ B3 ) @ C )
=> ( dvd_dvd_rat @ A3 @ C ) ) ).
% dvd_mult_left
thf(fact_4327_dvd__mult__left,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
=> ( dvd_dvd_nat @ A3 @ C ) ) ).
% dvd_mult_left
thf(fact_4328_dvd__mult__left,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
=> ( dvd_dvd_int @ A3 @ C ) ) ).
% dvd_mult_left
thf(fact_4329_dvd__triv__left,axiom,
! [A3: code_integer,B3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ A3 @ B3 ) ) ).
% dvd_triv_left
thf(fact_4330_dvd__triv__left,axiom,
! [A3: real,B3: real] : ( dvd_dvd_real @ A3 @ ( times_times_real @ A3 @ B3 ) ) ).
% dvd_triv_left
thf(fact_4331_dvd__triv__left,axiom,
! [A3: rat,B3: rat] : ( dvd_dvd_rat @ A3 @ ( times_times_rat @ A3 @ B3 ) ) ).
% dvd_triv_left
thf(fact_4332_dvd__triv__left,axiom,
! [A3: nat,B3: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ A3 @ B3 ) ) ).
% dvd_triv_left
thf(fact_4333_dvd__triv__left,axiom,
! [A3: int,B3: int] : ( dvd_dvd_int @ A3 @ ( times_times_int @ A3 @ B3 ) ) ).
% dvd_triv_left
thf(fact_4334_mult__dvd__mono,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer,D: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ C @ D )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4335_mult__dvd__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( dvd_dvd_real @ A3 @ B3 )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4336_mult__dvd__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( dvd_dvd_rat @ A3 @ B3 )
=> ( ( dvd_dvd_rat @ C @ D )
=> ( dvd_dvd_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4337_mult__dvd__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4338_mult__dvd__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4339_dvd__mult__right,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
=> ( dvd_dvd_Code_integer @ B3 @ C ) ) ).
% dvd_mult_right
thf(fact_4340_dvd__mult__right,axiom,
! [A3: real,B3: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ B3 ) @ C )
=> ( dvd_dvd_real @ B3 @ C ) ) ).
% dvd_mult_right
thf(fact_4341_dvd__mult__right,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A3 @ B3 ) @ C )
=> ( dvd_dvd_rat @ B3 @ C ) ) ).
% dvd_mult_right
thf(fact_4342_dvd__mult__right,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
=> ( dvd_dvd_nat @ B3 @ C ) ) ).
% dvd_mult_right
thf(fact_4343_dvd__mult__right,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
=> ( dvd_dvd_int @ B3 @ C ) ) ).
% dvd_mult_right
thf(fact_4344_dvd__triv__right,axiom,
! [A3: code_integer,B3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ A3 ) ) ).
% dvd_triv_right
thf(fact_4345_dvd__triv__right,axiom,
! [A3: real,B3: real] : ( dvd_dvd_real @ A3 @ ( times_times_real @ B3 @ A3 ) ) ).
% dvd_triv_right
thf(fact_4346_dvd__triv__right,axiom,
! [A3: rat,B3: rat] : ( dvd_dvd_rat @ A3 @ ( times_times_rat @ B3 @ A3 ) ) ).
% dvd_triv_right
thf(fact_4347_dvd__triv__right,axiom,
! [A3: nat,B3: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ A3 ) ) ).
% dvd_triv_right
thf(fact_4348_dvd__triv__right,axiom,
! [A3: int,B3: int] : ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ A3 ) ) ).
% dvd_triv_right
thf(fact_4349_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_4350_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_4351_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_4352_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_4353_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_4354_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_4355_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_4356_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_4357_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_4358_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_4359_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_4360_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_4361_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_4362_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_4363_option_Osel,axiom,
! [X2: nat] :
( ( the_nat @ ( some_nat @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_4364_option_Osel,axiom,
! [X2: product_prod_nat_nat] :
( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_4365_option_Osel,axiom,
! [X2: num] :
( ( the_num @ ( some_num @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_4366_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_4367_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_4368_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_4369_mult__ceiling__le,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A3 @ B3 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A3 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_4370_mult__ceiling__le,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A3 @ B3 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A3 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_4371_mult__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_4372_mult__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_4373_mult__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_4374_mult__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_4375_mult__mono_H,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_4376_mult__mono_H,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_4377_mult__mono_H,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_4378_mult__mono_H,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_4379_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_4380_zero__le__square,axiom,
! [A3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_4381_zero__le__square,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_4382_split__mult__pos__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_4383_split__mult__pos__le,axiom,
! [A3: rat,B3: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
| ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_4384_split__mult__pos__le,axiom,
! [A3: int,B3: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_4385_mult__left__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_4386_mult__left__mono__neg,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_4387_mult__left__mono__neg,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_4388_mult__nonpos__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_4389_mult__nonpos__nonpos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_4390_mult__nonpos__nonpos,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_4391_mult__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_4392_mult__left__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_4393_mult__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_4394_mult__left__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_4395_mult__right__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_4396_mult__right__mono__neg,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_4397_mult__right__mono__neg,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_4398_mult__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_4399_mult__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_4400_mult__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_4401_mult__right__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_4402_mult__le__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_4403_mult__le__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_4404_mult__le__0__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_4405_split__mult__neg__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_4406_split__mult__neg__le,axiom,
! [A3: rat,B3: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_4407_split__mult__neg__le,axiom,
! [A3: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_4408_split__mult__neg__le,axiom,
! [A3: int,B3: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_4409_mult__nonneg__nonneg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_4410_mult__nonneg__nonneg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_4411_mult__nonneg__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_4412_mult__nonneg__nonneg,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_4413_mult__nonneg__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_4414_mult__nonneg__nonpos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_4415_mult__nonneg__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_4416_mult__nonneg__nonpos,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_4417_mult__nonpos__nonneg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_4418_mult__nonpos__nonneg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_4419_mult__nonpos__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_4420_mult__nonpos__nonneg,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_4421_mult__nonneg__nonpos2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_4422_mult__nonneg__nonpos2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B3 @ A3 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_4423_mult__nonneg__nonpos2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_4424_mult__nonneg__nonpos2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B3 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_4425_zero__le__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_4426_zero__le__mult__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
| ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
& ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_4427_zero__le__mult__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_4428_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4429_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4430_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4431_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4432_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_4433_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_4434_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_4435_mult__neg__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_neg_neg
thf(fact_4436_mult__neg__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_rat @ B3 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).
% mult_neg_neg
thf(fact_4437_mult__neg__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% mult_neg_neg
thf(fact_4438_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_4439_not__square__less__zero,axiom,
! [A3: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A3 @ A3 ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_4440_not__square__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_4441_mult__less__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_4442_mult__less__0__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ B3 @ zero_zero_rat ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_4443_mult__less__0__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_4444_mult__neg__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_4445_mult__neg__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_4446_mult__neg__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_4447_mult__neg__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_4448_mult__pos__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_4449_mult__pos__neg,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ B3 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_4450_mult__pos__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_4451_mult__pos__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_4452_mult__pos__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_4453_mult__pos__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_4454_mult__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_4455_mult__pos__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_4456_mult__pos__neg2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_4457_mult__pos__neg2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ B3 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B3 @ A3 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_4458_mult__pos__neg2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_4459_mult__pos__neg2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B3 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_4460_zero__less__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_4461_zero__less__mult__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
& ( ord_less_rat @ zero_zero_rat @ B3 ) )
| ( ( ord_less_rat @ A3 @ zero_zero_rat )
& ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_4462_zero__less__mult__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_4463_zero__less__mult__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_4464_zero__less__mult__pos,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_4465_zero__less__mult__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_4466_zero__less__mult__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_4467_zero__less__mult__pos2,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_4468_zero__less__mult__pos2,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B3 @ A3 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_4469_zero__less__mult__pos2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_4470_zero__less__mult__pos2,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_4471_mult__less__cancel__left__neg,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_4472_mult__less__cancel__left__neg,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ord_less_rat @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_4473_mult__less__cancel__left__neg,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ord_less_int @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_4474_mult__less__cancel__left__pos,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_4475_mult__less__cancel__left__pos,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ord_less_rat @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_4476_mult__less__cancel__left__pos,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_4477_mult__strict__left__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_4478_mult__strict__left__mono__neg,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_4479_mult__strict__left__mono__neg,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_4480_mult__strict__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_4481_mult__strict__left__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_4482_mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_4483_mult__strict__left__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_4484_mult__less__cancel__left__disj,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_4485_mult__less__cancel__left__disj,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A3 @ B3 ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_4486_mult__less__cancel__left__disj,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B3 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_4487_mult__strict__right__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_4488_mult__strict__right__mono__neg,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_4489_mult__strict__right__mono__neg,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_4490_mult__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_4491_mult__strict__right__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_4492_mult__strict__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_4493_mult__strict__right__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_4494_mult__less__cancel__right__disj,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_4495_mult__less__cancel__right__disj,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A3 @ B3 ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_4496_mult__less__cancel__right__disj,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B3 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_4497_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4498_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4499_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4500_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B3: int,C: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4501_add__scale__eq__noteq,axiom,
! [R2: real,A3: real,B3: real,C: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A3 = B3 )
& ( C != D ) )
=> ( ( plus_plus_real @ A3 @ ( times_times_real @ R2 @ C ) )
!= ( plus_plus_real @ B3 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4502_add__scale__eq__noteq,axiom,
! [R2: rat,A3: rat,B3: rat,C: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A3 = B3 )
& ( C != D ) )
=> ( ( plus_plus_rat @ A3 @ ( times_times_rat @ R2 @ C ) )
!= ( plus_plus_rat @ B3 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4503_add__scale__eq__noteq,axiom,
! [R2: nat,A3: nat,B3: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A3 = B3 )
& ( C != D ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B3 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4504_add__scale__eq__noteq,axiom,
! [R2: int,A3: int,B3: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A3 = B3 )
& ( C != D ) )
=> ( ( plus_plus_int @ A3 @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B3 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4505_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_4506_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_4507_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_4508_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_4509_nonzero__eq__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( A3
= ( divide_divide_rat @ B3 @ C ) )
= ( ( times_times_rat @ A3 @ C )
= B3 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_4510_nonzero__eq__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( A3
= ( divide_divide_real @ B3 @ C ) )
= ( ( times_times_real @ A3 @ C )
= B3 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_4511_nonzero__divide__eq__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B3 @ C )
= A3 )
= ( B3
= ( times_times_rat @ A3 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_4512_nonzero__divide__eq__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B3 @ C )
= A3 )
= ( B3
= ( times_times_real @ A3 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_4513_eq__divide__imp,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A3 @ C )
= B3 )
=> ( A3
= ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_4514_eq__divide__imp,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= B3 )
=> ( A3
= ( divide_divide_real @ B3 @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_4515_divide__eq__imp,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( C != zero_zero_rat )
=> ( ( B3
= ( times_times_rat @ A3 @ C ) )
=> ( ( divide_divide_rat @ B3 @ C )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_4516_divide__eq__imp,axiom,
! [C: real,B3: real,A3: real] :
( ( C != zero_zero_real )
=> ( ( B3
= ( times_times_real @ A3 @ C ) )
=> ( ( divide_divide_real @ B3 @ C )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_4517_eq__divide__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( A3
= ( divide_divide_rat @ B3 @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A3 @ C )
= B3 ) )
& ( ( C = zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_4518_eq__divide__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( A3
= ( divide_divide_real @ B3 @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A3 @ C )
= B3 ) )
& ( ( C = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_4519_divide__eq__eq,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ( divide_divide_rat @ B3 @ C )
= A3 )
= ( ( ( C != zero_zero_rat )
=> ( B3
= ( times_times_rat @ A3 @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_4520_divide__eq__eq,axiom,
! [B3: real,C: real,A3: real] :
( ( ( divide_divide_real @ B3 @ C )
= A3 )
= ( ( ( C != zero_zero_real )
=> ( B3
= ( times_times_real @ A3 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_4521_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y )
= ( divide_divide_rat @ W2 @ Z ) )
= ( ( times_times_rat @ X @ Z )
= ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_4522_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W2 @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_4523_mult__numeral__1__right,axiom,
! [A3: rat] :
( ( times_times_rat @ A3 @ ( numeral_numeral_rat @ one ) )
= A3 ) ).
% mult_numeral_1_right
thf(fact_4524_mult__numeral__1__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ ( numeral_numeral_nat @ one ) )
= A3 ) ).
% mult_numeral_1_right
thf(fact_4525_mult__numeral__1__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ ( numeral_numeral_real @ one ) )
= A3 ) ).
% mult_numeral_1_right
thf(fact_4526_mult__numeral__1__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ ( numeral_numeral_int @ one ) )
= A3 ) ).
% mult_numeral_1_right
thf(fact_4527_mult__numeral__1__right,axiom,
! [A3: extended_enat] :
( ( times_7803423173614009249d_enat @ A3 @ ( numera1916890842035813515d_enat @ one ) )
= A3 ) ).
% mult_numeral_1_right
thf(fact_4528_mult__numeral__1__right,axiom,
! [A3: code_integer] :
( ( times_3573771949741848930nteger @ A3 @ ( numera6620942414471956472nteger @ one ) )
= A3 ) ).
% mult_numeral_1_right
thf(fact_4529_mult__numeral__1,axiom,
! [A3: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A3 )
= A3 ) ).
% mult_numeral_1
thf(fact_4530_mult__numeral__1,axiom,
! [A3: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A3 )
= A3 ) ).
% mult_numeral_1
thf(fact_4531_mult__numeral__1,axiom,
! [A3: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A3 )
= A3 ) ).
% mult_numeral_1
thf(fact_4532_mult__numeral__1,axiom,
! [A3: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A3 )
= A3 ) ).
% mult_numeral_1
thf(fact_4533_mult__numeral__1,axiom,
! [A3: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A3 )
= A3 ) ).
% mult_numeral_1
thf(fact_4534_mult__numeral__1,axiom,
! [A3: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A3 )
= A3 ) ).
% mult_numeral_1
thf(fact_4535_eq__add__iff1,axiom,
! [A3: real,E2: real,C: real,B3: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_4536_eq__add__iff1,axiom,
! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_4537_eq__add__iff1,axiom,
! [A3: int,E2: int,C: int,B3: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_4538_eq__add__iff2,axiom,
! [A3: real,E2: real,C: real,B3: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4539_eq__add__iff2,axiom,
! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
= ( C
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4540_eq__add__iff2,axiom,
! [A3: int,E2: int,C: int,B3: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4541_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4542_square__diff__square__factored,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4543_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4544_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_4545_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_4546_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_4547_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_4548_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_4549_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_4550_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_4551_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N: nat] :
( ( ( times_times_rat @ X @ Y )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_4552_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_4553_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_4554_unit__mult__right__cancel,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ B3 @ A3 )
= ( times_3573771949741848930nteger @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4555_unit__mult__right__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( times_times_nat @ B3 @ A3 )
= ( times_times_nat @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4556_unit__mult__right__cancel,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( ( times_times_int @ B3 @ A3 )
= ( times_times_int @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4557_unit__mult__left__cancel,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ A3 @ B3 )
= ( times_3573771949741848930nteger @ A3 @ C ) )
= ( B3 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4558_unit__mult__left__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B3 )
= ( times_times_nat @ A3 @ C ) )
= ( B3 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4559_unit__mult__left__cancel,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( ( times_times_int @ A3 @ B3 )
= ( times_times_int @ A3 @ C ) )
= ( B3 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4560_mult__unit__dvd__iff_H,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
= ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4561_mult__unit__dvd__iff_H,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( dvd_dvd_nat @ B3 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4562_mult__unit__dvd__iff_H,axiom,
! [A3: int,B3: int,C: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
= ( dvd_dvd_int @ B3 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4563_dvd__mult__unit__iff_H,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4564_dvd__mult__unit__iff_H,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4565_dvd__mult__unit__iff_H,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( dvd_dvd_int @ A3 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4566_mult__unit__dvd__iff,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
= ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4567_mult__unit__dvd__iff,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4568_mult__unit__dvd__iff,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
= ( dvd_dvd_int @ A3 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4569_dvd__mult__unit__iff,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) )
= ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4570_dvd__mult__unit__iff,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ C @ B3 ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4571_dvd__mult__unit__iff,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( dvd_dvd_int @ A3 @ ( times_times_int @ C @ B3 ) )
= ( dvd_dvd_int @ A3 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4572_is__unit__mult__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
& ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer ) ) ) ).
% is_unit_mult_iff
thf(fact_4573_is__unit__mult__iff,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
& ( dvd_dvd_nat @ B3 @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_4574_is__unit__mult__iff,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ one_one_int )
= ( ( dvd_dvd_int @ A3 @ one_one_int )
& ( dvd_dvd_int @ B3 @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_4575_power__Suc,axiom,
! [A3: complex,N: nat] :
( ( power_power_complex @ A3 @ ( suc @ N ) )
= ( times_times_complex @ A3 @ ( power_power_complex @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_4576_power__Suc,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_4577_power__Suc,axiom,
! [A3: rat,N: nat] :
( ( power_power_rat @ A3 @ ( suc @ N ) )
= ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_4578_power__Suc,axiom,
! [A3: nat,N: nat] :
( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_4579_power__Suc,axiom,
! [A3: int,N: nat] :
( ( power_power_int @ A3 @ ( suc @ N ) )
= ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_4580_power__Suc2,axiom,
! [A3: complex,N: nat] :
( ( power_power_complex @ A3 @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_4581_power__Suc2,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_4582_power__Suc2,axiom,
! [A3: rat,N: nat] :
( ( power_power_rat @ A3 @ ( suc @ N ) )
= ( times_times_rat @ ( power_power_rat @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_4583_power__Suc2,axiom,
! [A3: nat,N: nat] :
( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_4584_power__Suc2,axiom,
! [A3: int,N: nat] :
( ( power_power_int @ A3 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_4585_div__mult__div__if__dvd,axiom,
! [B3: nat,A3: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4586_div__mult__div__if__dvd,axiom,
! [B3: int,A3: int,D: int,C: int] :
( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4587_div__mult__div__if__dvd,axiom,
! [B3: code_integer,A3: code_integer,D: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( dvd_dvd_Code_integer @ D @ C )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ ( divide6298287555418463151nteger @ C @ D ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4588_dvd__mult__imp__div,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ B3 )
=> ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4589_dvd__mult__imp__div,axiom,
! [A3: int,C: int,B3: int] :
( ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ B3 )
=> ( dvd_dvd_int @ A3 @ ( divide_divide_int @ B3 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4590_dvd__mult__imp__div,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ B3 )
=> ( dvd_dvd_Code_integer @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4591_dvd__div__mult2__eq,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B3 @ C ) @ A3 )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4592_dvd__div__mult2__eq,axiom,
! [B3: int,C: int,A3: int] :
( ( dvd_dvd_int @ ( times_times_int @ B3 @ C ) @ A3 )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4593_dvd__div__mult2__eq,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B3 @ C ) @ A3 )
=> ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4594_div__div__eq__right,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B3 )
=> ( ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( divide_divide_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4595_div__div__eq__right,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ B3 )
=> ( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( divide_divide_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
= ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4596_div__div__eq__right,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( divide6298287555418463151nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4597_div__mult__swap,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B3 )
=> ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4598_div__mult__swap,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ B3 )
=> ( ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4599_div__mult__swap,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4600_dvd__div__mult,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B3 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B3 @ C ) @ A3 )
= ( divide_divide_nat @ ( times_times_nat @ B3 @ A3 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4601_dvd__div__mult,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ B3 )
=> ( ( times_times_int @ ( divide_divide_int @ B3 @ C ) @ A3 )
= ( divide_divide_int @ ( times_times_int @ B3 @ A3 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4602_dvd__div__mult,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B3 @ C ) @ A3 )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B3 @ A3 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4603_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_4604_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_4605_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_4606_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_4607_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_4608_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_4609_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_4610_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_4611_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_4612_bezout__add__strong__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ? [D4: nat,X4: nat,Y4: nat] :
( ( dvd_dvd_nat @ D4 @ A3 )
& ( dvd_dvd_nat @ D4 @ B3 )
& ( ( times_times_nat @ A3 @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_4613_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_4614_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_4615_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_4616_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_4617_power__odd__eq,axiom,
! [A3: complex,N: nat] :
( ( power_power_complex @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_complex @ A3 @ ( power_power_complex @ ( power_power_complex @ A3 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_4618_power__odd__eq,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A3 @ ( power_power_real @ ( power_power_real @ A3 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_4619_power__odd__eq,axiom,
! [A3: rat,N: nat] :
( ( power_power_rat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_rat @ A3 @ ( power_power_rat @ ( power_power_rat @ A3 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_4620_power__odd__eq,axiom,
! [A3: nat,N: nat] :
( ( power_power_nat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A3 @ ( power_power_nat @ ( power_power_nat @ A3 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_4621_power__odd__eq,axiom,
! [A3: int,N: nat] :
( ( power_power_int @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A3 @ ( power_power_int @ ( power_power_int @ A3 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_4622_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_4623_option_Oexhaust__sel,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( Option
= ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_4624_option_Oexhaust__sel,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( Option
= ( some_num @ ( the_num @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_4625_mult__le__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_4626_mult__le__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ B3 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_4627_mult__le__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B3 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_4628_mult__le__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_4629_mult__le__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ B3 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_4630_mult__le__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B3 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_4631_mult__left__less__imp__less,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_4632_mult__left__less__imp__less,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_4633_mult__left__less__imp__less,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_4634_mult__left__less__imp__less,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_4635_mult__strict__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_4636_mult__strict__mono,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_4637_mult__strict__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_4638_mult__strict__mono,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_4639_mult__less__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_4640_mult__less__cancel__left,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ B3 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_4641_mult__less__cancel__left,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B3 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_4642_mult__right__less__imp__less,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_4643_mult__right__less__imp__less,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_4644_mult__right__less__imp__less,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_4645_mult__right__less__imp__less,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_4646_mult__strict__mono_H,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_4647_mult__strict__mono_H,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_4648_mult__strict__mono_H,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_4649_mult__strict__mono_H,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_4650_mult__less__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_4651_mult__less__cancel__right,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ B3 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_4652_mult__less__cancel__right,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B3 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_4653_mult__le__cancel__left__neg,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_4654_mult__le__cancel__left__neg,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ord_less_eq_rat @ B3 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_4655_mult__le__cancel__left__neg,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ord_less_eq_int @ B3 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_4656_mult__le__cancel__left__pos,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_4657_mult__le__cancel__left__pos,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
= ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_4658_mult__le__cancel__left__pos,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_4659_mult__left__le__imp__le,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_4660_mult__left__le__imp__le,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_4661_mult__left__le__imp__le,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_4662_mult__left__le__imp__le,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_4663_mult__right__le__imp__le,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_4664_mult__right__le__imp__le,axiom,
! [A3: rat,C: rat,B3: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_4665_mult__right__le__imp__le,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_4666_mult__right__le__imp__le,axiom,
! [A3: int,C: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_4667_mult__le__less__imp__less,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_4668_mult__le__less__imp__less,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_4669_mult__le__less__imp__less,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_4670_mult__le__less__imp__less,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_4671_mult__less__le__imp__less,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_4672_mult__less__le__imp__less,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_4673_mult__less__le__imp__less,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_4674_mult__less__le__imp__less,axiom,
! [A3: int,B3: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_4675_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_4676_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_4677_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_4678_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_4679_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_4680_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_4681_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_4682_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_4683_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_4684_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_4685_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_4686_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_4687_mult__le__one,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_4688_mult__le__one,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_4689_mult__le__one,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_4690_mult__le__one,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_4691_mult__left__le,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_4692_mult__left__le,axiom,
! [C: rat,A3: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_4693_mult__left__le,axiom,
! [C: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_4694_mult__left__le,axiom,
! [C: int,A3: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_4695_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_4696_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_4697_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_4698_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_4699_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_4700_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_4701_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_4702_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_4703_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_4704_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_4705_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_4706_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_4707_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4708_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4709_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4710_divide__less__eq,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_4711_divide__less__eq,axiom,
! [B3: real,C: real,A3: real] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_4712_less__divide__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_4713_less__divide__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_4714_neg__divide__less__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
= ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).
% neg_divide_less_eq
thf(fact_4715_neg__divide__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).
% neg_divide_less_eq
thf(fact_4716_neg__less__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
= ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_4717_neg__less__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
= ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_4718_pos__divide__less__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
= ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_4719_pos__divide__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
= ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_4720_pos__less__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
= ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).
% pos_less_divide_eq
thf(fact_4721_pos__less__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
= ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).
% pos_less_divide_eq
thf(fact_4722_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_4723_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_4724_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_4725_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_4726_divide__strict__left__mono,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_4727_divide__strict__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_4728_divide__strict__left__mono__neg,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_4729_divide__strict__left__mono__neg,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_4730_divide__eq__eq__numeral_I1_J,axiom,
! [B3: rat,C: rat,W2: num] :
( ( ( divide_divide_rat @ B3 @ C )
= ( numeral_numeral_rat @ W2 ) )
= ( ( ( C != zero_zero_rat )
=> ( B3
= ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_4731_divide__eq__eq__numeral_I1_J,axiom,
! [B3: real,C: real,W2: num] :
( ( ( divide_divide_real @ B3 @ C )
= ( numeral_numeral_real @ W2 ) )
= ( ( ( C != zero_zero_real )
=> ( B3
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_4732_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B3: rat,C: rat] :
( ( ( numeral_numeral_rat @ W2 )
= ( divide_divide_rat @ B3 @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
= B3 ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_4733_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B3: real,C: real] :
( ( ( numeral_numeral_real @ W2 )
= ( divide_divide_real @ B3 @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
= B3 ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_4734_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: real,E2: real,C: real,B3: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4735_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4736_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: int,E2: int,C: int,B3: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4737_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: real,E2: real,C: real,B3: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4738_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
= ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4739_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: int,E2: int,C: int,B3: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4740_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_4741_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_4742_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_4743_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_4744_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_4745_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_4746_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_4747_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_4748_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_4749_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_4750_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A3: rat,B3: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_4751_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_4752_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A3: rat,B3: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
= ( divide_divide_rat @ ( plus_plus_rat @ A3 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_4753_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= ( divide_divide_real @ ( plus_plus_real @ A3 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_4754_less__add__iff2,axiom,
! [A3: real,E2: real,C: real,B3: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4755_less__add__iff2,axiom,
! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
= ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4756_less__add__iff2,axiom,
! [A3: int,E2: int,C: int,B3: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4757_less__add__iff1,axiom,
! [A3: real,E2: real,C: real,B3: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_4758_less__add__iff1,axiom,
! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_4759_less__add__iff1,axiom,
! [A3: int,E2: int,C: int,B3: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_4760_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_4761_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_4762_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_4763_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_4764_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4765_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4766_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A3: rat,B3: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4767_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= A3 ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4768_eq__minus__divide__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( A3
= ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A3 @ C )
= ( uminus_uminus_real @ B3 ) ) )
& ( ( C = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_4769_eq__minus__divide__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( A3
= ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A3 @ C )
= ( uminus_uminus_rat @ B3 ) ) )
& ( ( C = zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_4770_minus__divide__eq__eq,axiom,
! [B3: real,C: real,A3: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) )
= A3 )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B3 )
= ( times_times_real @ A3 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_4771_minus__divide__eq__eq,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) )
= A3 )
= ( ( ( C != zero_zero_rat )
=> ( ( uminus_uminus_rat @ B3 )
= ( times_times_rat @ A3 @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A3 = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_4772_nonzero__neg__divide__eq__eq,axiom,
! [B3: real,A3: real,C: real] :
( ( B3 != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
= C )
= ( ( uminus_uminus_real @ A3 )
= ( times_times_real @ C @ B3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_4773_nonzero__neg__divide__eq__eq,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( B3 != zero_zero_rat )
=> ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) )
= C )
= ( ( uminus_uminus_rat @ A3 )
= ( times_times_rat @ C @ B3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_4774_nonzero__neg__divide__eq__eq2,axiom,
! [B3: real,C: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) )
= ( ( times_times_real @ C @ B3 )
= ( uminus_uminus_real @ A3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_4775_nonzero__neg__divide__eq__eq2,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( C
= ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) ) )
= ( ( times_times_rat @ C @ B3 )
= ( uminus_uminus_rat @ A3 ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_4776_power__gt1__lemma,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_4777_power__gt1__lemma,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_4778_power__gt1__lemma,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_4779_power__gt1__lemma,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_4780_power__less__power__Suc,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ord_less_real @ ( power_power_real @ A3 @ N ) @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_4781_power__less__power__Suc,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A3 )
=> ( ord_less_rat @ ( power_power_rat @ A3 @ N ) @ ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_4782_power__less__power__Suc,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_4783_power__less__power__Suc,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_4784_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_4785_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_4786_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_4787_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_4788_mult__1s__ring__1_I2_J,axiom,
! [B3: real] :
( ( times_times_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
= ( uminus_uminus_real @ B3 ) ) ).
% mult_1s_ring_1(2)
thf(fact_4789_mult__1s__ring__1_I2_J,axiom,
! [B3: int] :
( ( times_times_int @ B3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B3 ) ) ).
% mult_1s_ring_1(2)
thf(fact_4790_mult__1s__ring__1_I2_J,axiom,
! [B3: code_integer] :
( ( times_3573771949741848930nteger @ B3 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
= ( uminus1351360451143612070nteger @ B3 ) ) ).
% mult_1s_ring_1(2)
thf(fact_4791_mult__1s__ring__1_I2_J,axiom,
! [B3: rat] :
( ( times_times_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
= ( uminus_uminus_rat @ B3 ) ) ).
% mult_1s_ring_1(2)
thf(fact_4792_mult__1s__ring__1_I1_J,axiom,
! [B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B3 )
= ( uminus_uminus_real @ B3 ) ) ).
% mult_1s_ring_1(1)
thf(fact_4793_mult__1s__ring__1_I1_J,axiom,
! [B3: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B3 )
= ( uminus_uminus_int @ B3 ) ) ).
% mult_1s_ring_1(1)
thf(fact_4794_mult__1s__ring__1_I1_J,axiom,
! [B3: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B3 )
= ( uminus1351360451143612070nteger @ B3 ) ) ).
% mult_1s_ring_1(1)
thf(fact_4795_mult__1s__ring__1_I1_J,axiom,
! [B3: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B3 )
= ( uminus_uminus_rat @ B3 ) ) ).
% mult_1s_ring_1(1)
thf(fact_4796_unit__dvdE,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ~ ( ( A3 != zero_z3403309356797280102nteger )
=> ! [C3: code_integer] :
( B3
!= ( times_3573771949741848930nteger @ A3 @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_4797_unit__dvdE,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ~ ( ( A3 != zero_zero_nat )
=> ! [C3: nat] :
( B3
!= ( times_times_nat @ A3 @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_4798_unit__dvdE,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ~ ( ( A3 != zero_zero_int )
=> ! [C3: int] :
( B3
!= ( times_times_int @ A3 @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_4799_unity__coeff__ex,axiom,
! [P: code_integer > $o,L: code_integer] :
( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
= ( ? [X3: code_integer] :
( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4800_unity__coeff__ex,axiom,
! [P: real > $o,L: real] :
( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
= ( ? [X3: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4801_unity__coeff__ex,axiom,
! [P: rat > $o,L: rat] :
( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
= ( ? [X3: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4802_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
= ( ? [X3: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4803_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
= ( ? [X3: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4804_dvd__div__eq__mult,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( ( divide_divide_nat @ B3 @ A3 )
= C )
= ( B3
= ( times_times_nat @ C @ A3 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4805_dvd__div__eq__mult,axiom,
! [A3: int,B3: int,C: int] :
( ( A3 != zero_zero_int )
=> ( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( ( divide_divide_int @ B3 @ A3 )
= C )
= ( B3
= ( times_times_int @ C @ A3 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4806_dvd__div__eq__mult,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( ( divide6298287555418463151nteger @ B3 @ A3 )
= C )
= ( B3
= ( times_3573771949741848930nteger @ C @ A3 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4807_div__dvd__iff__mult,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( B3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C )
= ( dvd_dvd_nat @ A3 @ ( times_times_nat @ C @ B3 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4808_div__dvd__iff__mult,axiom,
! [B3: int,A3: int,C: int] :
( ( B3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A3 @ B3 ) @ C )
= ( dvd_dvd_int @ A3 @ ( times_times_int @ C @ B3 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4809_div__dvd__iff__mult,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C )
= ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4810_dvd__div__iff__mult,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B3 )
=> ( ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ B3 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4811_dvd__div__iff__mult,axiom,
! [C: int,B3: int,A3: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B3 )
=> ( ( dvd_dvd_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ B3 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4812_dvd__div__iff__mult,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( dvd_dvd_Code_integer @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
= ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ B3 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4813_dvd__div__div__eq__mult,axiom,
! [A3: nat,C: nat,B3: nat,D: nat] :
( ( A3 != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B3 @ A3 )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B3 @ C )
= ( times_times_nat @ A3 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4814_dvd__div__div__eq__mult,axiom,
! [A3: int,C: int,B3: int,D: int] :
( ( A3 != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B3 @ A3 )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B3 @ C )
= ( times_times_int @ A3 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4815_dvd__div__div__eq__mult,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( dvd_dvd_Code_integer @ C @ D )
=> ( ( ( divide6298287555418463151nteger @ B3 @ A3 )
= ( divide6298287555418463151nteger @ D @ C ) )
= ( ( times_3573771949741848930nteger @ B3 @ C )
= ( times_3573771949741848930nteger @ A3 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4816_power__minus,axiom,
! [A3: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A3 @ N ) ) ) ).
% power_minus
thf(fact_4817_power__minus,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A3 @ N ) ) ) ).
% power_minus
thf(fact_4818_power__minus,axiom,
! [A3: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A3 @ N ) ) ) ).
% power_minus
thf(fact_4819_power__minus,axiom,
! [A3: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A3 @ N ) ) ) ).
% power_minus
thf(fact_4820_power__minus,axiom,
! [A3: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A3 @ N ) ) ) ).
% power_minus
thf(fact_4821_unit__eq__div1,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A3 @ B3 )
= C )
= ( A3
= ( times_times_nat @ C @ B3 ) ) ) ) ).
% unit_eq_div1
thf(fact_4822_unit__eq__div1,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( ( divide_divide_int @ A3 @ B3 )
= C )
= ( A3
= ( times_times_int @ C @ B3 ) ) ) ) ).
% unit_eq_div1
thf(fact_4823_unit__eq__div1,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A3 @ B3 )
= C )
= ( A3
= ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ).
% unit_eq_div1
thf(fact_4824_unit__eq__div2,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( A3
= ( divide_divide_nat @ C @ B3 ) )
= ( ( times_times_nat @ A3 @ B3 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4825_unit__eq__div2,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( A3
= ( divide_divide_int @ C @ B3 ) )
= ( ( times_times_int @ A3 @ B3 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4826_unit__eq__div2,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( A3
= ( divide6298287555418463151nteger @ C @ B3 ) )
= ( ( times_3573771949741848930nteger @ A3 @ B3 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4827_div__mult__unit2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B3 @ A3 )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4828_div__mult__unit2,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B3 @ A3 )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4829_div__mult__unit2,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B3 @ A3 )
=> ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4830_unit__div__commute,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ B3 ) ) ) ).
% unit_div_commute
thf(fact_4831_unit__div__commute,axiom,
! [B3: int,A3: int,C: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ C )
= ( divide_divide_int @ ( times_times_int @ A3 @ C ) @ B3 ) ) ) ).
% unit_div_commute
thf(fact_4832_unit__div__commute,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ B3 ) ) ) ).
% unit_div_commute
thf(fact_4833_unit__div__mult__swap,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4834_unit__div__mult__swap,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4835_unit__div__mult__swap,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4836_is__unit__div__mult2__eq,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4837_is__unit__div__mult2__eq,axiom,
! [B3: int,C: int,A3: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4838_is__unit__div__mult2__eq,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4839_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_4840_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_4841_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_4842_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_4843_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_4844_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_4845_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_4846_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_4847_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_4848_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_4849_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_4850_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ 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 @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_4851_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_4852_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ 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 @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_4853_dvd__minus__add,axiom,
! [Q4: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq_nat @ Q4 @ N )
=> ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q4 ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_4854_size__mult__mono,axiom,
! [B3: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A3 ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) ) ) ) ).
% size_mult_mono
thf(fact_4855_size__mult__mono,axiom,
! [B3: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A3 ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% size_mult_mono
thf(fact_4856_size__mult__mono,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A3 ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% size_mult_mono
thf(fact_4857_size__mult__mono_H,axiom,
! [B3: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A3 ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ B3 @ A3 ) ) ) ) ).
% size_mult_mono'
thf(fact_4858_size__mult__mono_H,axiom,
! [B3: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A3 ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ B3 @ A3 ) ) ) ) ).
% size_mult_mono'
thf(fact_4859_size__mult__mono_H,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A3 ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ B3 @ A3 ) ) ) ) ).
% size_mult_mono'
thf(fact_4860_euclidean__size__times__unit,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
= ( euclid6377331345833325938nteger @ B3 ) ) ) ).
% euclidean_size_times_unit
thf(fact_4861_euclidean__size__times__unit,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ( euclid4774559944035922753ze_int @ ( times_times_int @ A3 @ B3 ) )
= ( euclid4774559944035922753ze_int @ B3 ) ) ) ).
% euclidean_size_times_unit
thf(fact_4862_euclidean__size__times__unit,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A3 @ B3 ) )
= ( euclid4777050414544973029ze_nat @ B3 ) ) ) ).
% euclidean_size_times_unit
thf(fact_4863_field__le__mult__one__interval,axiom,
! [X: real,Y: 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 ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_4864_field__le__mult__one__interval,axiom,
! [X: rat,Y: 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 ) @ Y ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_4865_mult__le__cancel__left1,axiom,
! [C: real,B3: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_4866_mult__le__cancel__left1,axiom,
! [C: rat,B3: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B3 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B3 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_4867_mult__le__cancel__left1,axiom,
! [C: int,B3: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B3 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_4868_mult__le__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_4869_mult__le__cancel__left2,axiom,
! [C: rat,A3: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_4870_mult__le__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_4871_mult__le__cancel__right1,axiom,
! [C: real,B3: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_4872_mult__le__cancel__right1,axiom,
! [C: rat,B3: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B3 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_4873_mult__le__cancel__right1,axiom,
! [C: int,B3: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B3 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B3 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_4874_mult__le__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_4875_mult__le__cancel__right2,axiom,
! [A3: rat,C: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A3 @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_4876_mult__le__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_4877_mult__less__cancel__left1,axiom,
! [C: real,B3: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_4878_mult__less__cancel__left1,axiom,
! [C: rat,B3: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ C @ B3 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B3 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_4879_mult__less__cancel__left1,axiom,
! [C: int,B3: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B3 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_4880_mult__less__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_4881_mult__less__cancel__left2,axiom,
! [C: rat,A3: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_4882_mult__less__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_4883_mult__less__cancel__right1,axiom,
! [C: real,B3: real] :
( ( ord_less_real @ C @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_4884_mult__less__cancel__right1,axiom,
! [C: rat,B3: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ B3 @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B3 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_4885_mult__less__cancel__right1,axiom,
! [C: int,B3: int] :
( ( ord_less_int @ C @ ( times_times_int @ B3 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B3 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_4886_mult__less__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_4887_mult__less__cancel__right2,axiom,
! [A3: rat,C: rat] :
( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A3 @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_4888_mult__less__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_4889_divide__le__eq,axiom,
! [B3: real,C: real,A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_4890_divide__le__eq,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_4891_le__divide__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_4892_le__divide__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_4893_divide__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% divide_left_mono
thf(fact_4894_divide__left__mono,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% divide_left_mono
thf(fact_4895_neg__divide__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).
% neg_divide_le_eq
thf(fact_4896_neg__divide__le__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
= ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).
% neg_divide_le_eq
thf(fact_4897_neg__le__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
= ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_4898_neg__le__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
= ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_4899_pos__divide__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
= ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_4900_pos__divide__le__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
= ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_4901_pos__le__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).
% pos_le_divide_eq
thf(fact_4902_pos__le__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).
% pos_le_divide_eq
thf(fact_4903_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_4904_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_4905_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_4906_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_4907_divide__left__mono__neg,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_4908_divide__left__mono__neg,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_4909_convex__bound__le,axiom,
! [X: real,A3: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A3 )
=> ( ( ord_less_eq_real @ Y @ A3 )
=> ( ( 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 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_4910_convex__bound__le,axiom,
! [X: rat,A3: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A3 )
=> ( ( ord_less_eq_rat @ Y @ A3 )
=> ( ( 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 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_4911_convex__bound__le,axiom,
! [X: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A3 )
=> ( ( ord_less_eq_int @ Y @ A3 )
=> ( ( 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 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_4912_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B3: rat,C: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_4913_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B3: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_4914_divide__less__eq__numeral_I1_J,axiom,
! [B3: rat,C: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_4915_divide__less__eq__numeral_I1_J,axiom,
! [B3: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_4916_frac__le__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_4917_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_4918_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_4919_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_4920_power__Suc__less,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ A3 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N ) ) @ ( power_power_real @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_4921_power__Suc__less,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( ord_less_rat @ A3 @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N ) ) @ ( power_power_rat @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_4922_power__Suc__less,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_4923_power__Suc__less,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_4924_pos__minus__divide__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_4925_pos__minus__divide__less__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
= ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_4926_pos__less__minus__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_4927_pos__less__minus__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_4928_neg__minus__divide__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
= ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_4929_neg__minus__divide__less__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
= ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_4930_neg__less__minus__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_4931_neg__less__minus__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_4932_minus__divide__less__eq,axiom,
! [B3: real,C: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_4933_minus__divide__less__eq,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_4934_less__minus__divide__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_4935_less__minus__divide__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_4936_mult__2,axiom,
! [Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2
thf(fact_4937_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_4938_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_4939_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_4940_mult__2,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2
thf(fact_4941_mult__2,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
= ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).
% mult_2
thf(fact_4942_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_4943_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_4944_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_4945_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_4946_mult__2__right,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4947_mult__2__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4948_left__add__twice,axiom,
! [A3: rat,B3: rat] :
( ( plus_plus_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).
% left_add_twice
thf(fact_4949_left__add__twice,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).
% left_add_twice
thf(fact_4950_left__add__twice,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).
% left_add_twice
thf(fact_4951_left__add__twice,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).
% left_add_twice
thf(fact_4952_left__add__twice,axiom,
! [A3: extended_enat,B3: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A3 @ ( plus_p3455044024723400733d_enat @ A3 @ B3 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).
% left_add_twice
thf(fact_4953_left__add__twice,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ A3 @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).
% left_add_twice
thf(fact_4954_divide__eq__eq__numeral_I2_J,axiom,
! [B3: real,C: real,W2: num] :
( ( ( divide_divide_real @ B3 @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( C != zero_zero_real )
=> ( B3
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_4955_divide__eq__eq__numeral_I2_J,axiom,
! [B3: rat,C: rat,W2: num] :
( ( ( divide_divide_rat @ B3 @ C )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( C != zero_zero_rat )
=> ( B3
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_4956_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B3: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ B3 @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
= B3 ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_4957_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B3: rat,C: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ B3 @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
= B3 ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_4958_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_4959_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A3: rat,B3: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_4960_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_4961_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_4962_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= ( uminus_uminus_real @ B3 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_4963_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A3: rat,B3: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
= ( uminus_uminus_rat @ B3 ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A3 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_4964_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A3: real,B3: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= ( uminus_uminus_real @ B3 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
= ( divide_divide_real @ ( minus_minus_real @ A3 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_4965_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A3: rat,B3: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
= ( uminus_uminus_rat @ B3 ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
= ( divide_divide_rat @ ( minus_minus_rat @ A3 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_4966_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_4967_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_4968_is__unitE,axiom,
! [A3: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ~ ( ( A3 != zero_zero_nat )
=> ! [B2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A3 )
= B2 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B2 )
= A3 )
=> ( ( ( times_times_nat @ A3 @ B2 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A3 )
!= ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4969_is__unitE,axiom,
! [A3: int,C: int] :
( ( dvd_dvd_int @ A3 @ one_one_int )
=> ~ ( ( A3 != zero_zero_int )
=> ! [B2: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A3 )
= B2 )
=> ( ( ( divide_divide_int @ one_one_int @ B2 )
= A3 )
=> ( ( ( times_times_int @ A3 @ B2 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A3 )
!= ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4970_is__unitE,axiom,
! [A3: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ~ ( ( A3 != zero_z3403309356797280102nteger )
=> ! [B2: code_integer] :
( ( B2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 )
= B2 )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
= A3 )
=> ( ( ( times_3573771949741848930nteger @ A3 @ B2 )
= one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ C @ A3 )
!= ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4971_is__unit__div__mult__cancel__left,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ A3 @ B3 ) )
= ( divide_divide_nat @ one_one_nat @ B3 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4972_is__unit__div__mult__cancel__left,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ A3 @ B3 ) )
= ( divide_divide_int @ one_one_int @ B3 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4973_is__unit__div__mult__cancel__left,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4974_is__unit__div__mult__cancel__right,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ A3 ) )
= ( divide_divide_nat @ one_one_nat @ B3 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4975_is__unit__div__mult__cancel__right,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ A3 ) )
= ( divide_divide_int @ one_one_int @ B3 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4976_is__unit__div__mult__cancel__right,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ A3 ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4977_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_4978_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_4979_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_4980_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_4981_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_4982_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ 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 @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_4983_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_4984_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_4985_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_4986_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_4987_convex__bound__lt,axiom,
! [X: real,A3: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X @ A3 )
=> ( ( ord_less_real @ Y @ A3 )
=> ( ( 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 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_4988_convex__bound__lt,axiom,
! [X: rat,A3: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X @ A3 )
=> ( ( ord_less_rat @ Y @ A3 )
=> ( ( 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 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_4989_convex__bound__lt,axiom,
! [X: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A3 )
=> ( ( ord_less_int @ Y @ A3 )
=> ( ( 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 @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_4990_divide__le__eq__numeral_I1_J,axiom,
! [B3: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ 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 @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_4991_divide__le__eq__numeral_I1_J,axiom,
! [B3: rat,C: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ 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 @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_4992_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B3: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_4993_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B3: rat,C: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_4994_le__minus__divide__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_4995_le__minus__divide__eq,axiom,
! [A3: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_4996_minus__divide__le__eq,axiom,
! [B3: real,C: real,A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_4997_minus__divide__le__eq,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_4998_neg__le__minus__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_4999_neg__le__minus__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5000_neg__minus__divide__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5001_neg__minus__divide__le__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
= ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5002_pos__le__minus__divide__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5003_pos__le__minus__divide__eq,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5004_pos__minus__divide__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5005_pos__minus__divide__le__eq,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5006_scaling__mono,axiom,
! [U: real,V: real,R2: real,S3: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_5007_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S3: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ( ord_less_eq_rat @ R2 @ S3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_5008_divide__less__eq__numeral_I2_J,axiom,
! [B3: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ 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 @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5009_divide__less__eq__numeral_I2_J,axiom,
! [B3: rat,C: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ 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 @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5010_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B3: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5011_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B3: rat,C: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5012_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_5013_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_5014_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_5015_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_5016_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_5017_power__minus__mult,axiom,
! [N: nat,A3: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( power_power_complex @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_complex @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_5018_power__minus__mult,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_real @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_5019_power__minus__mult,axiom,
! [N: nat,A3: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_rat @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_5020_power__minus__mult,axiom,
! [N: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_nat @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_5021_power__minus__mult,axiom,
! [N: nat,A3: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_int @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_5022_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 ) )
| ? [Q5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_5023_euclidean__size__times__nonunit,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3 != zero_z3403309356797280102nteger )
=> ( ( B3 != zero_z3403309356797280102nteger )
=> ( ~ ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
=> ( ord_less_nat @ ( euclid6377331345833325938nteger @ B3 ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_5024_euclidean__size__times__nonunit,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( B3 != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ A3 @ one_one_int )
=> ( ord_less_nat @ ( euclid4774559944035922753ze_int @ B3 ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_5025_euclidean__size__times__nonunit,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ B3 ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_5026_divide__le__eq__numeral_I2_J,axiom,
! [B3: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ 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 @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5027_divide__le__eq__numeral_I2_J,axiom,
! [B3: rat,C: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ 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 @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5028_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B3: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5029_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B3: rat,C: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5030_oddE,axiom,
! [A3: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ~ ! [B2: nat] :
( A3
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_5031_oddE,axiom,
! [A3: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ~ ! [B2: int] :
( A3
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_5032_oddE,axiom,
! [A3: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ~ ! [B2: code_integer] :
( A3
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).
% oddE
thf(fact_5033_zero__le__even__power_H,axiom,
! [A3: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_5034_zero__le__even__power_H,axiom,
! [A3: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_5035_zero__le__even__power_H,axiom,
! [A3: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_5036_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_5037_triangle__def,axiom,
( nat_triangle
= ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_5038_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_5039_sum__squares__bound,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_5040_sum__squares__bound,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_5041_odd__0__le__power__imp__0__le,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_5042_odd__0__le__power__imp__0__le,axiom,
! [A3: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_5043_odd__0__le__power__imp__0__le,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_5044_odd__power__less__zero,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_5045_odd__power__less__zero,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_5046_odd__power__less__zero,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_5047_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_5048_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_5049_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_5050_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_5051_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_5052_arith__geo__mean,axiom,
! [U: real,X: real,Y: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_5053_arith__geo__mean,axiom,
! [U: rat,X: rat,Y: rat] :
( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ X @ Y ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_5054_even__mult__exp__div__exp__iff,axiom,
! [A3: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A3 @ ( 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 @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_5055_even__mult__exp__div__exp__iff,axiom,
! [A3: int,M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A3 @ ( 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 @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_5056_even__mult__exp__div__exp__iff,axiom,
! [A3: code_integer,M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ ( 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 @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_5057_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma2: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi2 != Ma2 )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi2 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_5058_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma2: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi2 != Ma2 )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi2 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_5059_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% mul_def
thf(fact_5060_mul__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( times_times_nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( some_nat @ Z ) ) ) ).
% mul_shift
thf(fact_5061_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N3: nat,TreeList3: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X3 @ N3 ) ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) ) ) ).
% in_children_def
thf(fact_5062_divmod__step__eq,axiom,
! [L: num,R2: code_integer,Q4: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_5063_divmod__step__eq,axiom,
! [L: num,R2: nat,Q4: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_5064_divmod__step__eq,axiom,
! [L: num,R2: int,Q4: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_5065_set__bit__0,axiom,
! [A3: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A3 )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_5066_set__bit__0,axiom,
! [A3: code_integer] :
( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A3 )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_5067_set__bit__0,axiom,
! [A3: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A3 )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_5068_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_5069_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_5070_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_5071_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_5072_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_5073_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_5074_unset__bit__0,axiom,
! [A3: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A3 )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_5075_unset__bit__0,axiom,
! [A3: code_integer] :
( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A3 )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_5076_unset__bit__0,axiom,
! [A3: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A3 )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_5077_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_5078_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_5079_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_5080_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_5081_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_5082_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_5083_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_5084_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_5085_div__mult2__numeral__eq,axiom,
! [A3: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_5086_div__mult2__numeral__eq,axiom,
! [A3: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_5087_div__mult2__numeral__eq,axiom,
! [A3: code_integer,K: num,L: num] :
( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ L ) )
= ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_5088_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_5089_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_5090_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_5091_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times_num @ X @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_5092_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_5093_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_5094_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_5095_zdiv__zmult2__eq,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_5096_unique__quotient__lemma__neg,axiom,
! [B3: int,Q6: int,R4: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ R2 )
=> ( ( ord_less_int @ B3 @ R4 )
=> ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_5097_unique__quotient__lemma,axiom,
! [B3: int,Q6: int,R4: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B3 )
=> ( ( ord_less_int @ R2 @ B3 )
=> ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_5098_zdiv__mono2__neg__lemma,axiom,
! [B3: int,Q4: int,R2: int,B7: int,Q6: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B3 )
=> ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_5099_zdiv__mono2__lemma,axiom,
! [B3: int,Q4: int,R2: int,B7: int,Q6: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B7 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B3 )
=> ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_5100_q__pos__lemma,axiom,
! [B7: int,Q6: int,R4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B7 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_5101_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ 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_5102_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ 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_5103_int__div__pos__eq,axiom,
! [A3: int,B3: int,Q4: int,R2: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B3 )
=> ( ( divide_divide_int @ A3 @ B3 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_5104_int__div__neg__eq,axiom,
! [A3: int,B3: int,Q4: int,R2: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ R2 )
=> ( ( divide_divide_int @ A3 @ B3 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_5105_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_5106_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_5107_cppi,axiom,
! [D6: int,P: int > $o,P4: int > $o,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D6 ) ) ) )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ( P4 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ A4 )
& ( P @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_5108_cpmi,axiom,
! [D6: int,P: int > $o,P4: int > $o,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D6 ) ) ) )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ( P4 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( P @ ( plus_plus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_5109_L2__set__mult__ineq__lemma,axiom,
! [A3: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A3 @ C ) ) @ ( times_times_real @ B3 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_5110_neg__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
= ( divide_divide_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_5111_pos__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
= ( divide_divide_int @ B3 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_5112_even__unset__bit__iff,axiom,
! [M: nat,A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A3 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_5113_even__unset__bit__iff,axiom,
! [M: nat,A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A3 ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_5114_even__unset__bit__iff,axiom,
! [M: nat,A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A3 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_5115_even__set__bit__iff,axiom,
! [M: nat,A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A3 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_5116_even__set__bit__iff,axiom,
! [M: nat,A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A3 ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_5117_even__set__bit__iff,axiom,
! [M: nat,A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A3 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_5118_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_5119_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_5120_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_5121_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_5122_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_5123_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_5124_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma2: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma2 )
=> ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ Mi2 @ X ) @ ( some_nat @ Mi2 ) @ 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 @ TreeList @ ( 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_5125_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma2: nat,Mi2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ Mi2 @ X ) @ ( some_nat @ Mi2 ) @ 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 @ TreeList @ ( 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_5126_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( Y != none_nat ) ) )
=> ( ! [A2: $o] :
( ? [Uw2: $o] :
( X
= ( vEBT_Leaf @ A2 @ Uw2 ) )
=> ( ( Xa2
= ( suc @ zero_zero_nat ) )
=> ~ ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( Y = none_nat ) ) ) ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ? [Va2: nat] :
( Xa2
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( B2
=> ( Y
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( Y = none_nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less_nat @ Ma @ Xa2 )
=> ( Y
= ( some_nat @ Ma ) ) )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( Y
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ Xa2 ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_5127_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A3 @ one_one_int ) @ B3 @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_5128_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X3: nat,N3: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% low_def
thf(fact_5129_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_5130_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_5131_mod__mod__trivial,axiom,
! [A3: nat,B3: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_mod_trivial
thf(fact_5132_mod__mod__trivial,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_mod_trivial
thf(fact_5133_mod__mod__trivial,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_mod_trivial
thf(fact_5134_finite__Collect__disjI,axiom,
! [P: real > $o,Q: real > $o] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X3: real] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_real @ ( collect_real @ P ) )
& ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5135_finite__Collect__disjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5136_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5137_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5138_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5139_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5140_finite__Collect__conjI,axiom,
! [P: real > $o,Q: real > $o] :
( ( ( finite_finite_real @ ( collect_real @ P ) )
| ( finite_finite_real @ ( collect_real @ Q ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X3: real] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5141_finite__Collect__conjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5142_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5143_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5144_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5145_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5146_bits__mod__0,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_5147_bits__mod__0,axiom,
! [A3: int] :
( ( modulo_modulo_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_5148_bits__mod__0,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A3 )
= zero_z3403309356797280102nteger ) ).
% bits_mod_0
thf(fact_5149_mod__self,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% mod_self
thf(fact_5150_mod__self,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ A3 )
= zero_zero_int ) ).
% mod_self
thf(fact_5151_mod__self,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ A3 @ A3 )
= zero_z3403309356797280102nteger ) ).
% mod_self
thf(fact_5152_mod__by__0,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% mod_by_0
thf(fact_5153_mod__by__0,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ zero_zero_int )
= A3 ) ).
% mod_by_0
thf(fact_5154_mod__by__0,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ A3 @ zero_z3403309356797280102nteger )
= A3 ) ).
% mod_by_0
thf(fact_5155_mod__0,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mod_0
thf(fact_5156_mod__0,axiom,
! [A3: int] :
( ( modulo_modulo_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mod_0
thf(fact_5157_mod__0,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A3 )
= zero_z3403309356797280102nteger ) ).
% mod_0
thf(fact_5158_mod__add__self1,axiom,
! [B3: nat,A3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_add_self1
thf(fact_5159_mod__add__self1,axiom,
! [B3: int,A3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_add_self1
thf(fact_5160_mod__add__self1,axiom,
! [B3: code_integer,A3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B3 @ A3 ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_add_self1
thf(fact_5161_mod__add__self2,axiom,
! [A3: nat,B3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_add_self2
thf(fact_5162_mod__add__self2,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_add_self2
thf(fact_5163_mod__add__self2,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_add_self2
thf(fact_5164_minus__mod__self2,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% minus_mod_self2
thf(fact_5165_minus__mod__self2,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% minus_mod_self2
thf(fact_5166_mod__minus__minus,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).
% mod_minus_minus
thf(fact_5167_mod__minus__minus,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( uminus1351360451143612070nteger @ B3 ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).
% mod_minus_minus
thf(fact_5168_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_5169_finite__Collect__subsets,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_5170_finite__Collect__subsets,axiom,
! [A4: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_5171_finite__Collect__subsets,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_5172_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_5173_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_5174_finite__interval__int1,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A3 @ I3 )
& ( ord_less_eq_int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int1
thf(fact_5175_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A3 @ I3 )
& ( ord_less_int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_5176_mod__mult__self2__is__0,axiom,
! [A3: nat,B3: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ B3 )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_5177_mod__mult__self2__is__0,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ B3 )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_5178_mod__mult__self2__is__0,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ B3 )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self2_is_0
thf(fact_5179_mod__mult__self1__is__0,axiom,
! [B3: nat,A3: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B3 @ A3 ) @ B3 )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_5180_mod__mult__self1__is__0,axiom,
! [B3: int,A3: int] :
( ( modulo_modulo_int @ ( times_times_int @ B3 @ A3 ) @ B3 )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_5181_mod__mult__self1__is__0,axiom,
! [B3: code_integer,A3: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B3 @ A3 ) @ B3 )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self1_is_0
thf(fact_5182_bits__mod__by__1,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_5183_bits__mod__by__1,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_5184_bits__mod__by__1,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ A3 @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_5185_mod__by__1,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_5186_mod__by__1,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_5187_mod__by__1,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ A3 @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% mod_by_1
thf(fact_5188_mod__div__trivial,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_5189_mod__div__trivial,axiom,
! [A3: int,B3: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_5190_mod__div__trivial,axiom,
! [A3: code_integer,B3: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 )
= zero_z3403309356797280102nteger ) ).
% mod_div_trivial
thf(fact_5191_bits__mod__div__trivial,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_5192_bits__mod__div__trivial,axiom,
! [A3: int,B3: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_5193_bits__mod__div__trivial,axiom,
! [A3: code_integer,B3: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 )
= zero_z3403309356797280102nteger ) ).
% bits_mod_div_trivial
thf(fact_5194_mod__mult__self1,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_mult_self1
thf(fact_5195_mod__mult__self1,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( times_times_int @ C @ B3 ) ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_mult_self1
thf(fact_5196_mod__mult__self1,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_mult_self1
thf(fact_5197_mod__mult__self2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_mult_self2
thf(fact_5198_mod__mult__self2,axiom,
! [A3: int,B3: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( times_times_int @ B3 @ C ) ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_mult_self2
thf(fact_5199_mod__mult__self2,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_mult_self2
thf(fact_5200_mod__mult__self3,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A3 ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_mult_self3
thf(fact_5201_mod__mult__self3,axiom,
! [C: int,B3: int,A3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A3 ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_mult_self3
thf(fact_5202_mod__mult__self3,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B3 ) @ A3 ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_mult_self3
thf(fact_5203_mod__mult__self4,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A3 ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% mod_mult_self4
thf(fact_5204_mod__mult__self4,axiom,
! [B3: int,C: int,A3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A3 ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% mod_mult_self4
thf(fact_5205_mod__mult__self4,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ C ) @ A3 ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% mod_mult_self4
thf(fact_5206_dvd__imp__mod__0,axiom,
! [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
=> ( ( modulo_modulo_nat @ B3 @ A3 )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_5207_dvd__imp__mod__0,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd_int @ A3 @ B3 )
=> ( ( modulo_modulo_int @ B3 @ A3 )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_5208_dvd__imp__mod__0,axiom,
! [A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ A3 @ B3 )
=> ( ( modulo364778990260209775nteger @ B3 @ A3 )
= zero_z3403309356797280102nteger ) ) ).
% dvd_imp_mod_0
thf(fact_5209_minus__mod__self1,axiom,
! [B3: int,A3: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B3 @ A3 ) @ B3 )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).
% minus_mod_self1
thf(fact_5210_minus__mod__self1,axiom,
! [B3: code_integer,A3: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B3 @ A3 ) @ B3 )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).
% minus_mod_self1
thf(fact_5211_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_5212_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_5213_finite__interval__int3,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A3 @ I3 )
& ( ord_less_eq_int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int3
thf(fact_5214_finite__interval__int2,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A3 @ I3 )
& ( ord_less_int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int2
thf(fact_5215_mod__minus1__right,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_5216_mod__minus1__right,axiom,
! [A3: code_integer] :
( ( modulo364778990260209775nteger @ A3 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_5217_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_5218_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_5219_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_5220_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_5221_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_5222_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_5223_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_5224_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_5225_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_5226_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_5227_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_5228_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_5229_not__mod__2__eq__0__eq__1,axiom,
! [A3: nat] :
( ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_5230_not__mod__2__eq__0__eq__1,axiom,
! [A3: int] :
( ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_5231_not__mod__2__eq__0__eq__1,axiom,
! [A3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_5232_not__mod__2__eq__1__eq__0,axiom,
! [A3: nat] :
( ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_5233_not__mod__2__eq__1__eq__0,axiom,
! [A3: int] :
( ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_5234_not__mod__2__eq__1__eq__0,axiom,
! [A3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer )
= ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_5235_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_5236_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_5237_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_5238_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_5239_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_5240_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_5241_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_5242_even__succ__mod__exp,axiom,
! [A3: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5243_even__succ__mod__exp,axiom,
! [A3: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5244_even__succ__mod__exp,axiom,
! [A3: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5245_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A: real] :
( ( member_real @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5246_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A: real] :
( ( member_real @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5247_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_complex,R: real > complex > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A: real] :
( ( member_real @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5248_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A: nat] :
( ( member_nat @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5249_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A: nat] :
( ( member_nat @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5250_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_complex,R: nat > complex > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A: nat] :
( ( member_nat @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5251_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A: int] :
( ( member_int @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5252_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A: int] :
( ( member_int @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5253_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_complex,R: int > complex > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A: int] :
( ( member_int @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5254_pigeonhole__infinite__rel,axiom,
! [A4: set_complex,B5: set_nat,R: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A: complex] :
( ( member_complex @ A @ A4 )
& ( R @ A @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5255_not__finite__existsD,axiom,
! [P: real > $o] :
( ~ ( finite_finite_real @ ( collect_real @ P ) )
=> ? [X_1: real] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_5256_not__finite__existsD,axiom,
! [P: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_5257_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_5258_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_5259_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_5260_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_1: complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_5261_less__set__def,axiom,
( ord_le7866589430770878221at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ord_le549003669493604880_nat_o
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_5262_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A6 )
@ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_5263_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_5264_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_5265_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A6 )
@ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_5266_subset__Collect__iff,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ B5
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5267_subset__Collect__iff,axiom,
! [B5: set_real,A4: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B5 @ A4 )
=> ( ( ord_less_eq_set_real @ B5
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5268_subset__Collect__iff,axiom,
! [B5: set_list_nat,A4: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ B5
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5269_subset__Collect__iff,axiom,
! [B5: set_set_nat,A4: set_set_nat,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ B5
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5270_subset__Collect__iff,axiom,
! [B5: set_nat,A4: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( ord_less_eq_set_nat @ B5
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5271_subset__Collect__iff,axiom,
! [B5: set_int,A4: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( ord_less_eq_set_int @ B5
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5272_subset__CollectI,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5273_subset__CollectI,axiom,
! [B5: set_real,A4: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B5 @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5274_subset__CollectI,axiom,
! [B5: set_list_nat,A4: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5275_subset__CollectI,axiom,
! [B5: set_set_nat,A4: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5276_subset__CollectI,axiom,
! [B5: set_nat,A4: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5277_subset__CollectI,axiom,
! [B5: set_int,A4: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5278_pred__subset__eq,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le704812498762024988_nat_o
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ R )
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ S2 ) )
= ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_5279_pred__subset__eq,axiom,
! [R: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ R )
@ ^ [X3: real] : ( member_real @ X3 @ S2 ) )
= ( ord_less_eq_set_real @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_5280_pred__subset__eq,axiom,
! [R: set_set_nat,S2: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S2 ) )
= ( ord_le6893508408891458716et_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_5281_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_5282_pred__subset__eq,axiom,
! [R: set_int,S2: set_int] :
( ( ord_less_eq_int_o
@ ^ [X3: int] : ( member_int @ X3 @ R )
@ ^ [X3: int] : ( member_int @ X3 @ S2 ) )
= ( ord_less_eq_set_int @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_5283_Collect__subset,axiom,
! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_5284_Collect__subset,axiom,
! [A4: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_5285_Collect__subset,axiom,
! [A4: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_5286_Collect__subset,axiom,
! [A4: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_5287_Collect__subset,axiom,
! [A4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_5288_Collect__subset,axiom,
! [A4: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_5289_less__eq__set__def,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ord_le704812498762024988_nat_o
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_5290_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A6 )
@ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_5291_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_5292_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_5293_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A6 )
@ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_5294_Collect__restrict,axiom,
! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_5295_Collect__restrict,axiom,
! [X7: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_5296_Collect__restrict,axiom,
! [X7: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_5297_Collect__restrict,axiom,
! [X7: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_5298_Collect__restrict,axiom,
! [X7: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_5299_Collect__restrict,axiom,
! [X7: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_5300_prop__restrict,axiom,
! [X: product_prod_nat_nat,Z6: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ Z6 )
=> ( ( ord_le3146513528884898305at_nat @ Z6
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_5301_prop__restrict,axiom,
! [X: real,Z6: set_real,X7: set_real,P: real > $o] :
( ( member_real @ X @ Z6 )
=> ( ( ord_less_eq_set_real @ Z6
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_5302_prop__restrict,axiom,
! [X: list_nat,Z6: set_list_nat,X7: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X @ Z6 )
=> ( ( ord_le6045566169113846134st_nat @ Z6
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_5303_prop__restrict,axiom,
! [X: set_nat,Z6: set_set_nat,X7: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ Z6 )
=> ( ( ord_le6893508408891458716et_nat @ Z6
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_5304_prop__restrict,axiom,
! [X: nat,Z6: set_nat,X7: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z6 )
=> ( ( ord_less_eq_set_nat @ Z6
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_5305_prop__restrict,axiom,
! [X: int,Z6: set_int,X7: set_int,P: int > $o] :
( ( member_int @ X @ Z6 )
=> ( ( ord_less_eq_set_int @ Z6
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_5306_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X3: list_nat] : $false ) ) ).
% empty_def
thf(fact_5307_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X3: set_nat] : $false ) ) ).
% empty_def
thf(fact_5308_empty__def,axiom,
( bot_bot_set_real
= ( collect_real
@ ^ [X3: real] : $false ) ) ).
% empty_def
thf(fact_5309_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_5310_empty__def,axiom,
( bot_bot_set_int
= ( collect_int
@ ^ [X3: int] : $false ) ) ).
% empty_def
thf(fact_5311_lambda__zero,axiom,
( ( ^ [H3: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_5312_lambda__zero,axiom,
( ( ^ [H3: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_5313_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_5314_lambda__zero,axiom,
( ( ^ [H3: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_5315_lambda__one,axiom,
( ( ^ [X3: complex] : X3 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_5316_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_5317_lambda__one,axiom,
( ( ^ [X3: rat] : X3 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_5318_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_5319_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_5320_subset__divisors__dvd,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [C4: real] : ( dvd_dvd_real @ C4 @ A3 ) )
@ ( collect_real
@ ^ [C4: real] : ( dvd_dvd_real @ C4 @ B3 ) ) )
= ( dvd_dvd_real @ A3 @ B3 ) ) ).
% subset_divisors_dvd
thf(fact_5321_subset__divisors__dvd,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A3 ) )
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B3 ) ) )
= ( dvd_dvd_nat @ A3 @ B3 ) ) ).
% subset_divisors_dvd
thf(fact_5322_subset__divisors__dvd,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le7084787975880047091nteger
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A3 ) )
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B3 ) ) )
= ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).
% subset_divisors_dvd
thf(fact_5323_subset__divisors__dvd,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A3 ) )
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B3 ) ) )
= ( dvd_dvd_int @ A3 @ B3 ) ) ).
% subset_divisors_dvd
thf(fact_5324_pred__subset__eq2,axiom,
! [R: set_Pr4811707699266497531nteger,S2: set_Pr4811707699266497531nteger] :
( ( ord_le3602516367967493612eger_o
@ ^ [X3: code_integer,Y3: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y3 ) @ R )
@ ^ [X3: code_integer,Y3: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y3 ) @ S2 ) )
= ( ord_le3725938330318615451nteger @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_5325_pred__subset__eq2,axiom,
! [R: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
( ( ord_le2162486998276636481er_o_o
@ ^ [X3: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X3 @ Y3 ) @ R )
@ ^ [X3: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X3 @ Y3 ) @ S2 ) )
= ( ord_le8980329558974975238eger_o @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_5326_pred__subset__eq2,axiom,
! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
( ( ord_le5604493270027003598_nat_o
@ ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ S2 ) )
= ( ord_le3000389064537975527at_nat @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_5327_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
= ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_5328_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X3: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
@ ^ [X3: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
= ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_5329_strict__subset__divisors__dvd,axiom,
! [A3: real,B3: real] :
( ( ord_less_set_real
@ ( collect_real
@ ^ [C4: real] : ( dvd_dvd_real @ C4 @ A3 ) )
@ ( collect_real
@ ^ [C4: real] : ( dvd_dvd_real @ C4 @ B3 ) ) )
= ( ( dvd_dvd_real @ A3 @ B3 )
& ~ ( dvd_dvd_real @ B3 @ A3 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5330_strict__subset__divisors__dvd,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A3 ) )
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B3 ) ) )
= ( ( dvd_dvd_nat @ A3 @ B3 )
& ~ ( dvd_dvd_nat @ B3 @ A3 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5331_strict__subset__divisors__dvd,axiom,
! [A3: int,B3: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A3 ) )
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B3 ) ) )
= ( ( dvd_dvd_int @ A3 @ B3 )
& ~ ( dvd_dvd_int @ B3 @ A3 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5332_strict__subset__divisors__dvd,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le1307284697595431911nteger
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A3 ) )
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B3 ) ) )
= ( ( dvd_dvd_Code_integer @ A3 @ B3 )
& ~ ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5333_bot__empty__eq2,axiom,
( bot_bo8134993004553108152eger_o
= ( ^ [X3: code_integer,Y3: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y3 ) @ bot_bo4276436098303576167nteger ) ) ) ).
% bot_empty_eq2
thf(fact_5334_bot__empty__eq2,axiom,
( bot_bo4731626569425807221er_o_o
= ( ^ [X3: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X3 @ Y3 ) @ bot_bo5379713665208646970eger_o ) ) ) ).
% bot_empty_eq2
thf(fact_5335_bot__empty__eq2,axiom,
( bot_bo4898103413517107610_nat_o
= ( ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ bot_bo5327735625951526323at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_5336_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_5337_bot__empty__eq2,axiom,
( bot_bot_int_int_o
= ( ^ [X3: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_5338_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_5339_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_5340_mod__mult__eq,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ).
% mod_mult_eq
thf(fact_5341_mod__mult__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ).
% mod_mult_eq
thf(fact_5342_mod__mult__eq,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ).
% mod_mult_eq
thf(fact_5343_mod__mult__cong,axiom,
! [A3: nat,C: nat,A7: nat,B3: nat,B7: nat] :
( ( ( modulo_modulo_nat @ A3 @ C )
= ( modulo_modulo_nat @ A7 @ C ) )
=> ( ( ( modulo_modulo_nat @ B3 @ C )
= ( modulo_modulo_nat @ B7 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5344_mod__mult__cong,axiom,
! [A3: int,C: int,A7: int,B3: int,B7: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ A7 @ C ) )
=> ( ( ( modulo_modulo_int @ B3 @ C )
= ( modulo_modulo_int @ B7 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5345_mod__mult__cong,axiom,
! [A3: code_integer,C: code_integer,A7: code_integer,B3: code_integer,B7: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ C )
= ( modulo364778990260209775nteger @ A7 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B3 @ C )
= ( modulo364778990260209775nteger @ B7 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5346_mod__mult__mult2,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5347_mod__mult__mult2,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A3 @ B3 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5348_mod__mult__mult2,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5349_mult__mod__right,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A3 @ B3 ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ).
% mult_mod_right
thf(fact_5350_mult__mod__right,axiom,
! [C: int,A3: int,B3: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A3 @ B3 ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ).
% mult_mod_right
thf(fact_5351_mult__mod__right,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) ) ) ).
% mult_mod_right
thf(fact_5352_mod__mult__left__eq,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A3 @ C ) @ B3 ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5353_mod__mult__left__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A3 @ C ) @ B3 ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5354_mod__mult__left__eq,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ B3 ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5355_mod__mult__right__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5356_mod__mult__right__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5357_mod__mult__right__eq,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5358_mod__add__eq,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C ) ) ).
% mod_add_eq
thf(fact_5359_mod__add__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).
% mod_add_eq
thf(fact_5360_mod__add__eq,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C ) ) ).
% mod_add_eq
thf(fact_5361_mod__add__cong,axiom,
! [A3: nat,C: nat,A7: nat,B3: nat,B7: nat] :
( ( ( modulo_modulo_nat @ A3 @ C )
= ( modulo_modulo_nat @ A7 @ C ) )
=> ( ( ( modulo_modulo_nat @ B3 @ C )
= ( modulo_modulo_nat @ B7 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_5362_mod__add__cong,axiom,
! [A3: int,C: int,A7: int,B3: int,B7: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ A7 @ C ) )
=> ( ( ( modulo_modulo_int @ B3 @ C )
= ( modulo_modulo_int @ B7 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_5363_mod__add__cong,axiom,
! [A3: code_integer,C: code_integer,A7: code_integer,B3: code_integer,B7: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ C )
= ( modulo364778990260209775nteger @ A7 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B3 @ C )
= ( modulo364778990260209775nteger @ B7 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_5364_mod__add__left__eq,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ B3 ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_5365_mod__add__left__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ B3 ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_5366_mod__add__left__eq,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ B3 ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_5367_mod__add__right__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_5368_mod__add__right__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_5369_mod__add__right__eq,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_5370_mod__diff__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).
% mod_diff_eq
thf(fact_5371_mod__diff__eq,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C ) ) ).
% mod_diff_eq
thf(fact_5372_mod__diff__cong,axiom,
! [A3: int,C: int,A7: int,B3: int,B7: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ A7 @ C ) )
=> ( ( ( modulo_modulo_int @ B3 @ C )
= ( modulo_modulo_int @ B7 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_5373_mod__diff__cong,axiom,
! [A3: code_integer,C: code_integer,A7: code_integer,B3: code_integer,B7: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ C )
= ( modulo364778990260209775nteger @ A7 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B3 @ C )
= ( modulo364778990260209775nteger @ B7 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A7 @ B7 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_5374_mod__diff__left__eq,axiom,
! [A3: int,C: int,B3: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A3 @ C ) @ B3 ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_5375_mod__diff__left__eq,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ B3 ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_5376_mod__diff__right__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_5377_mod__diff__right__eq,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_5378_mod__minus__eq,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ B3 ) ) @ B3 )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).
% mod_minus_eq
thf(fact_5379_mod__minus__eq,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ B3 )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).
% mod_minus_eq
thf(fact_5380_mod__minus__cong,axiom,
! [A3: int,B3: int,A7: int] :
( ( ( modulo_modulo_int @ A3 @ B3 )
= ( modulo_modulo_int @ A7 @ B3 ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A7 ) @ B3 ) ) ) ).
% mod_minus_cong
thf(fact_5381_mod__minus__cong,axiom,
! [A3: code_integer,B3: code_integer,A7: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ B3 )
= ( modulo364778990260209775nteger @ A7 @ B3 ) )
=> ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A7 ) @ B3 ) ) ) ).
% mod_minus_cong
thf(fact_5382_mod__minus__right,axiom,
! [A3: int,B3: int] :
( ( modulo_modulo_int @ A3 @ ( uminus_uminus_int @ B3 ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).
% mod_minus_right
thf(fact_5383_mod__minus__right,axiom,
! [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ) ).
% mod_minus_right
thf(fact_5384_power__mod,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ N ) @ B3 )
= ( modulo_modulo_nat @ ( power_power_nat @ A3 @ N ) @ B3 ) ) ).
% power_mod
thf(fact_5385_power__mod,axiom,
! [A3: int,B3: int,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A3 @ B3 ) @ N ) @ B3 )
= ( modulo_modulo_int @ ( power_power_int @ A3 @ N ) @ B3 ) ) ).
% power_mod
thf(fact_5386_power__mod,axiom,
! [A3: code_integer,B3: code_integer,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ N ) @ B3 )
= ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A3 @ N ) @ B3 ) ) ).
% power_mod
thf(fact_5387_dvd__mod__iff,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B3 )
=> ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A3 @ B3 ) )
= ( dvd_dvd_nat @ C @ A3 ) ) ) ).
% dvd_mod_iff
thf(fact_5388_dvd__mod__iff,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ B3 )
=> ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A3 @ B3 ) )
= ( dvd_dvd_int @ C @ A3 ) ) ) ).
% dvd_mod_iff
thf(fact_5389_dvd__mod__iff,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
= ( dvd_dvd_Code_integer @ C @ A3 ) ) ) ).
% dvd_mod_iff
thf(fact_5390_dvd__mod__imp__dvd,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A3 @ B3 ) )
=> ( ( dvd_dvd_nat @ C @ B3 )
=> ( dvd_dvd_nat @ C @ A3 ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_5391_dvd__mod__imp__dvd,axiom,
! [C: int,A3: int,B3: int] :
( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A3 @ B3 ) )
=> ( ( dvd_dvd_int @ C @ B3 )
=> ( dvd_dvd_int @ C @ A3 ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_5392_dvd__mod__imp__dvd,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
=> ( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( dvd_dvd_Code_integer @ C @ A3 ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_5393_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_5394_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_5395_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_5396_mod__mod__cancel,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B3 )
=> ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ C )
= ( modulo_modulo_nat @ A3 @ C ) ) ) ).
% mod_mod_cancel
thf(fact_5397_mod__mod__cancel,axiom,
! [C: int,B3: int,A3: int] :
( ( dvd_dvd_int @ C @ B3 )
=> ( ( modulo_modulo_int @ ( modulo_modulo_int @ A3 @ B3 ) @ C )
= ( modulo_modulo_int @ A3 @ C ) ) ) ).
% mod_mod_cancel
thf(fact_5398_mod__mod__cancel,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B3 )
=> ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ C )
= ( modulo364778990260209775nteger @ A3 @ C ) ) ) ).
% mod_mod_cancel
thf(fact_5399_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_5400_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_5401_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_5402_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_5403_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_5404_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_5405_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_5406_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_code(2)
thf(fact_5407_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_code(2)
thf(fact_5408_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_5409_finite__int__segment,axiom,
! [A3: real,B3: real] :
( finite_finite_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ ring_1_Ints_real )
& ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 ) ) ) ) ).
% finite_int_segment
thf(fact_5410_finite__int__segment,axiom,
! [A3: rat,B3: rat] :
( finite_finite_rat
@ ( collect_rat
@ ^ [X3: rat] :
( ( member_rat @ X3 @ ring_1_Ints_rat )
& ( ord_less_eq_rat @ A3 @ X3 )
& ( ord_less_eq_rat @ X3 @ B3 ) ) ) ) ).
% finite_int_segment
thf(fact_5411_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_5412_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_5413_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_5414_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ A3 ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5415_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ A3 ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5416_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ A3 @ B3 ) @ A3 ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5417_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5418_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5419_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5420_mod__eq__self__iff__div__eq__0,axiom,
! [A3: nat,B3: nat] :
( ( ( modulo_modulo_nat @ A3 @ B3 )
= A3 )
= ( ( divide_divide_nat @ A3 @ B3 )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_5421_mod__eq__self__iff__div__eq__0,axiom,
! [A3: int,B3: int] :
( ( ( modulo_modulo_int @ A3 @ B3 )
= A3 )
= ( ( divide_divide_int @ A3 @ B3 )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_5422_mod__eq__self__iff__div__eq__0,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ B3 )
= A3 )
= ( ( divide6298287555418463151nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_5423_mod__eqE,axiom,
! [A3: int,C: int,B3: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ B3 @ C ) )
=> ~ ! [D4: int] :
( B3
!= ( plus_plus_int @ A3 @ ( times_times_int @ C @ D4 ) ) ) ) ).
% mod_eqE
thf(fact_5424_mod__eqE,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ C )
= ( modulo364778990260209775nteger @ B3 @ C ) )
=> ~ ! [D4: code_integer] :
( B3
!= ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).
% mod_eqE
thf(fact_5425_mod__0__imp__dvd,axiom,
! [A3: nat,B3: nat] :
( ( ( modulo_modulo_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( dvd_dvd_nat @ B3 @ A3 ) ) ).
% mod_0_imp_dvd
thf(fact_5426_mod__0__imp__dvd,axiom,
! [A3: int,B3: int] :
( ( ( modulo_modulo_int @ A3 @ B3 )
= zero_zero_int )
=> ( dvd_dvd_int @ B3 @ A3 ) ) ).
% mod_0_imp_dvd
thf(fact_5427_mod__0__imp__dvd,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger )
=> ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).
% mod_0_imp_dvd
thf(fact_5428_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A: nat,B: nat] :
( ( modulo_modulo_nat @ B @ A )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_5429_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A: int,B: int] :
( ( modulo_modulo_int @ B @ A )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_5430_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_Code_integer
= ( ^ [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ B @ A )
= zero_z3403309356797280102nteger ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_5431_mod__eq__0__iff__dvd,axiom,
! [A3: nat,B3: nat] :
( ( ( modulo_modulo_nat @ A3 @ B3 )
= zero_zero_nat )
= ( dvd_dvd_nat @ B3 @ A3 ) ) ).
% mod_eq_0_iff_dvd
thf(fact_5432_mod__eq__0__iff__dvd,axiom,
! [A3: int,B3: int] :
( ( ( modulo_modulo_int @ A3 @ B3 )
= zero_zero_int )
= ( dvd_dvd_int @ B3 @ A3 ) ) ).
% mod_eq_0_iff_dvd
thf(fact_5433_mod__eq__0__iff__dvd,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger )
= ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).
% mod_eq_0_iff_dvd
thf(fact_5434_div__add1__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B3 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_5435_div__add1__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_5436_div__add1__eq,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_5437_dvd__minus__mod,axiom,
! [B3: nat,A3: nat] : ( dvd_dvd_nat @ B3 @ ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ).
% dvd_minus_mod
thf(fact_5438_dvd__minus__mod,axiom,
! [B3: int,A3: int] : ( dvd_dvd_int @ B3 @ ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).
% dvd_minus_mod
thf(fact_5439_dvd__minus__mod,axiom,
! [B3: code_integer,A3: code_integer] : ( dvd_dvd_Code_integer @ B3 @ ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).
% dvd_minus_mod
thf(fact_5440_mod__eq__dvd__iff,axiom,
! [A3: int,C: int,B3: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ B3 @ C ) )
= ( dvd_dvd_int @ C @ ( minus_minus_int @ A3 @ B3 ) ) ) ).
% mod_eq_dvd_iff
thf(fact_5441_mod__eq__dvd__iff,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer] :
( ( ( modulo364778990260209775nteger @ A3 @ C )
= ( modulo364778990260209775nteger @ B3 @ C ) )
= ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ).
% mod_eq_dvd_iff
thf(fact_5442_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_5443_mod__induct,axiom,
! [P: nat > $o,N: nat,P6: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P6 )
=> ( ( ord_less_nat @ M @ P6 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P6 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P6 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_5444_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_5445_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo_nat @ M3 @ N2 ) )
=> ( P @ M3 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_5446_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_5447_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_5448_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_5449_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N3 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).
% mod_if
thf(fact_5450_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_5451_finite__lists__length__eq,axiom,
! [A4: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs2: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
& ( ( size_s3451745648224563538omplex @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5452_finite__lists__length__eq,axiom,
! [A4: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs2: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
& ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5453_finite__lists__length__eq,axiom,
! [A4: set_o,N: nat] :
( ( finite_finite_o @ A4 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs2: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 )
& ( ( size_size_list_o @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5454_finite__lists__length__eq,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
& ( ( size_size_list_nat @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5455_finite__lists__length__eq,axiom,
! [A4: set_int,N: nat] :
( ( finite_finite_int @ A4 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs2: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
& ( ( size_size_list_int @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5456_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
=> ( ( modulo364778990260209775nteger @ A3 @ B3 )
= A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5457_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ B3 )
=> ( ( modulo_modulo_nat @ A3 @ B3 )
= A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5458_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ B3 )
=> ( ( modulo_modulo_int @ A3 @ B3 )
= A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5459_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5460_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5461_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5462_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q4: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_5463_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q4: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_5464_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q4: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
thf(fact_5465_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_5466_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_5467_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(1)
thf(fact_5468_finite__lists__length__le,axiom,
! [A4: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs2: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5469_finite__lists__length__le,axiom,
! [A4: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs2: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5470_finite__lists__length__le,axiom,
! [A4: set_o,N: nat] :
( ( finite_finite_o @ A4 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs2: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5471_finite__lists__length__le,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5472_finite__lists__length__le,axiom,
! [A4: set_int,N: nat] :
( ( finite_finite_int @ A4 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs2: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5473_cancel__div__mod__rules_I2_J,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) @ C )
= ( plus_plus_nat @ A3 @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5474_cancel__div__mod__rules_I2_J,axiom,
! [B3: int,A3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) @ ( modulo_modulo_int @ A3 @ B3 ) ) @ C )
= ( plus_plus_int @ A3 @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5475_cancel__div__mod__rules_I2_J,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ C )
= ( plus_p5714425477246183910nteger @ A3 @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5476_cancel__div__mod__rules_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) @ C )
= ( plus_plus_nat @ A3 @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5477_cancel__div__mod__rules_I1_J,axiom,
! [A3: int,B3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_int @ A3 @ B3 ) ) @ C )
= ( plus_plus_int @ A3 @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5478_cancel__div__mod__rules_I1_J,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ C )
= ( plus_p5714425477246183910nteger @ A3 @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5479_mod__div__decomp,axiom,
! [A3: nat,B3: nat] :
( A3
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ).
% mod_div_decomp
thf(fact_5480_mod__div__decomp,axiom,
! [A3: int,B3: int] :
( A3
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).
% mod_div_decomp
thf(fact_5481_mod__div__decomp,axiom,
! [A3: code_integer,B3: code_integer] :
( A3
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).
% mod_div_decomp
thf(fact_5482_div__mult__mod__eq,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_nat @ A3 @ B3 ) )
= A3 ) ).
% div_mult_mod_eq
thf(fact_5483_div__mult__mod__eq,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_int @ A3 @ B3 ) )
= A3 ) ).
% div_mult_mod_eq
thf(fact_5484_div__mult__mod__eq,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
= A3 ) ).
% div_mult_mod_eq
thf(fact_5485_mod__div__mult__eq,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) )
= A3 ) ).
% mod_div_mult_eq
thf(fact_5486_mod__div__mult__eq,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A3 @ B3 ) @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) )
= A3 ) ).
% mod_div_mult_eq
thf(fact_5487_mod__div__mult__eq,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) )
= A3 ) ).
% mod_div_mult_eq
thf(fact_5488_mod__mult__div__eq,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) )
= A3 ) ).
% mod_mult_div_eq
thf(fact_5489_mod__mult__div__eq,axiom,
! [A3: int,B3: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A3 @ B3 ) @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) )
= A3 ) ).
% mod_mult_div_eq
thf(fact_5490_mod__mult__div__eq,axiom,
! [A3: code_integer,B3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) )
= A3 ) ).
% mod_mult_div_eq
thf(fact_5491_mult__div__mod__eq,axiom,
! [B3: nat,A3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) @ ( modulo_modulo_nat @ A3 @ B3 ) )
= A3 ) ).
% mult_div_mod_eq
thf(fact_5492_mult__div__mod__eq,axiom,
! [B3: int,A3: int] :
( ( plus_plus_int @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) @ ( modulo_modulo_int @ A3 @ B3 ) )
= A3 ) ).
% mult_div_mod_eq
thf(fact_5493_mult__div__mod__eq,axiom,
! [B3: code_integer,A3: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
= A3 ) ).
% mult_div_mod_eq
thf(fact_5494_div__mult1__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A3 @ ( modulo_modulo_nat @ B3 @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5495_div__mult1__eq,axiom,
! [A3: int,B3: int,C: int] :
( ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5496_div__mult1__eq,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5497_unit__imp__mod__eq__0,axiom,
! [B3: nat,A3: nat] :
( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( modulo_modulo_nat @ A3 @ B3 )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_5498_unit__imp__mod__eq__0,axiom,
! [B3: int,A3: int] :
( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( modulo_modulo_int @ A3 @ B3 )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_5499_unit__imp__mod__eq__0,axiom,
! [B3: code_integer,A3: code_integer] :
( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( modulo364778990260209775nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
thf(fact_5500_minus__div__mult__eq__mod,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% minus_div_mult_eq_mod
thf(fact_5501_minus__div__mult__eq__mod,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% minus_div_mult_eq_mod
thf(fact_5502_minus__div__mult__eq__mod,axiom,
! [A3: code_integer,B3: code_integer] :
( ( minus_8373710615458151222nteger @ A3 @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% minus_div_mult_eq_mod
thf(fact_5503_minus__mod__eq__div__mult,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) )
= ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) ) ).
% minus_mod_eq_div_mult
thf(fact_5504_minus__mod__eq__div__mult,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) )
= ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) ) ).
% minus_mod_eq_div_mult
thf(fact_5505_minus__mod__eq__div__mult,axiom,
! [A3: code_integer,B3: code_integer] :
( ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) ) ).
% minus_mod_eq_div_mult
thf(fact_5506_minus__mod__eq__mult__div,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) )
= ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5507_minus__mod__eq__mult__div,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) )
= ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5508_minus__mod__eq__mult__div,axiom,
! [A3: code_integer,B3: code_integer] :
( ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
= ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5509_minus__mult__div__eq__mod,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ).
% minus_mult_div_eq_mod
thf(fact_5510_minus__mult__div__eq__mod,axiom,
! [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) )
= ( modulo_modulo_int @ A3 @ B3 ) ) ).
% minus_mult_div_eq_mod
thf(fact_5511_minus__mult__div__eq__mod,axiom,
! [A3: code_integer,B3: code_integer] :
( ( minus_8373710615458151222nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).
% minus_mult_div_eq_mod
thf(fact_5512_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_5513_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
= zero_z3403309356797280102nteger ) ) ).
% fact_mod
thf(fact_5514_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_5515_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_5516_mod__size__less,axiom,
! [B3: code_integer,A3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ord_less_nat @ ( euclid6377331345833325938nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ ( euclid6377331345833325938nteger @ B3 ) ) ) ).
% mod_size_less
thf(fact_5517_mod__size__less,axiom,
! [B3: int,A3: int] :
( ( B3 != zero_zero_int )
=> ( ord_less_nat @ ( euclid4774559944035922753ze_int @ ( modulo_modulo_int @ A3 @ B3 ) ) @ ( euclid4774559944035922753ze_int @ B3 ) ) ) ).
% mod_size_less
thf(fact_5518_mod__size__less,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ ( modulo_modulo_nat @ A3 @ B3 ) ) @ ( euclid4777050414544973029ze_nat @ B3 ) ) ) ).
% mod_size_less
thf(fact_5519_div__less__mono,axiom,
! [A4: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A4 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_5520_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_5521_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_5522_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_5523_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_5524_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q4: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
= ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_5525_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_5526_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M6: nat,N3: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).
% modulo_nat_def
thf(fact_5527_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_5528_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_5529_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_5530_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5531_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( modulo_modulo_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5532_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5533_even__iff__mod__2__eq__zero,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
= ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_5534_even__iff__mod__2__eq__zero,axiom,
! [A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
= ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_5535_even__iff__mod__2__eq__zero,axiom,
! [A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
= ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_5536_odd__iff__mod__2__eq__one,axiom,
! [A3: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
= ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_5537_odd__iff__mod__2__eq__one,axiom,
! [A3: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
= ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_5538_odd__iff__mod__2__eq__one,axiom,
! [A3: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
= ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_5539_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_5540_nth__rotate1,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
= ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_5541_nth__rotate1,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
= ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_5542_nth__rotate1,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
= ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_5543_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_5544_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_5545_divmod__digit__0_I2_J,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_5546_divmod__digit__0_I2_J,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
= ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_5547_divmod__digit__0_I2_J,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
=> ( ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_5548_bits__stable__imp__add__self,axiom,
! [A3: nat] :
( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( ( plus_plus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_nat ) ) ).
% bits_stable_imp_add_self
thf(fact_5549_bits__stable__imp__add__self,axiom,
! [A3: int] :
( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( ( plus_plus_int @ A3 @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= zero_zero_int ) ) ).
% bits_stable_imp_add_self
thf(fact_5550_bits__stable__imp__add__self,axiom,
! [A3: code_integer] :
( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A3 )
=> ( ( plus_p5714425477246183910nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= zero_z3403309356797280102nteger ) ) ).
% bits_stable_imp_add_self
thf(fact_5551_parity__cases,axiom,
! [A3: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_5552_parity__cases,axiom,
! [A3: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_5553_parity__cases,axiom,
! [A3: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger ) )
=> ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer ) ) ) ).
% parity_cases
thf(fact_5554_mod2__eq__if,axiom,
! [A3: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_5555_mod2__eq__if,axiom,
! [A3: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_5556_mod2__eq__if,axiom,
! [A3: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ) ).
% mod2_eq_if
thf(fact_5557_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S3 ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_5558_verit__le__mono__div,axiom,
! [A4: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_5559_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_5560_divmod__digit__0_I1_J,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_5561_divmod__digit__0_I1_J,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
= ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_5562_divmod__digit__0_I1_J,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_5563_vebt__member_Osimps_I5_J,axiom,
! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi2 )
=> ( ( X != Ma2 )
=> ( ~ ( ord_less_nat @ X @ Mi2 )
& ( ~ ( ord_less_nat @ X @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ X )
& ( ~ ( ord_less_nat @ Ma2 @ X )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_5564_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi2: nat,Ma2: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi2 )
| ( X = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_5565_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A3 @ ( 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_5566_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A3: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A3 @ ( 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_5567_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A3: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_5568_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( Y
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_5569_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_5570_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_5571_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( ( K
= ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_5572_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_5573_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_5574_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_5575_divmod__digit__1_I2_J,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
=> ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
= ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_5576_divmod__digit__1_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
=> ( ( minus_minus_nat @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo_nat @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_5577_divmod__digit__1_I2_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
=> ( ( minus_minus_int @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_5578_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi: nat,Ma: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_5579_unset__bit__Suc,axiom,
! [N: nat,A3: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A3 )
= ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5580_unset__bit__Suc,axiom,
! [N: nat,A3: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A3 )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5581_unset__bit__Suc,axiom,
! [N: nat,A3: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A3 )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5582_set__bit__Suc,axiom,
! [N: nat,A3: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A3 )
= ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5583_set__bit__Suc,axiom,
! [N: nat,A3: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A3 )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5584_set__bit__Suc,axiom,
! [N: nat,A3: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A3 )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5585_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_5586_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi )
=> ( ( Xa2 != Ma )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi )
& ( ~ ( ord_less_nat @ Xa2 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ Xa2 )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_5587_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ! [Mi: nat,Ma: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa2 = Mi )
| ( Xa2 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_5588_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi: nat,Ma: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_5589_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( Y
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [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 ) )
=> Y )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi )
=> ( ( Xa2 != Ma )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi )
& ( ~ ( ord_less_nat @ Xa2 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ Xa2 )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_5590_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa2 != Mi )
=> ( ( Xa2 != Ma )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi )
& ( ~ ( ord_less_nat @ Xa2 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ Xa2 )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_5591_divmod__digit__1_I1_J,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
=> ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_Code_integer )
= ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_5592_divmod__digit__1_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_nat )
= ( divide_divide_nat @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_5593_divmod__digit__1_I1_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_int )
= ( divide_divide_int @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_5594_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_5595_vebt__pred_Osimps_I7_J,axiom,
! [Ma2: nat,X: nat,Mi2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_nat @ Ma2 @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( some_nat @ Ma2 ) ) )
& ( ~ ( ord_less_nat @ Ma2 @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ TreeList ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ Mi2 @ X ) @ ( 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 @ 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 @ TreeList @ ( 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_5596_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_5597_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_5598_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( ( Y = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
=> ( ! [A2: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ Uw2 ) )
=> ( ( Xa2
= ( suc @ zero_zero_nat ) )
=> ( ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( Y = none_nat ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ! [Va2: nat] :
( ( Xa2
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( B2
=> ( Y
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A2
=> ( Y
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A2
=> ( Y = none_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
=> ( ( Y = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
=> ( ( Y = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less_nat @ Ma @ Xa2 )
=> ( Y
= ( some_nat @ Ma ) ) )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( Y
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ Xa2 ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_5599_product__nth,axiom,
! [N: nat,Xs: list_Code_integer,Ys3: list_Code_integer] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_s3445333598471063425nteger @ Ys3 ) ) )
=> ( ( nth_Pr2304437835452373666nteger @ ( produc8792966785426426881nteger @ Xs @ Ys3 ) @ N )
= ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N @ ( size_s3445333598471063425nteger @ Ys3 ) ) ) @ ( nth_Code_integer @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s3445333598471063425nteger @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5600_product__nth,axiom,
! [N: nat,Xs: list_int,Ys3: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys3 ) ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys3 ) @ N )
= ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5601_product__nth,axiom,
! [N: nat,Xs: list_int,Ys3: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
=> ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys3 ) @ N )
= ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5602_product__nth,axiom,
! [N: nat,Xs: list_int,Ys3: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys3 ) ) )
=> ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs @ Ys3 ) @ N )
= ( product_Pair_int_o @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5603_product__nth,axiom,
! [N: nat,Xs: list_Code_integer,Ys3: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys3 ) ) )
=> ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys3 ) @ N )
= ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5604_product__nth,axiom,
! [N: nat,Xs: list_int,Ys3: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) )
=> ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys3 ) @ N )
= ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5605_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys3: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys3 ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys3 ) @ N )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5606_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys3 ) @ N )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5607_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys3: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys3 ) ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys3 ) @ N )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5608_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys3: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys3 ) @ N )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_5609_flip__bit__Suc,axiom,
! [N: nat,A3: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A3 )
= ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5610_flip__bit__Suc,axiom,
! [N: nat,A3: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A3 )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5611_flip__bit__Suc,axiom,
! [N: nat,A3: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A3 )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5612_signed__take__bit__rec,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N3: nat,A: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_5613_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N3: nat,A: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_5614_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_5615_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_5616_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_5617_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_5618_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_5619_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_5620_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_5621_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_5622_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_5623_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_5624_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_5625_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_5626_signed__take__bit__0,axiom,
! [A3: code_integer] :
( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A3 )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5627_signed__take__bit__0,axiom,
! [A3: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A3 )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5628_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_5629_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_5630_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_5631_neg__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( modulo_modulo_int @ A3 @ B3 ) @ zero_zero_int )
& ( ord_less_int @ B3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).
% neg_mod_conj
thf(fact_5632_pos__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A3 @ B3 ) )
& ( ord_less_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 ) ) ) ).
% pos_mod_conj
thf(fact_5633_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo_int @ I @ K )
= I )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_5634_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_5635_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_5636_zdiv__mono__strict,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A4 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_5637_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
= ( plus_plus_int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_5638_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_5639_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
= ( ( dvd_dvd_int @ L @ K )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_5640_int__mod__pos__eq,axiom,
! [A3: int,B3: int,Q4: int,R2: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B3 )
=> ( ( modulo_modulo_int @ A3 @ B3 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_5641_int__mod__neg__eq,axiom,
! [A3: int,B3: int,Q4: int,R2: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ R2 )
=> ( ( modulo_modulo_int @ A3 @ B3 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_5642_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_5643_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
= ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_5644_zmod__minus1,axiom,
! [B3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
= ( minus_minus_int @ B3 @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_5645_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_5646_zmod__zmult2__eq,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A3 @ ( times_times_int @ B3 @ C ) )
= ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_5647_verit__le__mono__div__int,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_5648_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_5649_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_5650_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_5651_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_5652_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_5653_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_5654_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_5655_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_5656_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_5657_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_5658_pos__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_5659_neg__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A3 ) ) @ one_one_int ) ) ) ).
% neg_zmod_mult_2
thf(fact_5660_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_5661_even__flip__bit__iff,axiom,
! [M: nat,A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A3 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_5662_even__flip__bit__iff,axiom,
! [M: nat,A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A3 ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_5663_even__flip__bit__iff,axiom,
! [M: nat,A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A3 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_5664_signed__take__bit__Suc,axiom,
! [N: nat,A3: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A3 )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_5665_signed__take__bit__Suc,axiom,
! [N: nat,A3: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A3 )
= ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_5666_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 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( 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 ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi )
=> ( ( Xa2 != Ma )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi )
& ( ~ ( ord_less_nat @ Xa2 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ Xa2 )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_5667_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( Y
= ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi )
=> ( ( Xa2 != Ma )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi )
& ( ~ ( ord_less_nat @ Xa2 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ Xa2 )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_5668_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( Y
= ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_5669_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 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_5670_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 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_5671_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 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A2 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi )
=> ( ( Xa2 != Ma )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi )
& ( ~ ( ord_less_nat @ Xa2 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ Xa2 )
& ( ~ ( ord_less_nat @ Ma @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_5672_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi )
| ( Xa2 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_5673_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 ) )
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_5674_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 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi )
| ( Xa2 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_5675_artanh__def,axiom,
( artanh_real
= ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_5676_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_5677_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_5678_accp__subset,axiom,
! [R1: vEBT_VEBT > vEBT_VEBT > $o,R22: vEBT_VEBT > vEBT_VEBT > $o] :
( ( ord_le860153471104859278VEBT_o @ R1 @ R22 )
=> ( ord_le418104280809901481VEBT_o @ ( accp_VEBT_VEBT @ R22 ) @ ( accp_VEBT_VEBT @ R1 ) ) ) ).
% accp_subset
thf(fact_5679_accp__subset,axiom,
! [R1: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,R22: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o] :
( ( ord_le1077754993875142464_nat_o @ R1 @ R22 )
=> ( ord_le7812727212727832188_nat_o @ ( accp_P2887432264394892906BT_nat @ R22 ) @ ( accp_P2887432264394892906BT_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_5680_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_5681_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_5682_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_5683_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > complex,Y: 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 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5684_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > complex,Y: 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 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5685_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > complex,Y: 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 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5686_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > complex,Y: 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 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5687_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y: 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 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5688_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y: 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 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5689_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y: 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 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5690_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > real,Y: 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 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5691_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > rat,Y: 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 )
& ( ( Y @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5692_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > rat,Y: 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 )
& ( ( Y @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5693_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y: 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 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5694_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y: 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 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5695_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y: 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 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5696_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > real,Y: 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 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5697_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > rat,Y: 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 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5698_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > rat,Y: 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 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5699_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > rat,Y: int > rat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I5 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5700_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > rat,Y: complex > rat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I5 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5701_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > nat,Y: real > nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I5 )
& ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5702_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( X @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I5 )
& ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5703_insert__simp__excp,axiom,
! [Mi2: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma2: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_nat @ X @ Mi2 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma2 )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi2 @ Ma2 ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_5704_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_5705_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_5706_max__bot,axiom,
! [X: filter_nat] :
( ( ord_max_filter_nat @ bot_bot_filter_nat @ X )
= X ) ).
% max_bot
thf(fact_5707_max__bot,axiom,
! [X: set_real] :
( ( ord_max_set_real @ bot_bot_set_real @ X )
= X ) ).
% max_bot
thf(fact_5708_max__bot,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% max_bot
thf(fact_5709_max__bot,axiom,
! [X: set_int] :
( ( ord_max_set_int @ bot_bot_set_int @ X )
= X ) ).
% max_bot
thf(fact_5710_max__bot,axiom,
! [X: nat] :
( ( ord_max_nat @ bot_bot_nat @ X )
= X ) ).
% max_bot
thf(fact_5711_max__bot2,axiom,
! [X: filter_nat] :
( ( ord_max_filter_nat @ X @ bot_bot_filter_nat )
= X ) ).
% max_bot2
thf(fact_5712_max__bot2,axiom,
! [X: set_real] :
( ( ord_max_set_real @ X @ bot_bot_set_real )
= X ) ).
% max_bot2
thf(fact_5713_max__bot2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% max_bot2
thf(fact_5714_max__bot2,axiom,
! [X: set_int] :
( ( ord_max_set_int @ X @ bot_bot_set_int )
= X ) ).
% max_bot2
thf(fact_5715_max__bot2,axiom,
! [X: nat] :
( ( ord_max_nat @ X @ bot_bot_nat )
= X ) ).
% max_bot2
thf(fact_5716_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_5717_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_max_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
& ( B3 = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_5718_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_5719_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B3: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A3 @ B3 ) )
= ( ( A3 = zero_zero_nat )
& ( B3 = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_5720_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% max_nat.right_neutral
thf(fact_5721_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_5722_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_5723_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_5724_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_5725_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_5726_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_5727_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ V ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_5728_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_5729_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_5730_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_5731_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_5732_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_5733_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_5734_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_5735_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_5736_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
= ( numera6620942414471956472nteger @ X ) ) ).
% max_0_1(3)
thf(fact_5737_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_5738_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_5739_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_5740_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_5741_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_5742_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
= ( numera6620942414471956472nteger @ X ) ) ).
% max_0_1(4)
thf(fact_5743_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_5744_max__0__1_I1_J,axiom,
( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
= one_one_rat ) ).
% max_0_1(1)
thf(fact_5745_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_5746_max__0__1_I1_J,axiom,
( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
= one_one_Code_integer ) ).
% max_0_1(1)
thf(fact_5747_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_5748_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_5749_max__0__1_I2_J,axiom,
( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
= one_one_rat ) ).
% max_0_1(2)
thf(fact_5750_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_5751_max__0__1_I2_J,axiom,
( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
= one_one_Code_integer ) ).
% max_0_1(2)
thf(fact_5752_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_5753_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_5754_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_5755_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_5756_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_5757_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_5758_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_5759_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ X ) ) ).
% max_0_1(6)
thf(fact_5760_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_5761_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_5762_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_5763_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_5764_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_5765_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
= ( numera6620942414471956472nteger @ X ) ) ).
% max_0_1(5)
thf(fact_5766_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_5767_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_5768_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_5769_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_5770_list__update__beyond,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
=> ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_5771_list__update__beyond,axiom,
! [Xs: list_o,I: nat,X: $o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
=> ( ( list_update_o @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_5772_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_5773_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_5774_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_5775_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_5776_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_5777_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_5778_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_5779_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) )
= ( numera6620942414471956472nteger @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_5780_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_5781_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_5782_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_5783_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_5784_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_5785_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_5786_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_5787_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_5788_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_5789_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_5790_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_5791_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_5792_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_5793_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_5794_nth__list__update__eq,axiom,
! [I: nat,Xs: list_int,X: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_5795_nth__list__update__eq,axiom,
! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_5796_nth__list__update__eq,axiom,
! [I: nat,Xs: list_o,X: $o] :
( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_5797_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_5798_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_5799_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_5800_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Ma2: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_nat @ Mi2 @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma2 )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ ( ord_max_nat @ X @ Ma2 ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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_5801_set__swap,axiom,
! [I: nat,Xs: list_int,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
= ( set_int2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_5802_set__swap,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
= ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_5803_set__swap,axiom,
! [I: nat,Xs: list_o,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
=> ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
= ( set_o2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_5804_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_5805_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8557863876264182079omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5806_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5807_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5808_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5809_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5810_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_5811_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_5812_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_5813_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_5814_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_5815_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_5816_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_5817_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_5818_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_5819_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_5820_max__def,axiom,
( ord_max_Code_integer
= ( ^ [A: code_integer,B: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A @ B ) @ B @ A ) ) ) ).
% max_def
thf(fact_5821_max__def,axiom,
( ord_max_set_int
= ( ^ [A: set_int,B: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A @ B ) @ B @ A ) ) ) ).
% max_def
thf(fact_5822_max__def,axiom,
( ord_max_rat
= ( ^ [A: rat,B: rat] : ( if_rat @ ( ord_less_eq_rat @ A @ B ) @ B @ A ) ) ) ).
% max_def
thf(fact_5823_max__def,axiom,
( ord_max_num
= ( ^ [A: num,B: num] : ( if_num @ ( ord_less_eq_num @ A @ B ) @ B @ A ) ) ) ).
% max_def
thf(fact_5824_max__def,axiom,
( ord_max_nat
= ( ^ [A: nat,B: nat] : ( if_nat @ ( ord_less_eq_nat @ A @ B ) @ B @ A ) ) ) ).
% max_def
thf(fact_5825_max__def,axiom,
( ord_max_int
= ( ^ [A: int,B: int] : ( if_int @ ( ord_less_eq_int @ A @ B ) @ B @ A ) ) ) ).
% max_def
thf(fact_5826_max__absorb1,axiom,
! [Y: code_integer,X: code_integer] :
( ( ord_le3102999989581377725nteger @ Y @ X )
=> ( ( ord_max_Code_integer @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_5827_max__absorb1,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_max_set_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_5828_max__absorb1,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_max_rat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_5829_max__absorb1,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_max_num @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_5830_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_5831_max__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_max_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_5832_max__absorb2,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ Y )
=> ( ( ord_max_Code_integer @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_5833_max__absorb2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_max_set_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_5834_max__absorb2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_max_rat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_5835_max__absorb2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_max_num @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_5836_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_5837_max__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_max_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_5838_max__add__distrib__right,axiom,
! [X: real,Y: real,Z: real] :
( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
= ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5839_max__add__distrib__right,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5840_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5841_max__add__distrib__right,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
= ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5842_max__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
= ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5843_max__add__distrib__left,axiom,
! [X: real,Y: real,Z: real] :
( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5844_max__add__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5845_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5846_max__add__distrib__left,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
= ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5847_max__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5848_max__diff__distrib__left,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
= ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_5849_max__diff__distrib__left,axiom,
! [X: real,Y: real,Z: real] :
( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_5850_max__diff__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_5851_max__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_5852_nat__add__max__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).
% nat_add_max_right
thf(fact_5853_nat__add__max__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N @ Q4 ) ) ) ).
% nat_add_max_left
thf(fact_5854_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_5855_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_5856_max__def__raw,axiom,
( ord_max_Code_integer
= ( ^ [A: code_integer,B: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A @ B ) @ B @ A ) ) ) ).
% max_def_raw
thf(fact_5857_max__def__raw,axiom,
( ord_max_set_int
= ( ^ [A: set_int,B: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A @ B ) @ B @ A ) ) ) ).
% max_def_raw
thf(fact_5858_max__def__raw,axiom,
( ord_max_rat
= ( ^ [A: rat,B: rat] : ( if_rat @ ( ord_less_eq_rat @ A @ B ) @ B @ A ) ) ) ).
% max_def_raw
thf(fact_5859_max__def__raw,axiom,
( ord_max_num
= ( ^ [A: num,B: num] : ( if_num @ ( ord_less_eq_num @ A @ B ) @ B @ A ) ) ) ).
% max_def_raw
thf(fact_5860_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A: nat,B: nat] : ( if_nat @ ( ord_less_eq_nat @ A @ B ) @ B @ A ) ) ) ).
% max_def_raw
thf(fact_5861_max__def__raw,axiom,
( ord_max_int
= ( ^ [A: int,B: int] : ( if_int @ ( ord_less_eq_int @ A @ B ) @ B @ A ) ) ) ).
% max_def_raw
thf(fact_5862_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_5863_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_5864_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_5865_list__update__code_I3_J,axiom,
! [X: int,Xs: list_int,I: nat,Y: int] :
( ( list_update_int @ ( cons_int @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_int @ X @ ( list_update_int @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_5866_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_5867_list__update__code_I3_J,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_VEBT_VEBT @ X @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_5868_list__update__code_I2_J,axiom,
! [X: int,Xs: list_int,Y: int] :
( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_int @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_5869_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_5870_list__update__code_I2_J,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_VEBT_VEBT @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_5871_set__update__subsetI,axiom,
! [Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A4 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_5872_set__update__subsetI,axiom,
! [Xs: list_real,A4: set_real,X: real,I: nat] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_5873_set__update__subsetI,axiom,
! [Xs: list_set_nat,A4: set_set_nat,X: set_nat,I: nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 )
=> ( ( member_set_nat @ X @ A4 )
=> ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_5874_set__update__subsetI,axiom,
! [Xs: list_nat,A4: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_5875_set__update__subsetI,axiom,
! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
=> ( ( member_VEBT_VEBT @ X @ A4 )
=> ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_5876_set__update__subsetI,axiom,
! [Xs: list_int,A4: set_int,X: int,I: nat] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_5877_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_5878_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_5879_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_5880_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_5881_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_5882_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_5883_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_5884_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_5885_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_5886_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_5887_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_5888_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_5889_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_5890_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_5891_numeral__Bit1,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_Bit1
thf(fact_5892_numeral__Bit1,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).
% numeral_Bit1
thf(fact_5893_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_5894_set__update__memI,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5895_set__update__memI,axiom,
! [N: nat,Xs: list_real,X: real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5896_set__update__memI,axiom,
! [N: nat,Xs: list_set_nat,X: set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5897_set__update__memI,axiom,
! [N: nat,Xs: list_int,X: int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5898_set__update__memI,axiom,
! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5899_set__update__memI,axiom,
! [N: nat,Xs: list_o,X: $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5900_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5901_nth__list__update,axiom,
! [I: nat,Xs: list_int,J: nat,X: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
= ( nth_int @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_5902_nth__list__update,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_5903_nth__list__update,axiom,
! [I: nat,Xs: list_o,X: $o,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ J )
= ( ( ( I = J )
=> X )
& ( ( I != J )
=> ( nth_o @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_5904_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_5905_list__update__same__conv,axiom,
! [I: nat,Xs: list_int,X: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ( list_update_int @ Xs @ I @ X )
= Xs )
= ( ( nth_int @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_5906_list__update__same__conv,axiom,
! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
= Xs )
= ( ( nth_VEBT_VEBT @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_5907_list__update__same__conv,axiom,
! [I: nat,Xs: list_o,X: $o] :
( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
=> ( ( ( list_update_o @ Xs @ I @ X )
= Xs )
= ( ( nth_o @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_5908_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_5909_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_5910_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_5911_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_5912_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_5913_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_5914_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_code(3)
thf(fact_5915_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).
% numeral_code(3)
thf(fact_5916_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_5917_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_5918_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_5919_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_5920_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_5921_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_5922_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_5923_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_5924_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q4: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_5925_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q4: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_5926_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q4: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
!= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(3)
thf(fact_5927_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_5928_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_5929_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_5930_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_5931_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_5932_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size_num @ ( bit1 @ X33 ) )
= ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_5933_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_5934_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q4: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5935_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q4: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5936_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q4: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5937_cong__exp__iff__simps_I7_J,axiom,
! [Q4: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5938_cong__exp__iff__simps_I7_J,axiom,
! [Q4: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5939_cong__exp__iff__simps_I7_J,axiom,
! [Q4: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5940_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_5941_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_5942_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_5943_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_5944_vebt__insert_Osimps_I5_J,axiom,
! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ Mi2 @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ~ ( ( X = Mi2 )
| ( X = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ X @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ Mi2 @ X ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ Mi2 @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ Mi2 @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ Mi2 @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi2 ) @ Mi2 @ 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 @ Mi2 ) @ Mi2 @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_5945_accp__subset__induct,axiom,
! [D6: vEBT_VEBT > $o,R: vEBT_VEBT > vEBT_VEBT > $o,X: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ord_le418104280809901481VEBT_o @ D6 @ ( accp_VEBT_VEBT @ R ) )
=> ( ! [X4: vEBT_VEBT,Z3: vEBT_VEBT] :
( ( D6 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D6 @ Z3 ) ) )
=> ( ( D6 @ X )
=> ( ! [X4: vEBT_VEBT] :
( ( D6 @ X4 )
=> ( ! [Z5: vEBT_VEBT] :
( ( R @ Z5 @ X4 )
=> ( P @ Z5 ) )
=> ( P @ X4 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5946_accp__subset__induct,axiom,
! [D6: produc9072475918466114483BT_nat > $o,R: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,X: produc9072475918466114483BT_nat,P: produc9072475918466114483BT_nat > $o] :
( ( ord_le7812727212727832188_nat_o @ D6 @ ( accp_P2887432264394892906BT_nat @ R ) )
=> ( ! [X4: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
( ( D6 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D6 @ Z3 ) ) )
=> ( ( D6 @ X )
=> ( ! [X4: produc9072475918466114483BT_nat] :
( ( D6 @ X4 )
=> ( ! [Z5: produc9072475918466114483BT_nat] :
( ( R @ Z5 @ X4 )
=> ( P @ Z5 ) )
=> ( P @ X4 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5947_accp__subset__induct,axiom,
! [D6: nat > $o,R: nat > nat > $o,X: nat,P: nat > $o] :
( ( ord_less_eq_nat_o @ D6 @ ( accp_nat @ R ) )
=> ( ! [X4: nat,Z3: nat] :
( ( D6 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D6 @ Z3 ) ) )
=> ( ( D6 @ X )
=> ( ! [X4: nat] :
( ( D6 @ X4 )
=> ( ! [Z5: nat] :
( ( R @ Z5 @ X4 )
=> ( P @ Z5 ) )
=> ( P @ X4 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5948_accp__subset__induct,axiom,
! [D6: 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 @ D6 @ ( accp_P4275260045618599050at_nat @ R ) )
=> ( ! [X4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( D6 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D6 @ Z3 ) ) )
=> ( ( D6 @ X )
=> ( ! [X4: product_prod_nat_nat] :
( ( D6 @ X4 )
=> ( ! [Z5: product_prod_nat_nat] :
( ( R @ Z5 @ X4 )
=> ( P @ Z5 ) )
=> ( P @ X4 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5949_accp__subset__induct,axiom,
! [D6: 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 @ D6 @ ( accp_P1096762738010456898nt_int @ R ) )
=> ( ! [X4: product_prod_int_int,Z3: product_prod_int_int] :
( ( D6 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D6 @ Z3 ) ) )
=> ( ( D6 @ X )
=> ( ! [X4: product_prod_int_int] :
( ( D6 @ X4 )
=> ( ! [Z5: product_prod_int_int] :
( ( R @ Z5 @ X4 )
=> ( P @ Z5 ) )
=> ( P @ X4 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5950_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_5951_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
=> ( Y
!= ( 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 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Xa2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_5952_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_5953_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A2: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A2 @ B2 ) )
=> ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A2 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
=> ( ( Y
= ( 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 ) )
=> ( ( Y
= ( 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( Xa2 = Mi )
| ( Xa2 = Ma ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Xa2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi ) @ Mi @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_5954_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_5955_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( 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_5956_max_Oabsorb3,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ B3 @ A3 )
=> ( ( ord_max_Code_integer @ A3 @ B3 )
= A3 ) ) ).
% max.absorb3
thf(fact_5957_max_Oabsorb3,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_max_real @ A3 @ B3 )
= A3 ) ) ).
% max.absorb3
thf(fact_5958_max_Oabsorb3,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
=> ( ( ord_max_rat @ A3 @ B3 )
= A3 ) ) ).
% max.absorb3
thf(fact_5959_max_Oabsorb3,axiom,
! [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
=> ( ( ord_max_num @ A3 @ B3 )
= A3 ) ) ).
% max.absorb3
thf(fact_5960_max_Oabsorb3,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_max_nat @ A3 @ B3 )
= A3 ) ) ).
% max.absorb3
thf(fact_5961_max_Oabsorb3,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_max_int @ A3 @ B3 )
= A3 ) ) ).
% max.absorb3
thf(fact_5962_max_Oabsorb4,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le6747313008572928689nteger @ A3 @ B3 )
=> ( ( ord_max_Code_integer @ A3 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_5963_max_Oabsorb4,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_max_real @ A3 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_5964_max_Oabsorb4,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( ( ord_max_rat @ A3 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_5965_max_Oabsorb4,axiom,
! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( ( ord_max_num @ A3 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_5966_max_Oabsorb4,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_max_nat @ A3 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_5967_max_Oabsorb4,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_max_int @ A3 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_5968_max__less__iff__conj,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
= ( ( ord_le6747313008572928689nteger @ X @ Z )
& ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_5969_max__less__iff__conj,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_5970_max__less__iff__conj,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ( ord_less_rat @ X @ Z )
& ( ord_less_rat @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_5971_max__less__iff__conj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
= ( ( ord_less_num @ X @ Z )
& ( ord_less_num @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_5972_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
& ( ord_less_nat @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_5973_max__less__iff__conj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
& ( ord_less_int @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_5974_max_Obounded__iff,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 )
= ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
& ( ord_le3102999989581377725nteger @ C @ A3 ) ) ) ).
% max.bounded_iff
thf(fact_5975_max_Obounded__iff,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A3 )
= ( ( ord_less_eq_rat @ B3 @ A3 )
& ( ord_less_eq_rat @ C @ A3 ) ) ) ).
% max.bounded_iff
thf(fact_5976_max_Obounded__iff,axiom,
! [B3: num,C: num,A3: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A3 )
= ( ( ord_less_eq_num @ B3 @ A3 )
& ( ord_less_eq_num @ C @ A3 ) ) ) ).
% max.bounded_iff
thf(fact_5977_max_Obounded__iff,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A3 )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% max.bounded_iff
thf(fact_5978_max_Obounded__iff,axiom,
! [B3: int,C: int,A3: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A3 )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ C @ A3 ) ) ) ).
% max.bounded_iff
thf(fact_5979_max_Oabsorb1,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le3102999989581377725nteger @ B3 @ A3 )
=> ( ( ord_max_Code_integer @ A3 @ B3 )
= A3 ) ) ).
% max.absorb1
thf(fact_5980_max_Oabsorb1,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_max_rat @ A3 @ B3 )
= A3 ) ) ).
% max.absorb1
thf(fact_5981_max_Oabsorb1,axiom,
! [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
=> ( ( ord_max_num @ A3 @ B3 )
= A3 ) ) ).
% max.absorb1
thf(fact_5982_max_Oabsorb1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_max_nat @ A3 @ B3 )
= A3 ) ) ).
% max.absorb1
thf(fact_5983_max_Oabsorb1,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_max_int @ A3 @ B3 )
= A3 ) ) ).
% max.absorb1
thf(fact_5984_max_Oabsorb2,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ A3 @ B3 )
=> ( ( ord_max_Code_integer @ A3 @ B3 )
= B3 ) ) ).
% max.absorb2
thf(fact_5985_max_Oabsorb2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_max_rat @ A3 @ B3 )
= B3 ) ) ).
% max.absorb2
thf(fact_5986_max_Oabsorb2,axiom,
! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( ( ord_max_num @ A3 @ B3 )
= B3 ) ) ).
% max.absorb2
thf(fact_5987_max_Oabsorb2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_max_nat @ A3 @ B3 )
= B3 ) ) ).
% max.absorb2
thf(fact_5988_max_Oabsorb2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_max_int @ A3 @ B3 )
= B3 ) ) ).
% max.absorb2
thf(fact_5989_max_Omono,axiom,
! [C: code_integer,A3: code_integer,D: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ A3 )
=> ( ( ord_le3102999989581377725nteger @ D @ B3 )
=> ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C @ D ) @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ) ).
% max.mono
thf(fact_5990_max_Omono,axiom,
! [C: rat,A3: rat,D: rat,B3: rat] :
( ( ord_less_eq_rat @ C @ A3 )
=> ( ( ord_less_eq_rat @ D @ B3 )
=> ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A3 @ B3 ) ) ) ) ).
% max.mono
thf(fact_5991_max_Omono,axiom,
! [C: num,A3: num,D: num,B3: num] :
( ( ord_less_eq_num @ C @ A3 )
=> ( ( ord_less_eq_num @ D @ B3 )
=> ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A3 @ B3 ) ) ) ) ).
% max.mono
thf(fact_5992_max_Omono,axiom,
! [C: nat,A3: nat,D: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A3 )
=> ( ( ord_less_eq_nat @ D @ B3 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A3 @ B3 ) ) ) ) ).
% max.mono
thf(fact_5993_max_Omono,axiom,
! [C: int,A3: int,D: int,B3: int] :
( ( ord_less_eq_int @ C @ A3 )
=> ( ( ord_less_eq_int @ D @ B3 )
=> ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A3 @ B3 ) ) ) ) ).
% max.mono
thf(fact_5994_max_OorderE,axiom,
! [B3: code_integer,A3: code_integer] :
( ( ord_le3102999989581377725nteger @ B3 @ A3 )
=> ( A3
= ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).
% max.orderE
thf(fact_5995_max_OorderE,axiom,
! [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( A3
= ( ord_max_rat @ A3 @ B3 ) ) ) ).
% max.orderE
thf(fact_5996_max_OorderE,axiom,
! [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
=> ( A3
= ( ord_max_num @ A3 @ B3 ) ) ) ).
% max.orderE
thf(fact_5997_max_OorderE,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3
= ( ord_max_nat @ A3 @ B3 ) ) ) ).
% max.orderE
thf(fact_5998_max_OorderE,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( A3
= ( ord_max_int @ A3 @ B3 ) ) ) ).
% max.orderE
thf(fact_5999_max_OorderI,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3
= ( ord_max_Code_integer @ A3 @ B3 ) )
=> ( ord_le3102999989581377725nteger @ B3 @ A3 ) ) ).
% max.orderI
thf(fact_6000_max_OorderI,axiom,
! [A3: rat,B3: rat] :
( ( A3
= ( ord_max_rat @ A3 @ B3 ) )
=> ( ord_less_eq_rat @ B3 @ A3 ) ) ).
% max.orderI
thf(fact_6001_max_OorderI,axiom,
! [A3: num,B3: num] :
( ( A3
= ( ord_max_num @ A3 @ B3 ) )
=> ( ord_less_eq_num @ B3 @ A3 ) ) ).
% max.orderI
thf(fact_6002_max_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( ord_max_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% max.orderI
thf(fact_6003_max_OorderI,axiom,
! [A3: int,B3: int] :
( ( A3
= ( ord_max_int @ A3 @ B3 ) )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ).
% max.orderI
thf(fact_6004_max_OboundedE,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 )
=> ~ ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
=> ~ ( ord_le3102999989581377725nteger @ C @ A3 ) ) ) ).
% max.boundedE
thf(fact_6005_max_OboundedE,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_rat @ B3 @ A3 )
=> ~ ( ord_less_eq_rat @ C @ A3 ) ) ) ).
% max.boundedE
thf(fact_6006_max_OboundedE,axiom,
! [B3: num,C: num,A3: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_num @ B3 @ A3 )
=> ~ ( ord_less_eq_num @ C @ A3 ) ) ) ).
% max.boundedE
thf(fact_6007_max_OboundedE,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A3 )
=> ~ ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% max.boundedE
thf(fact_6008_max_OboundedE,axiom,
! [B3: int,C: int,A3: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_int @ B3 @ A3 )
=> ~ ( ord_less_eq_int @ C @ A3 ) ) ) ).
% max.boundedE
thf(fact_6009_max_OboundedI,axiom,
! [B3: code_integer,A3: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ B3 @ A3 )
=> ( ( ord_le3102999989581377725nteger @ C @ A3 )
=> ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 ) ) ) ).
% max.boundedI
thf(fact_6010_max_OboundedI,axiom,
! [B3: rat,A3: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
=> ( ( ord_less_eq_rat @ C @ A3 )
=> ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A3 ) ) ) ).
% max.boundedI
thf(fact_6011_max_OboundedI,axiom,
! [B3: num,A3: num,C: num] :
( ( ord_less_eq_num @ B3 @ A3 )
=> ( ( ord_less_eq_num @ C @ A3 )
=> ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A3 ) ) ) ).
% max.boundedI
thf(fact_6012_max_OboundedI,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A3 ) ) ) ).
% max.boundedI
thf(fact_6013_max_OboundedI,axiom,
! [B3: int,A3: int,C: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C @ A3 )
=> ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A3 ) ) ) ).
% max.boundedI
thf(fact_6014_max_Oorder__iff,axiom,
( ord_le3102999989581377725nteger
= ( ^ [B: code_integer,A: code_integer] :
( A
= ( ord_max_Code_integer @ A @ B ) ) ) ) ).
% max.order_iff
thf(fact_6015_max_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B: rat,A: rat] :
( A
= ( ord_max_rat @ A @ B ) ) ) ) ).
% max.order_iff
thf(fact_6016_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B: num,A: num] :
( A
= ( ord_max_num @ A @ B ) ) ) ) ).
% max.order_iff
thf(fact_6017_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( A
= ( ord_max_nat @ A @ B ) ) ) ) ).
% max.order_iff
thf(fact_6018_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( A
= ( ord_max_int @ A @ B ) ) ) ) ).
% max.order_iff
thf(fact_6019_max_Ocobounded1,axiom,
! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ A3 @ ( ord_max_Code_integer @ A3 @ B3 ) ) ).
% max.cobounded1
thf(fact_6020_max_Ocobounded1,axiom,
! [A3: rat,B3: rat] : ( ord_less_eq_rat @ A3 @ ( ord_max_rat @ A3 @ B3 ) ) ).
% max.cobounded1
thf(fact_6021_max_Ocobounded1,axiom,
! [A3: num,B3: num] : ( ord_less_eq_num @ A3 @ ( ord_max_num @ A3 @ B3 ) ) ).
% max.cobounded1
thf(fact_6022_max_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ A3 @ ( ord_max_nat @ A3 @ B3 ) ) ).
% max.cobounded1
thf(fact_6023_max_Ocobounded1,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ A3 @ ( ord_max_int @ A3 @ B3 ) ) ).
% max.cobounded1
thf(fact_6024_max_Ocobounded2,axiom,
! [B3: code_integer,A3: code_integer] : ( ord_le3102999989581377725nteger @ B3 @ ( ord_max_Code_integer @ A3 @ B3 ) ) ).
% max.cobounded2
thf(fact_6025_max_Ocobounded2,axiom,
! [B3: rat,A3: rat] : ( ord_less_eq_rat @ B3 @ ( ord_max_rat @ A3 @ B3 ) ) ).
% max.cobounded2
thf(fact_6026_max_Ocobounded2,axiom,
! [B3: num,A3: num] : ( ord_less_eq_num @ B3 @ ( ord_max_num @ A3 @ B3 ) ) ).
% max.cobounded2
thf(fact_6027_max_Ocobounded2,axiom,
! [B3: nat,A3: nat] : ( ord_less_eq_nat @ B3 @ ( ord_max_nat @ A3 @ B3 ) ) ).
% max.cobounded2
thf(fact_6028_max_Ocobounded2,axiom,
! [B3: int,A3: int] : ( ord_less_eq_int @ B3 @ ( ord_max_int @ A3 @ B3 ) ) ).
% max.cobounded2
thf(fact_6029_le__max__iff__disj,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
= ( ( ord_le3102999989581377725nteger @ Z @ X )
| ( ord_le3102999989581377725nteger @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_6030_le__max__iff__disj,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
= ( ( ord_less_eq_rat @ Z @ X )
| ( ord_less_eq_rat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_6031_le__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_eq_num @ Z @ X )
| ( ord_less_eq_num @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_6032_le__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z @ X )
| ( ord_less_eq_nat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_6033_le__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_eq_int @ Z @ X )
| ( ord_less_eq_int @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_6034_max_Oabsorb__iff1,axiom,
( ord_le3102999989581377725nteger
= ( ^ [B: code_integer,A: code_integer] :
( ( ord_max_Code_integer @ A @ B )
= A ) ) ) ).
% max.absorb_iff1
thf(fact_6035_max_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B: rat,A: rat] :
( ( ord_max_rat @ A @ B )
= A ) ) ) ).
% max.absorb_iff1
thf(fact_6036_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B: num,A: num] :
( ( ord_max_num @ A @ B )
= A ) ) ) ).
% max.absorb_iff1
thf(fact_6037_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_max_nat @ A @ B )
= A ) ) ) ).
% max.absorb_iff1
thf(fact_6038_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_max_int @ A @ B )
= A ) ) ) ).
% max.absorb_iff1
thf(fact_6039_max_Oabsorb__iff2,axiom,
( ord_le3102999989581377725nteger
= ( ^ [A: code_integer,B: code_integer] :
( ( ord_max_Code_integer @ A @ B )
= B ) ) ) ).
% max.absorb_iff2
thf(fact_6040_max_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A: rat,B: rat] :
( ( ord_max_rat @ A @ B )
= B ) ) ) ).
% max.absorb_iff2
thf(fact_6041_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A: num,B: num] :
( ( ord_max_num @ A @ B )
= B ) ) ) ).
% max.absorb_iff2
thf(fact_6042_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_max_nat @ A @ B )
= B ) ) ) ).
% max.absorb_iff2
thf(fact_6043_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_max_int @ A @ B )
= B ) ) ) ).
% max.absorb_iff2
thf(fact_6044_max_OcoboundedI1,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ A3 )
=> ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).
% max.coboundedI1
thf(fact_6045_max_OcoboundedI1,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_eq_rat @ C @ A3 )
=> ( ord_less_eq_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).
% max.coboundedI1
thf(fact_6046_max_OcoboundedI1,axiom,
! [C: num,A3: num,B3: num] :
( ( ord_less_eq_num @ C @ A3 )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).
% max.coboundedI1
thf(fact_6047_max_OcoboundedI1,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).
% max.coboundedI1
thf(fact_6048_max_OcoboundedI1,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_eq_int @ C @ A3 )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).
% max.coboundedI1
thf(fact_6049_max_OcoboundedI2,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ B3 )
=> ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).
% max.coboundedI2
thf(fact_6050_max_OcoboundedI2,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_eq_rat @ C @ B3 )
=> ( ord_less_eq_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).
% max.coboundedI2
thf(fact_6051_max_OcoboundedI2,axiom,
! [C: num,B3: num,A3: num] :
( ( ord_less_eq_num @ C @ B3 )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).
% max.coboundedI2
thf(fact_6052_max_OcoboundedI2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).
% max.coboundedI2
thf(fact_6053_max_OcoboundedI2,axiom,
! [C: int,B3: int,A3: int] :
( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).
% max.coboundedI2
thf(fact_6054_less__max__iff__disj,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
= ( ( ord_le6747313008572928689nteger @ Z @ X )
| ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_6055_less__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
= ( ( ord_less_real @ Z @ X )
| ( ord_less_real @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_6056_less__max__iff__disj,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
= ( ( ord_less_rat @ Z @ X )
| ( ord_less_rat @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_6057_less__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_num @ Z @ X )
| ( ord_less_num @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_6058_less__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
| ( ord_less_nat @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_6059_less__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
| ( ord_less_int @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_6060_max_Ostrict__boundedE,axiom,
! [B3: code_integer,C: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 )
=> ~ ( ( ord_le6747313008572928689nteger @ B3 @ A3 )
=> ~ ( ord_le6747313008572928689nteger @ C @ A3 ) ) ) ).
% max.strict_boundedE
thf(fact_6061_max_Ostrict__boundedE,axiom,
! [B3: real,C: real,A3: real] :
( ( ord_less_real @ ( ord_max_real @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ C @ A3 ) ) ) ).
% max.strict_boundedE
thf(fact_6062_max_Ostrict__boundedE,axiom,
! [B3: rat,C: rat,A3: rat] :
( ( ord_less_rat @ ( ord_max_rat @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_rat @ B3 @ A3 )
=> ~ ( ord_less_rat @ C @ A3 ) ) ) ).
% max.strict_boundedE
thf(fact_6063_max_Ostrict__boundedE,axiom,
! [B3: num,C: num,A3: num] :
( ( ord_less_num @ ( ord_max_num @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_num @ B3 @ A3 )
=> ~ ( ord_less_num @ C @ A3 ) ) ) ).
% max.strict_boundedE
thf(fact_6064_max_Ostrict__boundedE,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ C @ A3 ) ) ) ).
% max.strict_boundedE
thf(fact_6065_max_Ostrict__boundedE,axiom,
! [B3: int,C: int,A3: int] :
( ( ord_less_int @ ( ord_max_int @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_int @ B3 @ A3 )
=> ~ ( ord_less_int @ C @ A3 ) ) ) ).
% max.strict_boundedE
thf(fact_6066_max_Ostrict__order__iff,axiom,
( ord_le6747313008572928689nteger
= ( ^ [B: code_integer,A: code_integer] :
( ( A
= ( ord_max_Code_integer @ A @ B ) )
& ( A != B ) ) ) ) ).
% max.strict_order_iff
thf(fact_6067_max_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( A
= ( ord_max_real @ A @ B ) )
& ( A != B ) ) ) ) ).
% max.strict_order_iff
thf(fact_6068_max_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [B: rat,A: rat] :
( ( A
= ( ord_max_rat @ A @ B ) )
& ( A != B ) ) ) ) ).
% max.strict_order_iff
thf(fact_6069_max_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [B: num,A: num] :
( ( A
= ( ord_max_num @ A @ B ) )
& ( A != B ) ) ) ) ).
% max.strict_order_iff
thf(fact_6070_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
& ( A != B ) ) ) ) ).
% max.strict_order_iff
thf(fact_6071_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( A
= ( ord_max_int @ A @ B ) )
& ( A != B ) ) ) ) ).
% max.strict_order_iff
thf(fact_6072_max_Ostrict__coboundedI1,axiom,
! [C: code_integer,A3: code_integer,B3: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ A3 )
=> ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_6073_max_Ostrict__coboundedI1,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ A3 )
=> ( ord_less_real @ C @ ( ord_max_real @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_6074_max_Ostrict__coboundedI1,axiom,
! [C: rat,A3: rat,B3: rat] :
( ( ord_less_rat @ C @ A3 )
=> ( ord_less_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_6075_max_Ostrict__coboundedI1,axiom,
! [C: num,A3: num,B3: num] :
( ( ord_less_num @ C @ A3 )
=> ( ord_less_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_6076_max_Ostrict__coboundedI1,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ C @ A3 )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_6077_max_Ostrict__coboundedI1,axiom,
! [C: int,A3: int,B3: int] :
( ( ord_less_int @ C @ A3 )
=> ( ord_less_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_6078_max_Ostrict__coboundedI2,axiom,
! [C: code_integer,B3: code_integer,A3: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ B3 )
=> ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_6079_max_Ostrict__coboundedI2,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ ( ord_max_real @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_6080_max_Ostrict__coboundedI2,axiom,
! [C: rat,B3: rat,A3: rat] :
( ( ord_less_rat @ C @ B3 )
=> ( ord_less_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_6081_max_Ostrict__coboundedI2,axiom,
! [C: num,B3: num,A3: num] :
( ( ord_less_num @ C @ B3 )
=> ( ord_less_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_6082_max_Ostrict__coboundedI2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_6083_max_Ostrict__coboundedI2,axiom,
! [C: int,B3: int,A3: int] :
( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_6084_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A0: nat > nat > nat,A1: nat,A22: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A1 @ ( product_Pair_nat_nat @ A22 @ A32 ) ) ) )
=> ( ! [F2: nat > nat > nat,A2: nat,B2: nat,Acc: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) )
=> ( ( ~ ( ord_less_nat @ B2 @ A2 )
=> ( P @ F2 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F2 @ A2 @ Acc ) ) )
=> ( P @ F2 @ A2 @ B2 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_6085_VEBT__internal_Ooption__shift_Opelims,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2 = none_P5556105721700978146at_nat )
=> ( ( Y = none_P5556105721700978146at_nat )
=> ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
=> ( ! [V2: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ V2 ) )
=> ( ( Xb = none_P5556105721700978146at_nat )
=> ( ( Y = none_P5556105721700978146at_nat )
=> ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
=> ~ ! [A2: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ A2 ) )
=> ! [B2: product_prod_nat_nat] :
( ( Xb
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ ( X @ A2 @ B2 ) ) )
=> ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A2 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_6086_VEBT__internal_Ooption__shift_Opelims,axiom,
! [X: num > num > num,Xa2: option_num,Xb: option_num,Y: option_num] :
( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2 = none_num )
=> ( ( Y = none_num )
=> ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
=> ( ! [V2: num] :
( ( Xa2
= ( some_num @ V2 ) )
=> ( ( Xb = none_num )
=> ( ( Y = none_num )
=> ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
=> ~ ! [A2: num] :
( ( Xa2
= ( some_num @ A2 ) )
=> ! [B2: num] :
( ( Xb
= ( some_num @ B2 ) )
=> ( ( Y
= ( some_num @ ( X @ A2 @ B2 ) ) )
=> ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X @ ( produc8585076106096196333on_num @ ( some_num @ A2 ) @ ( some_num @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_6087_VEBT__internal_Ooption__shift_Opelims,axiom,
! [X: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2 = none_nat )
=> ( ( Y = none_nat )
=> ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
=> ( ! [V2: nat] :
( ( Xa2
= ( some_nat @ V2 ) )
=> ( ( Xb = none_nat )
=> ( ( Y = none_nat )
=> ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
=> ~ ! [A2: nat] :
( ( Xa2
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xb
= ( some_nat @ B2 ) )
=> ( ( Y
= ( some_nat @ ( X @ A2 @ B2 ) ) )
=> ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X @ ( produc5098337634421038937on_nat @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_6088_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [A2: real,B2: real,C3: real] :
( ( P @ A2 @ B2 )
=> ( ( P @ B2 @ C3 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C3 )
=> ( P @ A2 @ C3 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B3 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A2: real,B2: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_eq_real @ X4 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A2 ) @ D3 ) )
=> ( P @ A2 @ B2 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_6089_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_6090_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_6091_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_6092_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_6093_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_6094_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_6095_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_6096_abs__idempotent,axiom,
! [A3: real] :
( ( abs_abs_real @ ( abs_abs_real @ A3 ) )
= ( abs_abs_real @ A3 ) ) ).
% abs_idempotent
thf(fact_6097_abs__idempotent,axiom,
! [A3: int] :
( ( abs_abs_int @ ( abs_abs_int @ A3 ) )
= ( abs_abs_int @ A3 ) ) ).
% abs_idempotent
thf(fact_6098_abs__idempotent,axiom,
! [A3: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A3 ) )
= ( abs_abs_Code_integer @ A3 ) ) ).
% abs_idempotent
thf(fact_6099_abs__idempotent,axiom,
! [A3: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A3 ) )
= ( abs_abs_rat @ A3 ) ) ).
% abs_idempotent
thf(fact_6100_abs__abs,axiom,
! [A3: real] :
( ( abs_abs_real @ ( abs_abs_real @ A3 ) )
= ( abs_abs_real @ A3 ) ) ).
% abs_abs
thf(fact_6101_abs__abs,axiom,
! [A3: int] :
( ( abs_abs_int @ ( abs_abs_int @ A3 ) )
= ( abs_abs_int @ A3 ) ) ).
% abs_abs
thf(fact_6102_abs__abs,axiom,
! [A3: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A3 ) )
= ( abs_abs_Code_integer @ A3 ) ) ).
% abs_abs
thf(fact_6103_abs__abs,axiom,
! [A3: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A3 ) )
= ( abs_abs_rat @ A3 ) ) ).
% abs_abs
thf(fact_6104_abs__0,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_0
thf(fact_6105_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_6106_abs__0,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_0
thf(fact_6107_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_6108_abs__zero,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_zero
thf(fact_6109_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_6110_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_zero
thf(fact_6111_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_6112_abs__eq__0,axiom,
! [A3: code_integer] :
( ( ( abs_abs_Code_integer @ A3 )
= zero_z3403309356797280102nteger )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
thf(fact_6113_abs__eq__0,axiom,
! [A3: real] :
( ( ( abs_abs_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_6114_abs__eq__0,axiom,
! [A3: rat] :
( ( ( abs_abs_rat @ A3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% abs_eq_0
thf(fact_6115_abs__eq__0,axiom,
! [A3: int] :
( ( ( abs_abs_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_6116_abs__0__eq,axiom,
! [A3: code_integer] :
( ( zero_z3403309356797280102nteger
= ( abs_abs_Code_integer @ A3 ) )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
thf(fact_6117_abs__0__eq,axiom,
! [A3: real] :
( ( zero_zero_real
= ( abs_abs_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_6118_abs__0__eq,axiom,
! [A3: rat] :
( ( zero_zero_rat
= ( abs_abs_rat @ A3 ) )
= ( A3 = zero_zero_rat ) ) ).
% abs_0_eq
thf(fact_6119_abs__0__eq,axiom,
! [A3: int] :
( ( zero_zero_int
= ( abs_abs_int @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_6120_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_numeral
thf(fact_6121_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_numeral
thf(fact_6122_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_numeral
thf(fact_6123_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_numeral
thf(fact_6124_abs__mult__self__eq,axiom,
! [A3: code_integer] :
( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ A3 ) )
= ( times_3573771949741848930nteger @ A3 @ A3 ) ) ).
% abs_mult_self_eq
thf(fact_6125_abs__mult__self__eq,axiom,
! [A3: real] :
( ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ A3 ) )
= ( times_times_real @ A3 @ A3 ) ) ).
% abs_mult_self_eq
thf(fact_6126_abs__mult__self__eq,axiom,
! [A3: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ A3 ) )
= ( times_times_rat @ A3 @ A3 ) ) ).
% abs_mult_self_eq
thf(fact_6127_abs__mult__self__eq,axiom,
! [A3: int] :
( ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ A3 ) )
= ( times_times_int @ A3 @ A3 ) ) ).
% abs_mult_self_eq
thf(fact_6128_abs__1,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_1
thf(fact_6129_abs__1,axiom,
( ( abs_abs_complex @ one_one_complex )
= one_one_complex ) ).
% abs_1
thf(fact_6130_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_6131_abs__1,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_1
thf(fact_6132_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_6133_abs__add__abs,axiom,
! [A3: code_integer,B3: code_integer] :
( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) )
= ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% abs_add_abs
thf(fact_6134_abs__add__abs,axiom,
! [A3: real,B3: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).
% abs_add_abs
thf(fact_6135_abs__add__abs,axiom,
! [A3: rat,B3: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% abs_add_abs
thf(fact_6136_abs__add__abs,axiom,
! [A3: int,B3: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).
% abs_add_abs
thf(fact_6137_abs__minus__cancel,axiom,
! [A3: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A3 ) )
= ( abs_abs_real @ A3 ) ) ).
% abs_minus_cancel
thf(fact_6138_abs__minus__cancel,axiom,
! [A3: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A3 ) )
= ( abs_abs_int @ A3 ) ) ).
% abs_minus_cancel
thf(fact_6139_abs__minus__cancel,axiom,
! [A3: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A3 ) )
= ( abs_abs_Code_integer @ A3 ) ) ).
% abs_minus_cancel
thf(fact_6140_abs__minus__cancel,axiom,
! [A3: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A3 ) )
= ( abs_abs_rat @ A3 ) ) ).
% abs_minus_cancel
thf(fact_6141_abs__minus,axiom,
! [A3: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A3 ) )
= ( abs_abs_real @ A3 ) ) ).
% abs_minus
thf(fact_6142_abs__minus,axiom,
! [A3: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A3 ) )
= ( abs_abs_int @ A3 ) ) ).
% abs_minus
thf(fact_6143_abs__minus,axiom,
! [A3: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A3 ) )
= ( abs_abs_Code_integer @ A3 ) ) ).
% abs_minus
thf(fact_6144_abs__minus,axiom,
! [A3: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A3 ) )
= ( abs_abs_rat @ A3 ) ) ).
% abs_minus
thf(fact_6145_dvd__abs__iff,axiom,
! [M: real,K: real] :
( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
= ( dvd_dvd_real @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_6146_dvd__abs__iff,axiom,
! [M: int,K: int] :
( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
= ( dvd_dvd_int @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_6147_dvd__abs__iff,axiom,
! [M: code_integer,K: code_integer] :
( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
= ( dvd_dvd_Code_integer @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_6148_dvd__abs__iff,axiom,
! [M: rat,K: rat] :
( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
= ( dvd_dvd_rat @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_6149_abs__dvd__iff,axiom,
! [M: real,K: real] :
( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
= ( dvd_dvd_real @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_6150_abs__dvd__iff,axiom,
! [M: int,K: int] :
( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
= ( dvd_dvd_int @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_6151_abs__dvd__iff,axiom,
! [M: code_integer,K: code_integer] :
( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
= ( dvd_dvd_Code_integer @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_6152_abs__dvd__iff,axiom,
! [M: rat,K: rat] :
( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
= ( dvd_dvd_rat @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_6153_abs__le__zero__iff,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ zero_z3403309356797280102nteger )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
thf(fact_6154_abs__le__zero__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_6155_abs__le__zero__iff,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% abs_le_zero_iff
thf(fact_6156_abs__le__zero__iff,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_6157_abs__le__self__iff,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ A3 )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).
% abs_le_self_iff
thf(fact_6158_abs__le__self__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ A3 )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% abs_le_self_iff
thf(fact_6159_abs__le__self__iff,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ A3 )
= ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).
% abs_le_self_iff
thf(fact_6160_abs__le__self__iff,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ A3 )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% abs_le_self_iff
thf(fact_6161_abs__of__nonneg,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ( abs_abs_Code_integer @ A3 )
= A3 ) ) ).
% abs_of_nonneg
thf(fact_6162_abs__of__nonneg,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( abs_abs_real @ A3 )
= A3 ) ) ).
% abs_of_nonneg
thf(fact_6163_abs__of__nonneg,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( abs_abs_rat @ A3 )
= A3 ) ) ).
% abs_of_nonneg
thf(fact_6164_abs__of__nonneg,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( abs_abs_int @ A3 )
= A3 ) ) ).
% abs_of_nonneg
thf(fact_6165_zero__less__abs__iff,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A3 ) )
= ( A3 != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
thf(fact_6166_zero__less__abs__iff,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A3 ) )
= ( A3 != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_6167_zero__less__abs__iff,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A3 ) )
= ( A3 != zero_zero_rat ) ) ).
% zero_less_abs_iff
thf(fact_6168_zero__less__abs__iff,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A3 ) )
= ( A3 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_6169_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_6170_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_6171_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_neg_numeral
thf(fact_6172_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_6173_abs__neg__one,axiom,
( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
= one_one_real ) ).
% abs_neg_one
thf(fact_6174_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_6175_abs__neg__one,axiom,
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= one_one_Code_integer ) ).
% abs_neg_one
thf(fact_6176_abs__neg__one,axiom,
( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= one_one_rat ) ).
% abs_neg_one
thf(fact_6177_zero__le__divide__abs__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ ( abs_abs_real @ B3 ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A3 )
| ( B3 = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_6178_zero__le__divide__abs__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ ( abs_abs_rat @ B3 ) ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
| ( B3 = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_6179_divide__le__0__abs__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ ( abs_abs_real @ B3 ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A3 @ zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_6180_divide__le__0__abs__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ ( abs_abs_rat @ B3 ) ) @ zero_zero_rat )
= ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
| ( B3 = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
thf(fact_6181_abs__of__nonpos,axiom,
! [A3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( abs_abs_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ) ).
% abs_of_nonpos
thf(fact_6182_abs__of__nonpos,axiom,
! [A3: code_integer] :
( ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A3 )
= ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% abs_of_nonpos
thf(fact_6183_abs__of__nonpos,axiom,
! [A3: rat] :
( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
=> ( ( abs_abs_rat @ A3 )
= ( uminus_uminus_rat @ A3 ) ) ) ).
% abs_of_nonpos
thf(fact_6184_abs__of__nonpos,axiom,
! [A3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( abs_abs_int @ A3 )
= ( uminus_uminus_int @ A3 ) ) ) ).
% abs_of_nonpos
thf(fact_6185_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_6186_zero__less__power__abs__iff,axiom,
! [A3: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ N ) )
= ( ( A3 != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6187_zero__less__power__abs__iff,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A3 ) @ N ) )
= ( ( A3 != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6188_zero__less__power__abs__iff,axiom,
! [A3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A3 ) @ N ) )
= ( ( A3 != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6189_zero__less__power__abs__iff,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N ) )
= ( ( A3 != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6190_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_6191_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique3479559517661332726nteger @ M @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_6192_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_6193_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_6194_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6195_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_6196_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_6197_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6198_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_6199_abs__ge__self,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ ( abs_abs_real @ A3 ) ) ).
% abs_ge_self
thf(fact_6200_abs__ge__self,axiom,
! [A3: code_integer] : ( ord_le3102999989581377725nteger @ A3 @ ( abs_abs_Code_integer @ A3 ) ) ).
% abs_ge_self
thf(fact_6201_abs__ge__self,axiom,
! [A3: rat] : ( ord_less_eq_rat @ A3 @ ( abs_abs_rat @ A3 ) ) ).
% abs_ge_self
thf(fact_6202_abs__ge__self,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ ( abs_abs_int @ A3 ) ) ).
% abs_ge_self
thf(fact_6203_abs__le__D1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% abs_le_D1
thf(fact_6204_abs__le__D1,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
=> ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ).
% abs_le_D1
thf(fact_6205_abs__le__D1,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 )
=> ( ord_less_eq_rat @ A3 @ B3 ) ) ).
% abs_le_D1
thf(fact_6206_abs__le__D1,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% abs_le_D1
thf(fact_6207_abs__eq__0__iff,axiom,
! [A3: code_integer] :
( ( ( abs_abs_Code_integer @ A3 )
= zero_z3403309356797280102nteger )
= ( A3 = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0_iff
thf(fact_6208_abs__eq__0__iff,axiom,
! [A3: real] :
( ( ( abs_abs_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_6209_abs__eq__0__iff,axiom,
! [A3: rat] :
( ( ( abs_abs_rat @ A3 )
= zero_zero_rat )
= ( A3 = zero_zero_rat ) ) ).
% abs_eq_0_iff
thf(fact_6210_abs__eq__0__iff,axiom,
! [A3: int] :
( ( ( abs_abs_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_6211_abs__mult,axiom,
! [A3: code_integer,B3: code_integer] :
( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% abs_mult
thf(fact_6212_abs__mult,axiom,
! [A3: real,B3: real] :
( ( abs_abs_real @ ( times_times_real @ A3 @ B3 ) )
= ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).
% abs_mult
thf(fact_6213_abs__mult,axiom,
! [A3: rat,B3: rat] :
( ( abs_abs_rat @ ( times_times_rat @ A3 @ B3 ) )
= ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% abs_mult
thf(fact_6214_abs__mult,axiom,
! [A3: int,B3: int] :
( ( abs_abs_int @ ( times_times_int @ A3 @ B3 ) )
= ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).
% abs_mult
thf(fact_6215_abs__one,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_one
thf(fact_6216_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_6217_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_one
thf(fact_6218_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_6219_abs__minus__commute,axiom,
! [A3: code_integer,B3: code_integer] :
( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) )
= ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A3 ) ) ) ).
% abs_minus_commute
thf(fact_6220_abs__minus__commute,axiom,
! [A3: real,B3: real] :
( ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( abs_abs_real @ ( minus_minus_real @ B3 @ A3 ) ) ) ).
% abs_minus_commute
thf(fact_6221_abs__minus__commute,axiom,
! [A3: rat,B3: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A3 ) ) ) ).
% abs_minus_commute
thf(fact_6222_abs__minus__commute,axiom,
! [A3: int,B3: int] :
( ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) )
= ( abs_abs_int @ ( minus_minus_int @ B3 @ A3 ) ) ) ).
% abs_minus_commute
thf(fact_6223_abs__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( abs_abs_real @ X )
= ( abs_abs_real @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_real @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6224_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( abs_abs_int @ X )
= ( abs_abs_int @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6225_abs__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( abs_abs_Code_integer @ X )
= ( abs_abs_Code_integer @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6226_abs__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( ( abs_abs_rat @ X )
= ( abs_abs_rat @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_rat @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6227_dvd__if__abs__eq,axiom,
! [L: real,K: real] :
( ( ( abs_abs_real @ L )
= ( abs_abs_real @ K ) )
=> ( dvd_dvd_real @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_6228_dvd__if__abs__eq,axiom,
! [L: int,K: int] :
( ( ( abs_abs_int @ L )
= ( abs_abs_int @ K ) )
=> ( dvd_dvd_int @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_6229_dvd__if__abs__eq,axiom,
! [L: code_integer,K: code_integer] :
( ( ( abs_abs_Code_integer @ L )
= ( abs_abs_Code_integer @ K ) )
=> ( dvd_dvd_Code_integer @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_6230_dvd__if__abs__eq,axiom,
! [L: rat,K: rat] :
( ( ( abs_abs_rat @ L )
= ( abs_abs_rat @ K ) )
=> ( dvd_dvd_rat @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_6231_abs__ge__zero,axiom,
! [A3: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A3 ) ) ).
% abs_ge_zero
thf(fact_6232_abs__ge__zero,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A3 ) ) ).
% abs_ge_zero
thf(fact_6233_abs__ge__zero,axiom,
! [A3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A3 ) ) ).
% abs_ge_zero
thf(fact_6234_abs__ge__zero,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A3 ) ) ).
% abs_ge_zero
thf(fact_6235_abs__not__less__zero,axiom,
! [A3: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A3 ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_6236_abs__not__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( abs_abs_real @ A3 ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_6237_abs__not__less__zero,axiom,
! [A3: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A3 ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_6238_abs__not__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( abs_abs_int @ A3 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_6239_abs__of__pos,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 )
=> ( ( abs_abs_Code_integer @ A3 )
= A3 ) ) ).
% abs_of_pos
thf(fact_6240_abs__of__pos,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( abs_abs_real @ A3 )
= A3 ) ) ).
% abs_of_pos
thf(fact_6241_abs__of__pos,axiom,
! [A3: rat] :
( ( ord_less_rat @ zero_zero_rat @ A3 )
=> ( ( abs_abs_rat @ A3 )
= A3 ) ) ).
% abs_of_pos
thf(fact_6242_abs__of__pos,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( abs_abs_int @ A3 )
= A3 ) ) ).
% abs_of_pos
thf(fact_6243_abs__triangle__ineq,axiom,
! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% abs_triangle_ineq
thf(fact_6244_abs__triangle__ineq,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A3 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).
% abs_triangle_ineq
thf(fact_6245_abs__triangle__ineq,axiom,
! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A3 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% abs_triangle_ineq
thf(fact_6246_abs__triangle__ineq,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A3 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).
% abs_triangle_ineq
thf(fact_6247_abs__mult__less,axiom,
! [A3: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A3 ) @ C )
=> ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B3 ) @ D )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6248_abs__mult__less,axiom,
! [A3: real,C: real,B3: real,D: real] :
( ( ord_less_real @ ( abs_abs_real @ A3 ) @ C )
=> ( ( ord_less_real @ ( abs_abs_real @ B3 ) @ D )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6249_abs__mult__less,axiom,
! [A3: rat,C: rat,B3: rat,D: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A3 ) @ C )
=> ( ( ord_less_rat @ ( abs_abs_rat @ B3 ) @ D )
=> ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6250_abs__mult__less,axiom,
! [A3: int,C: int,B3: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A3 ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B3 ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6251_abs__triangle__ineq2__sym,axiom,
! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A3 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6252_abs__triangle__ineq2__sym,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ A3 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6253_abs__triangle__ineq2__sym,axiom,
! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A3 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6254_abs__triangle__ineq2__sym,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ A3 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6255_abs__triangle__ineq3,axiom,
! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq3
thf(fact_6256_abs__triangle__ineq3,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq3
thf(fact_6257_abs__triangle__ineq3,axiom,
! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq3
thf(fact_6258_abs__triangle__ineq3,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq3
thf(fact_6259_abs__triangle__ineq2,axiom,
! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq2
thf(fact_6260_abs__triangle__ineq2,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq2
thf(fact_6261_abs__triangle__ineq2,axiom,
! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq2
thf(fact_6262_abs__triangle__ineq2,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) ) ) ).
% abs_triangle_ineq2
thf(fact_6263_nonzero__abs__divide,axiom,
! [B3: rat,A3: rat] :
( ( B3 != zero_zero_rat )
=> ( ( abs_abs_rat @ ( divide_divide_rat @ A3 @ B3 ) )
= ( divide_divide_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).
% nonzero_abs_divide
thf(fact_6264_nonzero__abs__divide,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A3 @ B3 ) )
= ( divide_divide_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ) ).
% nonzero_abs_divide
thf(fact_6265_abs__leI,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
=> ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 ) ) ) ).
% abs_leI
thf(fact_6266_abs__leI,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ A3 @ B3 )
=> ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 ) ) ) ).
% abs_leI
thf(fact_6267_abs__leI,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 ) ) ) ).
% abs_leI
thf(fact_6268_abs__leI,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 )
=> ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 ) ) ) ).
% abs_leI
thf(fact_6269_abs__le__D2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).
% abs_le_D2
thf(fact_6270_abs__le__D2,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).
% abs_le_D2
thf(fact_6271_abs__le__D2,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ).
% abs_le_D2
thf(fact_6272_abs__le__D2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).
% abs_le_D2
thf(fact_6273_abs__le__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 )
= ( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ) ).
% abs_le_iff
thf(fact_6274_abs__le__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
= ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
& ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ) ).
% abs_le_iff
thf(fact_6275_abs__le__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 )
= ( ( ord_less_eq_rat @ A3 @ B3 )
& ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ) ).
% abs_le_iff
thf(fact_6276_abs__le__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 )
= ( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).
% abs_le_iff
thf(fact_6277_abs__ge__minus__self,axiom,
! [A3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ ( abs_abs_real @ A3 ) ) ).
% abs_ge_minus_self
thf(fact_6278_abs__ge__minus__self,axiom,
! [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( abs_abs_Code_integer @ A3 ) ) ).
% abs_ge_minus_self
thf(fact_6279_abs__ge__minus__self,axiom,
! [A3: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ ( abs_abs_rat @ A3 ) ) ).
% abs_ge_minus_self
thf(fact_6280_abs__ge__minus__self,axiom,
! [A3: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ ( abs_abs_int @ A3 ) ) ).
% abs_ge_minus_self
thf(fact_6281_abs__less__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( abs_abs_real @ A3 ) @ B3 )
= ( ( ord_less_real @ A3 @ B3 )
& ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ) ).
% abs_less_iff
thf(fact_6282_abs__less__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( abs_abs_int @ A3 ) @ B3 )
= ( ( ord_less_int @ A3 @ B3 )
& ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).
% abs_less_iff
thf(fact_6283_abs__less__iff,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
= ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
& ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ) ).
% abs_less_iff
thf(fact_6284_abs__less__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A3 ) @ B3 )
= ( ( ord_less_rat @ A3 @ B3 )
& ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ) ).
% abs_less_iff
thf(fact_6285_dense__eq0__I,axiom,
! [X: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_6286_dense__eq0__I,axiom,
! [X: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
thf(fact_6287_abs__eq__mult,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
| ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger ) )
& ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
| ( ord_le3102999989581377725nteger @ B3 @ zero_z3403309356797280102nteger ) ) )
=> ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ) ).
% abs_eq_mult
thf(fact_6288_abs__eq__mult,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
| ( ord_less_eq_real @ A3 @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B3 )
| ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A3 @ B3 ) )
= ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ) ).
% abs_eq_mult
thf(fact_6289_abs__eq__mult,axiom,
! [A3: rat,B3: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
| ( ord_less_eq_rat @ A3 @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
| ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A3 @ B3 ) )
= ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).
% abs_eq_mult
thf(fact_6290_abs__eq__mult,axiom,
! [A3: int,B3: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
| ( ord_less_eq_int @ A3 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B3 )
| ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A3 @ B3 ) )
= ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ) ).
% abs_eq_mult
thf(fact_6291_abs__mult__pos,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
= ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6292_abs__mult__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
= ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6293_abs__mult__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
= ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6294_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6295_abs__minus__le__zero,axiom,
! [A3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A3 ) ) @ zero_zero_real ) ).
% abs_minus_le_zero
thf(fact_6296_abs__minus__le__zero,axiom,
! [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A3 ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_6297_abs__minus__le__zero,axiom,
! [A3: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A3 ) ) @ zero_zero_rat ) ).
% abs_minus_le_zero
thf(fact_6298_abs__minus__le__zero,axiom,
! [A3: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A3 ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_6299_eq__abs__iff_H,axiom,
! [A3: real,B3: real] :
( ( A3
= ( abs_abs_real @ B3 ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus_uminus_real @ A3 ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6300_eq__abs__iff_H,axiom,
! [A3: code_integer,B3: code_integer] :
( ( A3
= ( abs_abs_Code_integer @ B3 ) )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus1351360451143612070nteger @ A3 ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6301_eq__abs__iff_H,axiom,
! [A3: rat,B3: rat] :
( ( A3
= ( abs_abs_rat @ B3 ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus_uminus_rat @ A3 ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6302_eq__abs__iff_H,axiom,
! [A3: int,B3: int] :
( ( A3
= ( abs_abs_int @ B3 ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus_uminus_int @ A3 ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6303_abs__eq__iff_H,axiom,
! [A3: real,B3: real] :
( ( ( abs_abs_real @ A3 )
= B3 )
= ( ( ord_less_eq_real @ zero_zero_real @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_real @ B3 ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6304_abs__eq__iff_H,axiom,
! [A3: code_integer,B3: code_integer] :
( ( ( abs_abs_Code_integer @ A3 )
= B3 )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus1351360451143612070nteger @ B3 ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6305_abs__eq__iff_H,axiom,
! [A3: rat,B3: rat] :
( ( ( abs_abs_rat @ A3 )
= B3 )
= ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_rat @ B3 ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6306_abs__eq__iff_H,axiom,
! [A3: int,B3: int] :
( ( ( abs_abs_int @ A3 )
= B3 )
= ( ( ord_less_eq_int @ zero_zero_int @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus_uminus_int @ B3 ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6307_abs__div__pos,axiom,
! [Y: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
= ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_6308_abs__div__pos,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
= ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_6309_zero__le__power__abs,axiom,
! [A3: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6310_zero__le__power__abs,axiom,
! [A3: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A3 ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6311_zero__le__power__abs,axiom,
! [A3: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A3 ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6312_zero__le__power__abs,axiom,
! [A3: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6313_abs__if__raw,axiom,
( abs_abs_real
= ( ^ [A: real] : ( if_real @ ( ord_less_real @ A @ zero_zero_real ) @ ( uminus_uminus_real @ A ) @ A ) ) ) ).
% abs_if_raw
thf(fact_6314_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A: int] : ( if_int @ ( ord_less_int @ A @ zero_zero_int ) @ ( uminus_uminus_int @ A ) @ A ) ) ) ).
% abs_if_raw
thf(fact_6315_abs__if__raw,axiom,
( abs_abs_Code_integer
= ( ^ [A: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A ) @ A ) ) ) ).
% abs_if_raw
thf(fact_6316_abs__if__raw,axiom,
( abs_abs_rat
= ( ^ [A: rat] : ( if_rat @ ( ord_less_rat @ A @ zero_zero_rat ) @ ( uminus_uminus_rat @ A ) @ A ) ) ) ).
% abs_if_raw
thf(fact_6317_abs__of__neg,axiom,
! [A3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( abs_abs_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ) ).
% abs_of_neg
thf(fact_6318_abs__of__neg,axiom,
! [A3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( abs_abs_int @ A3 )
= ( uminus_uminus_int @ A3 ) ) ) ).
% abs_of_neg
thf(fact_6319_abs__of__neg,axiom,
! [A3: code_integer] :
( ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A3 )
= ( uminus1351360451143612070nteger @ A3 ) ) ) ).
% abs_of_neg
thf(fact_6320_abs__of__neg,axiom,
! [A3: rat] :
( ( ord_less_rat @ A3 @ zero_zero_rat )
=> ( ( abs_abs_rat @ A3 )
= ( uminus_uminus_rat @ A3 ) ) ) ).
% abs_of_neg
thf(fact_6321_abs__if,axiom,
( abs_abs_real
= ( ^ [A: real] : ( if_real @ ( ord_less_real @ A @ zero_zero_real ) @ ( uminus_uminus_real @ A ) @ A ) ) ) ).
% abs_if
thf(fact_6322_abs__if,axiom,
( abs_abs_int
= ( ^ [A: int] : ( if_int @ ( ord_less_int @ A @ zero_zero_int ) @ ( uminus_uminus_int @ A ) @ A ) ) ) ).
% abs_if
thf(fact_6323_abs__if,axiom,
( abs_abs_Code_integer
= ( ^ [A: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A ) @ A ) ) ) ).
% abs_if
thf(fact_6324_abs__if,axiom,
( abs_abs_rat
= ( ^ [A: rat] : ( if_rat @ ( ord_less_rat @ A @ zero_zero_rat ) @ ( uminus_uminus_rat @ A ) @ A ) ) ) ).
% abs_if
thf(fact_6325_abs__triangle__ineq4,axiom,
! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% abs_triangle_ineq4
thf(fact_6326_abs__triangle__ineq4,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).
% abs_triangle_ineq4
thf(fact_6327_abs__triangle__ineq4,axiom,
! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% abs_triangle_ineq4
thf(fact_6328_abs__triangle__ineq4,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).
% abs_triangle_ineq4
thf(fact_6329_abs__diff__triangle__ineq,axiom,
! [A3: code_integer,B3: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6330_abs__diff__triangle__ineq,axiom,
! [A3: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A3 @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6331_abs__diff__triangle__ineq,axiom,
! [A3: rat,B3: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6332_abs__diff__triangle__ineq,axiom,
! [A3: int,B3: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A3 @ B3 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A3 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6333_abs__diff__le__iff,axiom,
! [X: code_integer,A3: code_integer,R2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A3 ) ) @ R2 )
= ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A3 @ R2 ) @ X )
& ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A3 @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6334_abs__diff__le__iff,axiom,
! [X: real,A3: real,R2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A3 ) ) @ R2 )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A3 @ R2 ) @ X )
& ( ord_less_eq_real @ X @ ( plus_plus_real @ A3 @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6335_abs__diff__le__iff,axiom,
! [X: rat,A3: rat,R2: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A3 ) ) @ R2 )
= ( ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ R2 ) @ X )
& ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A3 @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6336_abs__diff__le__iff,axiom,
! [X: int,A3: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A3 ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A3 @ R2 ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A3 @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6337_abs__diff__less__iff,axiom,
! [X: code_integer,A3: code_integer,R2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A3 ) ) @ R2 )
= ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A3 @ R2 ) @ X )
& ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A3 @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6338_abs__diff__less__iff,axiom,
! [X: real,A3: real,R2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A3 ) ) @ R2 )
= ( ( ord_less_real @ ( minus_minus_real @ A3 @ R2 ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ A3 @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6339_abs__diff__less__iff,axiom,
! [X: rat,A3: rat,R2: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A3 ) ) @ R2 )
= ( ( ord_less_rat @ ( minus_minus_rat @ A3 @ R2 ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ A3 @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6340_abs__diff__less__iff,axiom,
! [X: int,A3: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A3 ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A3 @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A3 @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6341_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A: real] : ( if_real @ ( ord_less_real @ A @ zero_zero_real ) @ ( uminus_uminus_real @ A ) @ A ) ) ) ).
% abs_real_def
thf(fact_6342_sin__bound__lemma,axiom,
! [X: real,Y: real,U: real,V: real] :
( ( X = Y )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_6343_finite__abs__int__segment,axiom,
! [A3: real] :
( finite_finite_real
@ ( collect_real
@ ^ [K3: real] :
( ( member_real @ K3 @ ring_1_Ints_real )
& ( ord_less_eq_real @ ( abs_abs_real @ K3 ) @ A3 ) ) ) ) ).
% finite_abs_int_segment
thf(fact_6344_finite__abs__int__segment,axiom,
! [A3: rat] :
( finite_finite_rat
@ ( collect_rat
@ ^ [K3: rat] :
( ( member_rat @ K3 @ ring_1_Ints_rat )
& ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A3 ) ) ) ) ).
% finite_abs_int_segment
thf(fact_6345_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_6346_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_6347_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_6348_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_6349_Ints__nonzero__abs__ge1,axiom,
! [X: code_integer] :
( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
=> ( ( X != zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_6350_Ints__nonzero__abs__ge1,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( X != zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_6351_Ints__nonzero__abs__ge1,axiom,
! [X: rat] :
( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( X != zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_6352_Ints__nonzero__abs__ge1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( X != zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_6353_Ints__nonzero__abs__less1,axiom,
! [X: code_integer] :
( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
=> ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
=> ( X = zero_z3403309356797280102nteger ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_6354_Ints__nonzero__abs__less1,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( X = zero_zero_real ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_6355_Ints__nonzero__abs__less1,axiom,
! [X: rat] :
( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
=> ( X = zero_zero_rat ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_6356_Ints__nonzero__abs__less1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
=> ( X = zero_zero_int ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_6357_Ints__eq__abs__less1,axiom,
! [X: code_integer,Y: code_integer] :
( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
=> ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
=> ( ( X = Y )
= ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_6358_Ints__eq__abs__less1,axiom,
! [X: real,Y: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ Y @ ring_1_Ints_real )
=> ( ( X = Y )
= ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_6359_Ints__eq__abs__less1,axiom,
! [X: rat,Y: rat] :
( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( member_rat @ Y @ ring_1_Ints_rat )
=> ( ( X = Y )
= ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_6360_Ints__eq__abs__less1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( member_int @ Y @ ring_1_Ints_int )
=> ( ( X = Y )
= ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_6361_abs__le__square__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
= ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6362_abs__le__square__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
= ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6363_abs__le__square__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
= ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6364_abs__le__square__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
= ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6365_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_6366_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_6367_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_6368_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_6369_abs__sqrt__wlog,axiom,
! [P: code_integer > code_integer > $o,X: code_integer] :
( ! [X4: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
=> ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6370_abs__sqrt__wlog,axiom,
! [P: real > real > $o,X: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6371_abs__sqrt__wlog,axiom,
! [P: rat > rat > $o,X: rat] :
( ! [X4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
=> ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6372_abs__sqrt__wlog,axiom,
! [P: int > int > $o,X: int] :
( ! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6373_power2__le__iff__abs__le,axiom,
! [Y: code_integer,X: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6374_power2__le__iff__abs__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6375_power2__le__iff__abs__le,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6376_power2__le__iff__abs__le,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6377_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_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_power__mono__even,axiom,
! [N: nat,A3: code_integer,B3: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N ) @ ( power_8256067586552552935nteger @ B3 @ N ) ) ) ) ).
% power_mono_even
thf(fact_6386_power__mono__even,axiom,
! [N: nat,A3: real,B3: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ).
% power_mono_even
thf(fact_6387_power__mono__even,axiom,
! [N: nat,A3: rat,B3: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) )
=> ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ).
% power_mono_even
thf(fact_6388_power__mono__even,axiom,
! [N: nat,A3: int,B3: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ).
% power_mono_even
thf(fact_6389_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M6: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6390_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M6: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6391_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M6: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6392_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_6393_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_6394_lemma__interval,axiom,
! [A3: real,X: real,B3: real] :
( ( ord_less_real @ A3 @ X )
=> ( ( ord_less_real @ X @ B3 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A3 @ Y5 )
& ( ord_less_eq_real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_6395_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F: nat > nat > nat,A3: nat,B3: nat,Acc2: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc2 ) ) ) )
=> ( ( ( ord_less_nat @ B3 @ A3 )
=> ( ( set_fo2584398358068434914at_nat @ F @ A3 @ B3 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less_nat @ B3 @ A3 )
=> ( ( set_fo2584398358068434914at_nat @ F @ A3 @ B3 @ Acc2 )
= ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_6396_fold__atLeastAtMost__nat_Opelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) )
=> ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_6397_lemma__interval__lt,axiom,
! [A3: real,X: real,B3: real] :
( ( ord_less_real @ A3 @ X )
=> ( ( ord_less_real @ X @ B3 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_real @ A3 @ Y5 )
& ( ord_less_real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_6398_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_6399_in__measure,axiom,
! [X: code_integer,Y: code_integer,F: code_integer > nat] :
( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measure_Code_integer @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_6400_in__measure,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur8038558561449204169at_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_6401_in__measure,axiom,
! [X: nat,Y: nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_6402_in__measure,axiom,
! [X: int,Y: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_6403_tanh__0,axiom,
( ( tanh_real @ zero_zero_real )
= zero_zero_real ) ).
% tanh_0
thf(fact_6404_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_6405_tanh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% tanh_real_le_iff
thf(fact_6406_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_6407_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% tanh_real_pos_iff
thf(fact_6408_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% tanh_real_neg_iff
thf(fact_6409_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_6410_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_6411_infinite__int__iff__unbounded__le,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M6: int] :
? [N3: int] :
( ( ord_less_eq_int @ M6 @ ( abs_abs_int @ N3 ) )
& ( member_int @ N3 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_6412_infinite__int__iff__unbounded,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M6: int] :
? [N3: int] :
( ( ord_less_int @ M6 @ ( abs_abs_int @ N3 ) )
& ( member_int @ N3 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_6413_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% tanh_real_lt_1
thf(fact_6414_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_6415_dvd__imp__le__int,axiom,
! [I: int,D: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_6416_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_6417_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_6418_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_6419_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F3: nat > nat > nat,A: nat,B: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B @ A ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F3 @ A @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_6420_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_6421_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_6422_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_6423_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_6424_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_6425_in__finite__psubset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B5 ) @ finite_psubset_nat )
= ( ( ord_less_set_nat @ A4 @ B5 )
& ( finite_finite_nat @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_6426_in__finite__psubset,axiom,
! [A4: set_int,B5: set_int] :
( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A4 @ B5 ) @ finite_psubset_int )
= ( ( ord_less_set_int @ A4 @ B5 )
& ( finite_finite_int @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_6427_in__finite__psubset,axiom,
! [A4: set_complex,B5: set_complex] :
( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A4 @ B5 ) @ finite8643634255014194347omplex )
= ( ( ord_less_set_complex @ A4 @ B5 )
& ( finite3207457112153483333omplex @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_6428_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_6429_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( 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 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_6430_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_6431_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_6432_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_6433_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_6434_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_6435_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( 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_6436_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6437_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( 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_6438_flip__bit__0,axiom,
! [A3: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A3 )
= ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6439_flip__bit__0,axiom,
! [A3: code_integer] :
( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A3 )
= ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6440_flip__bit__0,axiom,
! [A3: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A3 )
= ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6441_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_6442_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_6443_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_6444_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_6445_of__bool__eq_I1_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $false )
= zero_zero_rat ) ).
% of_bool_eq(1)
thf(fact_6446_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_6447_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_6448_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= zero_zero_real )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_6449_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= zero_zero_rat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_6450_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_6451_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= zero_zero_int )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_6452_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_6453_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_6454_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_6455_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_6456_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= one_one_complex )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6457_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= one_one_real )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6458_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= one_one_rat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6459_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6460_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6461_of__bool__eq_I2_J,axiom,
( ( zero_n1201886186963655149omplex @ $true )
= one_one_complex ) ).
% of_bool_eq(2)
thf(fact_6462_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_6463_of__bool__eq_I2_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $true )
= one_one_rat ) ).
% of_bool_eq(2)
thf(fact_6464_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_6465_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_6466_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% abs_bool_eq
thf(fact_6467_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% abs_bool_eq
thf(fact_6468_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% abs_bool_eq
thf(fact_6469_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% abs_bool_eq
thf(fact_6470_of__int__fact,axiom,
! [N: nat] :
( ( ring_18347121197199848620nteger @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri3624122377584611663nteger @ N ) ) ).
% of_int_fact
thf(fact_6471_of__int__fact,axiom,
! [N: nat] :
( ( ring_1_of_int_rat @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri773545260158071498ct_rat @ N ) ) ).
% of_int_fact
thf(fact_6472_of__int__fact,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri2265585572941072030t_real @ N ) ) ).
% of_int_fact
thf(fact_6473_of__bool__or__iff,axiom,
! [P: $o,Q: $o] :
( ( zero_n356916108424825756nteger
@ ( P
| Q ) )
= ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% of_bool_or_iff
thf(fact_6474_of__bool__or__iff,axiom,
! [P: $o,Q: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P
| Q ) )
= ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% of_bool_or_iff
thf(fact_6475_of__bool__or__iff,axiom,
! [P: $o,Q: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P
| Q ) )
= ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% of_bool_or_iff
thf(fact_6476_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ zero_zero_nat @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_6477_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_6478_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_6479_of__int__0,axiom,
( ( ring_18347121197199848620nteger @ zero_zero_int )
= zero_z3403309356797280102nteger ) ).
% of_int_0
thf(fact_6480_of__int__0,axiom,
( ( ring_1_of_int_rat @ zero_zero_int )
= zero_zero_rat ) ).
% of_int_0
thf(fact_6481_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6482_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6483_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_z3403309356797280102nteger
= ( ring_18347121197199848620nteger @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6484_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_rat
= ( ring_1_of_int_rat @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6485_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6486_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6487_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_18347121197199848620nteger @ Z )
= zero_z3403309356797280102nteger )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6488_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= zero_zero_rat )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6489_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_6490_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_6491_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_6492_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_6493_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_6494_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_6495_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_6496_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_6497_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_6498_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_6499_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_6500_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_6501_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_6502_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_6503_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_6504_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_6505_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n1201886186963655149omplex @ ~ P )
= ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6506_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n3304061248610475627l_real @ ~ P )
= ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6507_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n2052037380579107095ol_rat @ ~ P )
= ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6508_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n2684676970156552555ol_int @ ~ P )
= ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6509_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_17405671764205052669omplex @ Z )
= one_one_complex )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6510_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6511_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6512_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_18347121197199848620nteger @ Z )
= one_one_Code_integer )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6513_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= one_one_rat )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6514_of__int__1,axiom,
( ( ring_17405671764205052669omplex @ one_one_int )
= one_one_complex ) ).
% of_int_1
thf(fact_6515_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_6516_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_6517_of__int__1,axiom,
( ( ring_18347121197199848620nteger @ one_one_int )
= one_one_Code_integer ) ).
% of_int_1
thf(fact_6518_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_6519_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_6520_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% arctan_less_zero_iff
thf(fact_6521_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_arctan_iff
thf(fact_6522_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_6523_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_6524_frac__of__int,axiom,
! [Z: int] :
( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
= zero_zero_real ) ).
% frac_of_int
thf(fact_6525_frac__of__int,axiom,
! [Z: int] :
( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
= zero_zero_rat ) ).
% frac_of_int
thf(fact_6526_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_6527_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
= ( ord_less_int @ L @ zero_zero_int ) ) ).
% concat_bit_negative_iff
thf(fact_6528_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_6529_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6530_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_6531_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_6532_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_6533_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6534_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_6535_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_6536_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6537_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_6538_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_6539_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_6540_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6541_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_6542_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_6543_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_6544_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_6545_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6546_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_6547_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_6548_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_6549_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6550_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_6551_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_6552_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_6553_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_6554_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_6555_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6556_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_6557_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_6558_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_6559_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6560_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_6561_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6562_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_6563_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_6564_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_6565_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6566_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_6567_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_6568_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6569_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_6570_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_6571_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_6572_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6573_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_6574_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_6575_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_6576_of__int__le__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W2 ) @ ( ring_18347121197199848620nteger @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6577_of__int__le__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6578_of__int__le__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6579_of__int__le__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6580_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6581_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6582_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6583_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6584_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6585_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6586_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6587_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B3: int,W2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B3 @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6588_of__int__less__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W2 ) @ ( ring_18347121197199848620nteger @ X ) )
= ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6589_of__int__less__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6590_of__int__less__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6591_of__int__less__of__int__power__cancel__iff,axiom,
! [B3: int,W2: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6592_of__bool__half__eq__0,axiom,
! [B3: $o] :
( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% of_bool_half_eq_0
thf(fact_6593_of__bool__half__eq__0,axiom,
! [B3: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_6594_of__bool__half__eq__0,axiom,
! [B3: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_6595_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6596_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6597_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6598_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6599_of__int__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6600_of__int__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6601_of__int__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6602_of__int__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6603_of__int__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6604_of__int__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6605_of__int__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6606_of__int__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6607_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6608_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6609_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6610_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6611_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_6612_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_6613_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_6614_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_6615_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_6616_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_6617_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_6618_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_6619_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_6620_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6621_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6622_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6623_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6624_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6625_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6626_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6627_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6628_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6629_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6630_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6631_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6632_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6633_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6634_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6635_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A3 ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6636_of__bool__eq__iff,axiom,
! [P6: $o,Q4: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P6 )
= ( zero_n2687167440665602831ol_nat @ Q4 ) )
= ( P6 = Q4 ) ) ).
% of_bool_eq_iff
thf(fact_6637_of__bool__eq__iff,axiom,
! [P6: $o,Q4: $o] :
( ( ( zero_n2684676970156552555ol_int @ P6 )
= ( zero_n2684676970156552555ol_int @ Q4 ) )
= ( P6 = Q4 ) ) ).
% of_bool_eq_iff
thf(fact_6638_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_6639_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_6640_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_6641_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_6642_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_6643_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_6644_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_6645_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_6646_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_6647_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_6648_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_6649_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_6650_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone'
thf(fact_6651_arctan__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% arctan_le_iff
thf(fact_6652_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_6653_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_6654_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_6655_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_6656_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_6657_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_6658_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_6659_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_6660_split__of__bool__asm,axiom,
! [P: complex > $o,P6: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_complex ) )
| ( ~ P6
& ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6661_split__of__bool__asm,axiom,
! [P: real > $o,P6: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_real ) )
| ( ~ P6
& ~ ( P @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6662_split__of__bool__asm,axiom,
! [P: rat > $o,P6: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_rat ) )
| ( ~ P6
& ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6663_split__of__bool__asm,axiom,
! [P: nat > $o,P6: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_nat ) )
| ( ~ P6
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6664_split__of__bool__asm,axiom,
! [P: int > $o,P6: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_int ) )
| ( ~ P6
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6665_split__of__bool,axiom,
! [P: complex > $o,P6: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
= ( ( P6
=> ( P @ one_one_complex ) )
& ( ~ P6
=> ( P @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_6666_split__of__bool,axiom,
! [P: real > $o,P6: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
= ( ( P6
=> ( P @ one_one_real ) )
& ( ~ P6
=> ( P @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_6667_split__of__bool,axiom,
! [P: rat > $o,P6: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
= ( ( P6
=> ( P @ one_one_rat ) )
& ( ~ P6
=> ( P @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_6668_split__of__bool,axiom,
! [P: nat > $o,P6: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
= ( ( P6
=> ( P @ one_one_nat ) )
& ( ~ P6
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_6669_split__of__bool,axiom,
! [P: int > $o,P6: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
= ( ( P6
=> ( P @ one_one_int ) )
& ( ~ P6
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_6670_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_6671_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_6672_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_6673_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_6674_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_6675_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_6676_le__of__int__ceiling,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_6677_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L3: num] :
( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_6678_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_6679_ceiling__le__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_6680_ceiling__le,axiom,
! [X: real,A3: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A3 ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A3 ) ) ).
% ceiling_le
thf(fact_6681_ceiling__le,axiom,
! [X: rat,A3: int] :
( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A3 ) )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A3 ) ) ).
% ceiling_le
thf(fact_6682_less__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_6683_less__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_6684_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_6685_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_6686_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% of_int_nonneg
thf(fact_6687_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_6688_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_6689_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_6690_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_6691_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_6692_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_6693_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% of_int_pos
thf(fact_6694_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_6695_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_6696_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_6697_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_6698_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_6699_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_6700_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_6701_floor__exists1,axiom,
! [X: real] :
? [X4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
=> ( Y5 = X4 ) ) ) ).
% floor_exists1
thf(fact_6702_floor__exists1,axiom,
! [X: rat] :
? [X4: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
=> ( Y5 = X4 ) ) ) ).
% floor_exists1
thf(fact_6703_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_6704_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_6705_of__int__ceiling__le__add__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% of_int_ceiling_le_add_one
thf(fact_6706_of__int__ceiling__le__add__one,axiom,
! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% of_int_ceiling_le_add_one
thf(fact_6707_of__int__ceiling__diff__one__le,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% of_int_ceiling_diff_one_le
thf(fact_6708_of__int__ceiling__diff__one__le,axiom,
! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% of_int_ceiling_diff_one_le
thf(fact_6709_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N3: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_6710_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N3: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% int_less_real_le
thf(fact_6711_ceiling__correct,axiom,
! [X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% ceiling_correct
thf(fact_6712_ceiling__correct,axiom,
! [X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% ceiling_correct
thf(fact_6713_ceiling__unique,axiom,
! [Z: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
=> ( ( archim7802044766580827645g_real @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_6714_ceiling__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_6715_ceiling__eq__iff,axiom,
! [X: real,A3: int] :
( ( ( archim7802044766580827645g_real @ X )
= A3 )
= ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A3 ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A3 ) ) ) ) ).
% ceiling_eq_iff
thf(fact_6716_ceiling__eq__iff,axiom,
! [X: rat,A3: int] :
( ( ( archim2889992004027027881ng_rat @ X )
= A3 )
= ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A3 ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A3 ) ) ) ) ).
% ceiling_eq_iff
thf(fact_6717_ceiling__split,axiom,
! [P: int > $o,T: real] :
( ( P @ ( archim7802044766580827645g_real @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
& ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_6718_ceiling__split,axiom,
! [P: int > $o,T: rat] :
( ( P @ ( archim2889992004027027881ng_rat @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
& ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_6719_ceiling__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% ceiling_less_iff
thf(fact_6720_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% ceiling_less_iff
thf(fact_6721_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% le_ceiling_iff
thf(fact_6722_le__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% le_ceiling_iff
thf(fact_6723_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_6724_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_6725_ceiling__divide__upper,axiom,
! [Q4: real,P6: real] :
( ( ord_less_real @ zero_zero_real @ Q4 )
=> ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q4 ) ) ) @ Q4 ) ) ) ).
% ceiling_divide_upper
thf(fact_6726_ceiling__divide__upper,axiom,
! [Q4: rat,P6: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q4 ) ) ) @ Q4 ) ) ) ).
% ceiling_divide_upper
thf(fact_6727_mult__ceiling__le__Ints,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A3 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A3 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6728_mult__ceiling__le__Ints,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim7802044766580827645g_real @ ( times_times_real @ A3 @ B3 ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim7802044766580827645g_real @ A3 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6729_mult__ceiling__le__Ints,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A3 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A3 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6730_mult__ceiling__le__Ints,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( member_real @ A3 @ ring_1_Ints_real )
=> ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A3 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A3 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6731_mult__ceiling__le__Ints,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A3 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A3 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6732_mult__ceiling__le__Ints,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A3 @ B3 ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A3 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6733_mult__ceiling__le__Ints,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A3 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A3 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6734_mult__ceiling__le__Ints,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
=> ( ( member_rat @ A3 @ ring_1_Ints_rat )
=> ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A3 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A3 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_6735_ceiling__divide__lower,axiom,
! [Q4: rat,P6: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P6 ) ) ).
% ceiling_divide_lower
thf(fact_6736_ceiling__divide__lower,axiom,
! [Q4: real,P6: real] :
( ( ord_less_real @ zero_zero_real @ Q4 )
=> ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) @ P6 ) ) ).
% ceiling_divide_lower
thf(fact_6737_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_6738_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_6739_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_6740_ceiling__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim7802044766580827645g_real @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_6741_ceiling__eq,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_6742_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L3: num] :
( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_6743_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_6744_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_6745_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_6746_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_6747_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L3: num] :
( produc4245557441103728435nt_int
@ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_6748_divmod__step__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L3: num] :
( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_6749_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L3: num] :
( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_6750_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L3: num] :
( produc4245557441103728435nt_int
@ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_6751_round__unique,axiom,
! [X: real,Y: int] :
( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= Y ) ) ) ).
% round_unique
thf(fact_6752_round__unique,axiom,
! [X: rat,Y: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= Y ) ) ) ).
% round_unique
thf(fact_6753_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_6754_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_6755_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_6756_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_6757_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M6: nat,N3: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N3 = zero_zero_nat )
| ( ord_less_nat @ M6 @ N3 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
@ ( produc2626176000494625587at_nat
@ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_6758_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_6759_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_6760_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_6761_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_6762_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_6763_round__0,axiom,
( ( archim7778729529865785530nd_rat @ zero_zero_rat )
= zero_zero_int ) ).
% round_0
thf(fact_6764_round__1,axiom,
( ( archim8280529875227126926d_real @ one_one_real )
= one_one_int ) ).
% round_1
thf(fact_6765_round__1,axiom,
( ( archim7778729529865785530nd_rat @ one_one_rat )
= one_one_int ) ).
% round_1
thf(fact_6766_divmod__integer_H__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M6: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_6767_less__eq__integer__code_I1_J,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% less_eq_integer_code(1)
thf(fact_6768_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
= K ) ).
% plus_integer_code(1)
thf(fact_6769_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_6770_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(1)
thf(fact_6771_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(2)
thf(fact_6772_minus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
= K ) ).
% minus_integer_code(1)
thf(fact_6773_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
= ( uminus1351360451143612070nteger @ L ) ) ).
% minus_integer_code(2)
thf(fact_6774_round__mono,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% round_mono
thf(fact_6775_ceiling__ge__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% ceiling_ge_round
thf(fact_6776_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_6777_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_6778_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_6779_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_6780_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_6781_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_6782_binomial__code,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K3 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_6783_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_6784_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_6785_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_6786_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : X3
@ ( 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_6787_take__bit__rec,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N3: nat,A: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_6788_take__bit__rec,axiom,
( bit_se1745604003318907178nteger
= ( ^ [N3: nat,A: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_6789_take__bit__rec,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N3: nat,A: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_6790_integer__of__int__eq__of__int,axiom,
code_integer_of_int = ring_18347121197199848620nteger ).
% integer_of_int_eq_of_int
thf(fact_6791_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% take_bit_of_0
thf(fact_6792_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% take_bit_of_0
thf(fact_6793_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% take_bit_of_0
thf(fact_6794_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_6795_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_6796_sum_Oneutral__const,axiom,
! [A4: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu3: int] : zero_zero_int
@ A4 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_6797_sum_Oneutral__const,axiom,
! [A4: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu3: complex] : zero_zero_complex
@ A4 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_6798_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu3: nat] : zero_zero_nat
@ A4 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_6799_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu3: nat] : zero_zero_real
@ A4 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_6800_sum_Oempty,axiom,
! [G: real > real] :
( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_6801_sum_Oempty,axiom,
! [G: real > rat] :
( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
= zero_zero_rat ) ).
% sum.empty
thf(fact_6802_sum_Oempty,axiom,
! [G: real > nat] :
( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_6803_sum_Oempty,axiom,
! [G: real > int] :
( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_6804_sum_Oempty,axiom,
! [G: nat > rat] :
( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty
thf(fact_6805_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_6806_sum_Oempty,axiom,
! [G: int > real] :
( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
= zero_zero_real ) ).
% sum.empty
thf(fact_6807_sum_Oempty,axiom,
! [G: int > rat] :
( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
= zero_zero_rat ) ).
% sum.empty
thf(fact_6808_sum_Oempty,axiom,
! [G: int > nat] :
( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
= zero_zero_nat ) ).
% sum.empty
thf(fact_6809_sum_Oempty,axiom,
! [G: int > int] :
( ( groups4538972089207619220nt_int @ G @ bot_bot_set_int )
= zero_zero_int ) ).
% sum.empty
thf(fact_6810_sum_Oinfinite,axiom,
! [A4: set_int,G: int > real] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_6811_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_6812_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_6813_sum_Oinfinite,axiom,
! [A4: set_int,G: int > rat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_6814_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > rat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_6815_sum_Oinfinite,axiom,
! [A4: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_6816_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_6817_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_6818_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > int] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_6819_sum_Oinfinite,axiom,
! [A4: set_int,G: int > int] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_6820_sum__eq__0__iff,axiom,
! [F4: set_int,F: int > nat] :
( ( finite_finite_int @ F4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ F4 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_6821_sum__eq__0__iff,axiom,
! [F4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ F4 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_6822_sum__eq__0__iff,axiom,
! [F4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ F4 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_6823_take__bit__0,axiom,
! [A3: nat] :
( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% take_bit_0
thf(fact_6824_take__bit__0,axiom,
! [A3: code_integer] :
( ( bit_se1745604003318907178nteger @ zero_zero_nat @ A3 )
= zero_z3403309356797280102nteger ) ).
% take_bit_0
thf(fact_6825_take__bit__0,axiom,
! [A3: int] :
( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A3 )
= zero_zero_int ) ).
% take_bit_0
thf(fact_6826_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_Suc_1
thf(fact_6827_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% take_bit_Suc_1
thf(fact_6828_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% take_bit_Suc_1
thf(fact_6829_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_numeral_1
thf(fact_6830_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% take_bit_numeral_1
thf(fact_6831_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
= one_one_int ) ).
% take_bit_numeral_1
thf(fact_6832_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_6833_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_6834_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_6835_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_6836_pochhammer__0,axiom,
! [A3: complex] :
( ( comm_s2602460028002588243omplex @ A3 @ zero_zero_nat )
= one_one_complex ) ).
% pochhammer_0
thf(fact_6837_pochhammer__0,axiom,
! [A3: real] :
( ( comm_s7457072308508201937r_real @ A3 @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_6838_pochhammer__0,axiom,
! [A3: rat] :
( ( comm_s4028243227959126397er_rat @ A3 @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_6839_pochhammer__0,axiom,
! [A3: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_6840_pochhammer__0,axiom,
! [A3: int] :
( ( comm_s4660882817536571857er_int @ A3 @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_6841_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_6842_sum_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_6843_sum_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_6844_sum_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_6845_sum_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_6846_sum_Odelta,axiom,
! [S2: set_nat,A3: nat,B3: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_6847_sum_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_6848_sum_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_6849_sum_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_6850_sum_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_6851_sum_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K3: complex] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K3: complex] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_6852_sum_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_6853_sum_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_6854_sum_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_6855_sum_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_6856_sum_Odelta_H,axiom,
! [S2: set_nat,A3: nat,B3: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_6857_sum_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_6858_sum_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_6859_sum_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_6860_sum_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_6861_sum_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K3: complex] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K3: complex] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_6862_sum__abs,axiom,
! [F: int > int,A4: set_int] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs
thf(fact_6863_sum__abs,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs
thf(fact_6864_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_6865_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
= zero_z3403309356797280102nteger )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_6866_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_6867_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_6868_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_6869_sum__abs__ge__zero,axiom,
! [F: int > int,A4: set_int] :
( ord_less_eq_int @ zero_zero_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs_ge_zero
thf(fact_6870_sum__abs__ge__zero,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs_ge_zero
thf(fact_6871_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_6872_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_6873_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_6874_even__take__bit__eq,axiom,
! [N: nat,A3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A3 ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ).
% even_take_bit_eq
thf(fact_6875_even__take__bit__eq,axiom,
! [N: nat,A3: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A3 ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ).
% even_take_bit_eq
thf(fact_6876_even__take__bit__eq,axiom,
! [N: nat,A3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A3 ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ).
% even_take_bit_eq
thf(fact_6877_take__bit__Suc__0,axiom,
! [A3: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A3 )
= ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_6878_take__bit__Suc__0,axiom,
! [A3: code_integer] :
( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A3 )
= ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_6879_take__bit__Suc__0,axiom,
! [A3: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A3 )
= ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_6880_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6881_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6882_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6883_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6884_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6885_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6886_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s7457072308508201937r_real @ ( ring_1_of_int_real @ X ) @ N )
= ( ring_1_of_int_real @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_6887_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s8582702949713902594nteger @ ( ring_18347121197199848620nteger @ X ) @ N )
= ( ring_18347121197199848620nteger @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_6888_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( ring_1_of_int_rat @ X ) @ N )
= ( ring_1_of_int_rat @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_6889_euclidean__size__integer_Oabs__eq,axiom,
! [X: int] :
( ( euclid6377331345833325938nteger @ ( code_integer_of_int @ X ) )
= ( euclid4774559944035922753ze_int @ X ) ) ).
% euclidean_size_integer.abs_eq
thf(fact_6890_take__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se1745604003318907178nteger @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2923211474154528505it_int @ Xa2 @ X ) ) ) ).
% take_bit_integer.abs_eq
thf(fact_6891_uminus__integer__code_I1_J,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% uminus_integer_code(1)
thf(fact_6892_less__integer__code_I1_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% less_integer_code(1)
thf(fact_6893_abs__integer__code,axiom,
( abs_abs_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_6894_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A4: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6895_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A4: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6896_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A4: set_real] :
( ( ( groups1300246762558778688al_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6897_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A4: set_nat] :
( ( ( groups2906978787729119204at_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6898_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A4: set_int] :
( ( ( groups3906332499630173760nt_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6899_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A4: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A4 )
!= zero_zero_nat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6900_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A4: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A4 )
!= zero_zero_nat )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6901_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A4: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A4 )
!= zero_zero_int )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6902_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A4: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A4 )
!= zero_zero_int )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6903_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > int,A4: set_int] :
( ( ( groups4538972089207619220nt_int @ G @ A4 )
!= zero_zero_int )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_6904_sum_Oneutral,axiom,
! [A4: set_int,G: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_6905_sum_Oneutral,axiom,
! [A4: set_complex,G: complex > complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_6906_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_6907_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_6908_take__bit__tightened,axiom,
! [N: nat,A3: nat,B3: nat,M: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ A3 )
= ( bit_se2925701944663578781it_nat @ N @ B3 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( bit_se2925701944663578781it_nat @ M @ A3 )
= ( bit_se2925701944663578781it_nat @ M @ B3 ) ) ) ) ).
% take_bit_tightened
thf(fact_6909_take__bit__tightened,axiom,
! [N: nat,A3: code_integer,B3: code_integer,M: nat] :
( ( ( bit_se1745604003318907178nteger @ N @ A3 )
= ( bit_se1745604003318907178nteger @ N @ B3 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( bit_se1745604003318907178nteger @ M @ A3 )
= ( bit_se1745604003318907178nteger @ M @ B3 ) ) ) ) ).
% take_bit_tightened
thf(fact_6910_take__bit__tightened,axiom,
! [N: nat,A3: int,B3: int,M: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ A3 )
= ( bit_se2923211474154528505it_int @ N @ B3 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( bit_se2923211474154528505it_int @ M @ A3 )
= ( bit_se2923211474154528505it_int @ M @ B3 ) ) ) ) ).
% take_bit_tightened
thf(fact_6911_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N @ Q4 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_6912_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_6913_zero__integer__def,axiom,
( zero_z3403309356797280102nteger
= ( code_integer_of_int @ zero_zero_int ) ) ).
% zero_integer_def
thf(fact_6914_less__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( ord_less_int @ Xa2 @ X ) ) ).
% less_integer.abs_eq
thf(fact_6915_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_6916_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus1351360451143612070nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus_int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_6917_divide__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide_int @ Xa2 @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_6918_abs__integer_Oabs__eq,axiom,
! [X: int] :
( ( abs_abs_Code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( abs_abs_int @ X ) ) ) ).
% abs_integer.abs_eq
thf(fact_6919_modulo__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( modulo_modulo_int @ Xa2 @ X ) ) ) ).
% modulo_integer.abs_eq
thf(fact_6920_sum__mono,axiom,
! [K4: set_real,F: real > rat,G: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ K4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6921_sum__mono,axiom,
! [K4: set_nat,F: nat > rat,G: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6922_sum__mono,axiom,
! [K4: set_int,F: int > rat,G: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ K4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6923_sum__mono,axiom,
! [K4: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6924_sum__mono,axiom,
! [K4: set_int,F: int > nat,G: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6925_sum__mono,axiom,
! [K4: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ K4 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6926_sum__mono,axiom,
! [K4: set_nat,F: nat > int,G: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K4 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6927_sum__mono,axiom,
! [K4: set_int,F: int > int,G: int > int] :
( ! [I2: int] :
( ( member_int @ I2 @ K4 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K4 ) @ ( groups4538972089207619220nt_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6928_sum__mono,axiom,
! [K4: set_nat,F: nat > nat,G: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K4 ) @ ( groups3542108847815614940at_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6929_sum__mono,axiom,
! [K4: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K4 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K4 ) @ ( groups6591440286371151544t_real @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_6930_unset__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se8260200283734997820nteger @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa2 @ X ) ) ) ).
% unset_bit_integer.abs_eq
thf(fact_6931_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_int,G: real > int > int,R: real > int > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups1932886352136224148al_int
@ ^ [X3: real] :
( groups4538972089207619220nt_int @ ( G @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y3: int] :
( groups1932886352136224148al_int
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6932_sum_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_int,G: nat > int > int,R: nat > int > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups3539618377306564664at_int
@ ^ [X3: nat] :
( groups4538972089207619220nt_int @ ( G @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y3: int] :
( groups3539618377306564664at_int
@ ^ [X3: nat] : ( G @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6933_sum_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups5690904116761175830ex_int
@ ^ [X3: complex] :
( groups4538972089207619220nt_int @ ( G @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y3: int] :
( groups5690904116761175830ex_int
@ ^ [X3: complex] : ( G @ X3 @ Y3 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6934_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_complex,G: real > complex > complex,R: real > complex > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups5754745047067104278omplex
@ ^ [X3: real] :
( groups7754918857620584856omplex @ ( G @ X3 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y3: complex] :
( groups5754745047067104278omplex
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6935_sum_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups2073611262835488442omplex
@ ^ [X3: nat] :
( groups7754918857620584856omplex @ ( G @ X3 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y3: complex] :
( groups2073611262835488442omplex
@ ^ [X3: nat] : ( G @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6936_sum_Oswap__restrict,axiom,
! [A4: set_int,B5: set_complex,G: int > complex > complex,R: int > complex > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups3049146728041665814omplex
@ ^ [X3: int] :
( groups7754918857620584856omplex @ ( G @ X3 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y3: complex] :
( groups3049146728041665814omplex
@ ^ [X3: int] : ( G @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6937_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > nat,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups1935376822645274424al_nat
@ ^ [X3: real] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups1935376822645274424al_nat
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6938_sum_Oswap__restrict,axiom,
! [A4: set_int,B5: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( G @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6939_sum_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X3: complex] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups5693394587270226106ex_nat
@ ^ [X3: complex] : ( G @ X3 @ Y3 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6940_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > real,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups8097168146408367636l_real
@ ^ [X3: real] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups6591440286371151544t_real
@ ^ [Y3: nat] :
( groups8097168146408367636l_real
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_6941_set__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se2793503036327961859nteger @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se7879613467334960850it_int @ Xa2 @ X ) ) ) ).
% set_bit_integer.abs_eq
thf(fact_6942_mod__sum__eq,axiom,
! [F: int > int,A3: int,A4: set_int] :
( ( modulo_modulo_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A3 ) ) ).
% mod_sum_eq
thf(fact_6943_mod__sum__eq,axiom,
! [F: nat > nat,A3: nat,A4: set_nat] :
( ( modulo_modulo_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A3 ) ) ).
% mod_sum_eq
thf(fact_6944_flip__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se1345352211410354436nteger @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2159334234014336723it_int @ Xa2 @ X ) ) ) ).
% flip_bit_integer.abs_eq
thf(fact_6945_sum__nonpos,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_6946_sum__nonpos,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_6947_sum__nonpos,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6948_sum__nonpos,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6949_sum__nonpos,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_6950_sum__nonpos,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_6951_sum__nonpos,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_6952_sum__nonpos,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_6953_sum__nonpos,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_6954_sum__nonpos,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_6955_sum__nonneg,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6956_sum__nonneg,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6957_sum__nonneg,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6958_sum__nonneg,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6959_sum__nonneg,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6960_sum__nonneg,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6961_sum__nonneg,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6962_sum__nonneg,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6963_sum__nonneg,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6964_sum__nonneg,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_6965_sum__mono__inv,axiom,
! [F: real > rat,I5: set_real,G: real > rat,I: real] :
( ( ( groups1300246762558778688al_rat @ F @ I5 )
= ( groups1300246762558778688al_rat @ G @ I5 ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_real @ I @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6966_sum__mono__inv,axiom,
! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
( ( ( groups2906978787729119204at_rat @ F @ I5 )
= ( groups2906978787729119204at_rat @ G @ I5 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6967_sum__mono__inv,axiom,
! [F: int > rat,I5: set_int,G: int > rat,I: int] :
( ( ( groups3906332499630173760nt_rat @ F @ I5 )
= ( groups3906332499630173760nt_rat @ G @ I5 ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_int @ I @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6968_sum__mono__inv,axiom,
! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
( ( ( groups5058264527183730370ex_rat @ F @ I5 )
= ( groups5058264527183730370ex_rat @ G @ I5 ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_complex @ I @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6969_sum__mono__inv,axiom,
! [F: real > nat,I5: set_real,G: real > nat,I: real] :
( ( ( groups1935376822645274424al_nat @ F @ I5 )
= ( groups1935376822645274424al_nat @ G @ I5 ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_real @ I @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6970_sum__mono__inv,axiom,
! [F: int > nat,I5: set_int,G: int > nat,I: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I5 )
= ( groups4541462559716669496nt_nat @ G @ I5 ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_int @ I @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6971_sum__mono__inv,axiom,
! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I5 )
= ( groups5693394587270226106ex_nat @ G @ I5 ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_complex @ I @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6972_sum__mono__inv,axiom,
! [F: real > int,I5: set_real,G: real > int,I: real] :
( ( ( groups1932886352136224148al_int @ F @ I5 )
= ( groups1932886352136224148al_int @ G @ I5 ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_real @ I @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6973_sum__mono__inv,axiom,
! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
( ( ( groups3539618377306564664at_int @ F @ I5 )
= ( groups3539618377306564664at_int @ G @ I5 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6974_sum__mono__inv,axiom,
! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I5 )
= ( groups5690904116761175830ex_int @ G @ I5 ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_complex @ I @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_6975_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_6976_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_6977_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_6978_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_6979_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_6980_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_6981_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_6982_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_6983_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [N: nat,A3: code_integer,B3: code_integer] :
( ( ( bit_ri6519982836138164636nteger @ N @ A3 )
= ( bit_ri6519982836138164636nteger @ N @ B3 ) )
= ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A3 )
= ( bit_se1745604003318907178nteger @ ( suc @ N ) @ B3 ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_6984_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [N: nat,A3: int,B3: int] :
( ( ( bit_ri631733984087533419it_int @ N @ A3 )
= ( bit_ri631733984087533419it_int @ N @ B3 ) )
= ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 )
= ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B3 ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_6985_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_6986_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_6987_signed__take__bit__take__bit,axiom,
! [M: nat,N: nat,A3: code_integer] :
( ( bit_ri6519982836138164636nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A3 ) )
= ( if_Cod4779417660136461971nteger @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se1745604003318907178nteger @ N ) @ ( bit_ri6519982836138164636nteger @ M ) @ A3 ) ) ).
% signed_take_bit_take_bit
thf(fact_6988_signed__take__bit__take__bit,axiom,
! [M: nat,N: nat,A3: int] :
( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A3 ) )
= ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A3 ) ) ).
% signed_take_bit_take_bit
thf(fact_6989_take__bit__unset__bit__eq,axiom,
! [N: nat,M: nat,A3: int] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A3 ) )
= ( bit_se2923211474154528505it_int @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A3 ) )
= ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A3 ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_6990_take__bit__unset__bit__eq,axiom,
! [N: nat,M: nat,A3: code_integer] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A3 ) )
= ( bit_se1745604003318907178nteger @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A3 ) )
= ( bit_se8260200283734997820nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A3 ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_6991_take__bit__unset__bit__eq,axiom,
! [N: nat,M: nat,A3: nat] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A3 ) )
= ( bit_se2925701944663578781it_nat @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A3 ) )
= ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A3 ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_6992_take__bit__set__bit__eq,axiom,
! [N: nat,M: nat,A3: nat] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A3 ) )
= ( bit_se2925701944663578781it_nat @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A3 ) )
= ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A3 ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_6993_take__bit__set__bit__eq,axiom,
! [N: nat,M: nat,A3: code_integer] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A3 ) )
= ( bit_se1745604003318907178nteger @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A3 ) )
= ( bit_se2793503036327961859nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A3 ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_6994_take__bit__set__bit__eq,axiom,
! [N: nat,M: nat,A3: int] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A3 ) )
= ( bit_se2923211474154528505it_int @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A3 ) )
= ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A3 ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_6995_take__bit__flip__bit__eq,axiom,
! [N: nat,M: nat,A3: nat] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A3 ) )
= ( bit_se2925701944663578781it_nat @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A3 ) )
= ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A3 ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_6996_take__bit__flip__bit__eq,axiom,
! [N: nat,M: nat,A3: code_integer] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A3 ) )
= ( bit_se1745604003318907178nteger @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A3 ) )
= ( bit_se1345352211410354436nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A3 ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_6997_take__bit__flip__bit__eq,axiom,
! [N: nat,M: nat,A3: int] :
( ( ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A3 ) )
= ( bit_se2923211474154528505it_int @ N @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A3 ) )
= ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A3 ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_6998_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_6999_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_7000_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_7001_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_7002_pochhammer__neq__0__mono,axiom,
! [A3: real,M: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A3 @ M )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A3 @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7003_pochhammer__neq__0__mono,axiom,
! [A3: rat,M: nat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A3 @ M )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A3 @ N )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7004_pochhammer__eq__0__mono,axiom,
! [A3: real,N: nat,M: nat] :
( ( ( comm_s7457072308508201937r_real @ A3 @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A3 @ M )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7005_pochhammer__eq__0__mono,axiom,
! [A3: rat,N: nat,M: nat] :
( ( ( comm_s4028243227959126397er_rat @ A3 @ N )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A3 @ M )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7006_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_7007_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_7008_one__integer__def,axiom,
( one_one_Code_integer
= ( code_integer_of_int @ one_one_int ) ) ).
% one_integer_def
thf(fact_7009_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_7010_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7011_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7012_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7013_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > rat,P: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups1300246762558778688al_rat
@ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7014_sum_Ointer__filter,axiom,
! [A4: set_nat,G: nat > rat,P: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7015_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > rat,P: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7016_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > rat,P: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups5058264527183730370ex_rat
@ ^ [X3: complex] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7017_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1935376822645274424al_nat @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7018_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > nat,P: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7019_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups5693394587270226106ex_nat
@ ^ [X3: complex] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_7020_minus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_7021_pochhammer__fact,axiom,
( semiri5044797733671781792omplex
= ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% pochhammer_fact
thf(fact_7022_pochhammer__fact,axiom,
( semiri773545260158071498ct_rat
= ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% pochhammer_fact
thf(fact_7023_pochhammer__fact,axiom,
( semiri1406184849735516958ct_int
= ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% pochhammer_fact
thf(fact_7024_pochhammer__fact,axiom,
( semiri1408675320244567234ct_nat
= ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% pochhammer_fact
thf(fact_7025_pochhammer__fact,axiom,
( semiri2265585572941072030t_real
= ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% pochhammer_fact
thf(fact_7026_sum__le__included,axiom,
! [S3: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7027_sum__le__included,axiom,
! [S3: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7028_sum__le__included,axiom,
! [S3: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7029_sum__le__included,axiom,
! [S3: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7030_sum__le__included,axiom,
! [S3: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7031_sum__le__included,axiom,
! [S3: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7032_sum__le__included,axiom,
! [S3: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7033_sum__le__included,axiom,
! [S3: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7034_sum__le__included,axiom,
! [S3: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7035_sum__le__included,axiom,
! [S3: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_7036_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > real] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A4 )
= zero_zero_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7037_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A4 )
= zero_zero_real )
= ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7038_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A4 )
= zero_zero_real )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7039_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7040_sum__nonneg__eq__0__iff,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7041_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7042_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7043_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7044_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7045_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A4 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_7046_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7047_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7048_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7049_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7050_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7051_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7052_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7053_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7054_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7055_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_7056_sum_Orelated,axiom,
! [R: real > real > $o,S2: set_int,H: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7057_sum_Orelated,axiom,
! [R: real > real > $o,S2: set_complex,H: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7058_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_nat,H: nat > rat,G: nat > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2906978787729119204at_rat @ H @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7059_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_int,H: int > rat,G: int > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3906332499630173760nt_rat @ H @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7060_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_complex,H: complex > rat,G: complex > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5058264527183730370ex_rat @ H @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7061_sum_Orelated,axiom,
! [R: nat > nat > $o,S2: set_int,H: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7062_sum_Orelated,axiom,
! [R: nat > nat > $o,S2: set_complex,H: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7063_sum_Orelated,axiom,
! [R: int > int > $o,S2: set_nat,H: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y23: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7064_sum_Orelated,axiom,
! [R: int > int > $o,S2: set_complex,H: complex > int,G: complex > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y23: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5690904116761175830ex_int @ H @ S2 ) @ ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7065_sum_Orelated,axiom,
! [R: int > int > $o,S2: set_int,H: int > int,G: int > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y23: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups4538972089207619220nt_int @ H @ S2 ) @ ( groups4538972089207619220nt_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_7066_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7067_sum__strict__mono,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7068_sum__strict__mono,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7069_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7070_sum__strict__mono,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7071_sum__strict__mono,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7072_sum__strict__mono,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7073_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7074_sum__strict__mono,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7075_sum__strict__mono,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_7076_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_7077_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7078_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T2: set_int,G: real > real,H: int > real] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_int @ ( J @ A2 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7079_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T2: set_complex,G: real > real,H: complex > real] :
( ( finite_finite_real @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_complex @ ( J @ A2 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7080_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T2: set_real,G: int > real,H: real > real] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7081_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > real,H: int > real] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( member_int @ ( J @ A2 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7082_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T2: set_complex,G: int > real,H: complex > real] :
( ( finite_finite_int @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( member_complex @ ( J @ A2 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7083_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T2: set_real,G: complex > real,H: real > real] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B2 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7084_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T2: set_int,G: complex > real,H: int > real] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( member_int @ ( J @ A2 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B2 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7085_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T2: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( member_complex @ ( J @ A2 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B2 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7086_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ S2 )
= ( groups1300246762558778688al_rat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_7087_Suc__times__binomial__add,axiom,
! [A3: nat,B3: nat] :
( ( times_times_nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B3 ) ) @ ( suc @ A3 ) ) )
= ( times_times_nat @ ( suc @ B3 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B3 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_7088_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_7089_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_7090_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_7091_take__bit__signed__take__bit,axiom,
! [M: nat,N: nat,A3: code_integer] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( bit_se1745604003318907178nteger @ M @ ( bit_ri6519982836138164636nteger @ N @ A3 ) )
= ( bit_se1745604003318907178nteger @ M @ A3 ) ) ) ).
% take_bit_signed_take_bit
thf(fact_7092_take__bit__signed__take__bit,axiom,
! [M: nat,N: nat,A3: int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A3 ) )
= ( bit_se2923211474154528505it_int @ M @ A3 ) ) ) ).
% take_bit_signed_take_bit
thf(fact_7093_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_7094_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_7095_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_7096_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_7097_sum__nonneg__leq__bound,axiom,
! [S3: set_real,F: real > real,B5: real,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S3 )
= B5 )
=> ( ( member_real @ I @ S3 )
=> ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7098_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > real,B5: real,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S3 )
= B5 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7099_sum__nonneg__leq__bound,axiom,
! [S3: set_complex,F: complex > real,B5: real,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S3 )
= B5 )
=> ( ( member_complex @ I @ S3 )
=> ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7100_sum__nonneg__leq__bound,axiom,
! [S3: set_real,F: real > rat,B5: rat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S3 )
= B5 )
=> ( ( member_real @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7101_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F: nat > rat,B5: rat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S3 )
= B5 )
=> ( ( member_nat @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7102_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > rat,B5: rat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
= B5 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7103_sum__nonneg__leq__bound,axiom,
! [S3: set_complex,F: complex > rat,B5: rat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
= B5 )
=> ( ( member_complex @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7104_sum__nonneg__leq__bound,axiom,
! [S3: set_real,F: real > nat,B5: nat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S3 )
= B5 )
=> ( ( member_real @ I @ S3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7105_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > nat,B5: nat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
= B5 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7106_sum__nonneg__leq__bound,axiom,
! [S3: set_complex,F: complex > nat,B5: nat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
= B5 )
=> ( ( member_complex @ I @ S3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_7107_sum__nonneg__0,axiom,
! [S3: set_real,F: real > real,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S3 )
= zero_zero_real )
=> ( ( member_real @ I @ S3 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7108_sum__nonneg__0,axiom,
! [S3: set_int,F: int > real,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S3 )
= zero_zero_real )
=> ( ( member_int @ I @ S3 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7109_sum__nonneg__0,axiom,
! [S3: set_complex,F: complex > real,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S3 )
= zero_zero_real )
=> ( ( member_complex @ I @ S3 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7110_sum__nonneg__0,axiom,
! [S3: set_real,F: real > rat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_real @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7111_sum__nonneg__0,axiom,
! [S3: set_nat,F: nat > rat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_nat @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7112_sum__nonneg__0,axiom,
! [S3: set_int,F: int > rat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_int @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7113_sum__nonneg__0,axiom,
! [S3: set_complex,F: complex > rat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_complex @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7114_sum__nonneg__0,axiom,
! [S3: set_real,F: real > nat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_real @ I @ S3 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7115_sum__nonneg__0,axiom,
! [S3: set_int,F: int > nat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_int @ I @ S3 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7116_sum__nonneg__0,axiom,
! [S3: set_complex,F: complex > nat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_complex @ I @ S3 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_7117_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% pochhammer_0_left
thf(fact_7118_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_7119_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_7120_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_7121_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_7122_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= zero_zero_real ) ) ) )
= ( groups8097168146408367636l_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7123_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= zero_zero_real ) ) ) )
= ( groups8778361861064173332t_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7124_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X3: complex] :
( ( G @ X3 )
= zero_zero_real ) ) ) )
= ( groups5808333547571424918x_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7125_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= zero_zero_rat ) ) ) )
= ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7126_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= zero_zero_rat ) ) ) )
= ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7127_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X3: complex] :
( ( G @ X3 )
= zero_zero_rat ) ) ) )
= ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7128_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( groups1935376822645274424al_nat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= zero_zero_nat ) ) ) )
= ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7129_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7130_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X3: complex] :
( ( G @ X3 )
= zero_zero_nat ) ) ) )
= ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7131_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( groups1932886352136224148al_int @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= zero_zero_int ) ) ) )
= ( groups1932886352136224148al_int @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_7132_sum__pos2,axiom,
! [I5: set_real,I: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7133_sum__pos2,axiom,
! [I5: set_int,I: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7134_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7135_sum__pos2,axiom,
! [I5: set_real,I: real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7136_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7137_sum__pos2,axiom,
! [I5: set_int,I: int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7138_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7139_sum__pos2,axiom,
! [I5: set_real,I: real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7140_sum__pos2,axiom,
! [I5: set_int,I: int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7141_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_7142_sum__pos,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7143_sum__pos,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7144_sum__pos,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7145_sum__pos,axiom,
! [I5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7146_sum__pos,axiom,
! [I5: set_real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7147_sum__pos,axiom,
! [I5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7148_sum__pos,axiom,
! [I5: set_int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7149_sum__pos,axiom,
! [I5: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7150_sum__pos,axiom,
! [I5: set_real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7151_sum__pos,axiom,
! [I5: set_int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_7152_sum_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ A4 )
= ( groups8097168146408367636l_real @ H @ B5 ) )
= ( ( groups8097168146408367636l_real @ G @ C2 )
= ( groups8097168146408367636l_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7153_sum_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ A4 )
= ( groups5808333547571424918x_real @ H @ B5 ) )
= ( ( groups5808333547571424918x_real @ G @ C2 )
= ( groups5808333547571424918x_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7154_sum_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( groups1300246762558778688al_rat @ H @ B5 ) )
= ( ( groups1300246762558778688al_rat @ G @ C2 )
= ( groups1300246762558778688al_rat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7155_sum_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H: complex > rat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( groups5058264527183730370ex_rat @ H @ B5 ) )
= ( ( groups5058264527183730370ex_rat @ G @ C2 )
= ( groups5058264527183730370ex_rat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7156_sum_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( groups1935376822645274424al_nat @ H @ B5 ) )
= ( ( groups1935376822645274424al_nat @ G @ C2 )
= ( groups1935376822645274424al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7157_sum_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H: complex > nat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( groups5693394587270226106ex_nat @ H @ B5 ) )
= ( ( groups5693394587270226106ex_nat @ G @ C2 )
= ( groups5693394587270226106ex_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7158_sum_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ A4 )
= ( groups1932886352136224148al_int @ H @ B5 ) )
= ( ( groups1932886352136224148al_int @ G @ C2 )
= ( groups1932886352136224148al_int @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7159_sum_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H: complex > int] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( groups5690904116761175830ex_int @ H @ B5 ) )
= ( ( groups5690904116761175830ex_int @ G @ C2 )
= ( groups5690904116761175830ex_int @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7160_sum_Osame__carrier,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H: nat > rat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups2906978787729119204at_rat @ G @ A4 )
= ( groups2906978787729119204at_rat @ H @ B5 ) )
= ( ( groups2906978787729119204at_rat @ G @ C2 )
= ( groups2906978787729119204at_rat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7161_sum_Osame__carrier,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_int ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ( ( groups3539618377306564664at_int @ G @ A4 )
= ( groups3539618377306564664at_int @ H @ B5 ) )
= ( ( groups3539618377306564664at_int @ G @ C2 )
= ( groups3539618377306564664at_int @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_7162_sum_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ C2 )
= ( groups8097168146408367636l_real @ H @ C2 ) )
=> ( ( groups8097168146408367636l_real @ G @ A4 )
= ( groups8097168146408367636l_real @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7163_sum_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ C2 )
= ( groups5808333547571424918x_real @ H @ C2 ) )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( groups5808333547571424918x_real @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7164_sum_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ C2 )
= ( groups1300246762558778688al_rat @ H @ C2 ) )
=> ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( groups1300246762558778688al_rat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7165_sum_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H: complex > rat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ C2 )
= ( groups5058264527183730370ex_rat @ H @ C2 ) )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( groups5058264527183730370ex_rat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7166_sum_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ C2 )
= ( groups1935376822645274424al_nat @ H @ C2 ) )
=> ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( groups1935376822645274424al_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7167_sum_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H: complex > nat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ C2 )
= ( groups5693394587270226106ex_nat @ H @ C2 ) )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( groups5693394587270226106ex_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7168_sum_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ C2 )
= ( groups1932886352136224148al_int @ H @ C2 ) )
=> ( ( groups1932886352136224148al_int @ G @ A4 )
= ( groups1932886352136224148al_int @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7169_sum_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H: complex > int] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ C2 )
= ( groups5690904116761175830ex_int @ H @ C2 ) )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( groups5690904116761175830ex_int @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7170_sum_Osame__carrierI,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H: nat > rat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_rat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_rat ) )
=> ( ( ( groups2906978787729119204at_rat @ G @ C2 )
= ( groups2906978787729119204at_rat @ H @ C2 ) )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= ( groups2906978787729119204at_rat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7171_sum_Osame__carrierI,axiom,
! [C2: set_nat,A4: set_nat,B5: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A4 ) )
=> ( ( G @ A2 )
= zero_zero_int ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B5 ) )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ( ( groups3539618377306564664at_int @ G @ C2 )
= ( groups3539618377306564664at_int @ H @ C2 ) )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= ( groups3539618377306564664at_int @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_7172_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7173_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S2 )
= ( groups5058264527183730370ex_rat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7174_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7175_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ S2 )
= ( groups5690904116761175830ex_int @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7176_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ S2 )
= ( groups2906978787729119204at_rat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7177_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7178_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8778361861064173332t_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7179_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > rat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ S2 )
= ( groups3906332499630173760nt_rat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7180_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > nat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7181_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > int] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_7182_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ T2 )
= ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7183_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T2 )
= ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7184_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T2 )
= ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7185_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ T2 )
= ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7186_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ T2 )
= ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7187_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ T2 )
= ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7188_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ T2 )
= ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7189_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > rat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ T2 )
= ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7190_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > nat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G @ T2 )
= ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7191_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > int] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ T2 )
= ( groups4538972089207619220nt_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_7192_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > real,G: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7193_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7194_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > rat,G: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ S2 )
= ( groups1300246762558778688al_rat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7195_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S2 )
= ( groups5058264527183730370ex_rat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7196_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > nat,G: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7197_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7198_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > int,G: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ S2 )
= ( groups1932886352136224148al_int @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7199_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ S2 )
= ( groups5690904116761175830ex_int @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7200_sum_Omono__neutral__cong__left,axiom,
! [T2: set_nat,S2: set_nat,H: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups2906978787729119204at_rat @ G @ S2 )
= ( groups2906978787729119204at_rat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7201_sum_Omono__neutral__cong__left,axiom,
! [T2: set_nat,S2: set_nat,H: nat > int,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( H @ X4 )
= zero_zero_int ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_7202_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ T2 )
= ( groups8097168146408367636l_real @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7203_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ T2 )
= ( groups5808333547571424918x_real @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7204_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ T2 )
= ( groups1300246762558778688al_rat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7205_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat,H: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T2 )
= ( groups5058264527183730370ex_rat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7206_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ T2 )
= ( groups1935376822645274424al_nat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7207_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat,H: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T2 )
= ( groups5693394587270226106ex_nat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7208_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ T2 )
= ( groups1932886352136224148al_int @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7209_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int,H: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ T2 )
= ( groups5690904116761175830ex_int @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7210_sum_Omono__neutral__cong__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat,H: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups2906978787729119204at_rat @ G @ T2 )
= ( groups2906978787729119204at_rat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7211_sum_Omono__neutral__cong__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ T2 )
= ( groups3539618377306564664at_int @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_7212_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7213_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7214_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7215_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7216_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7217_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7218_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7219_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7220_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7221_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_7222_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7223_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7224_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7225_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7226_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7227_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7228_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7229_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7230_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7231_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_7232_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_7233_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_7234_pochhammer__rec,axiom,
! [A3: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A3 @ ( suc @ N ) )
= ( times_times_complex @ A3 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7235_pochhammer__rec,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ A3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A3 @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7236_pochhammer__rec,axiom,
! [A3: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A3 @ ( suc @ N ) )
= ( times_times_rat @ A3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7237_pochhammer__rec,axiom,
! [A3: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ A3 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7238_pochhammer__rec,axiom,
! [A3: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A3 @ ( suc @ N ) )
= ( times_times_int @ A3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A3 @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7239_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_7240_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
= ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_7241_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_7242_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_7243_binomial__mono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K5 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% binomial_mono
thf(fact_7244_binomial__antimono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K5 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
=> ( ( ord_less_eq_nat @ K5 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_7245_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_7246_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7247_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7248_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7249_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7250_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7251_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7252_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7253_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7254_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7255_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_7256_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_7257_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_7258_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_7259_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_7260_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_7261_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_7262_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_7263_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_7264_take__bit__eq__0__iff,axiom,
! [N: nat,A3: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ A3 )
= zero_zero_nat )
= ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A3 ) ) ).
% take_bit_eq_0_iff
thf(fact_7265_take__bit__eq__0__iff,axiom,
! [N: nat,A3: code_integer] :
( ( ( bit_se1745604003318907178nteger @ N @ A3 )
= zero_z3403309356797280102nteger )
= ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A3 ) ) ).
% take_bit_eq_0_iff
thf(fact_7266_take__bit__eq__0__iff,axiom,
! [N: nat,A3: int] :
( ( ( bit_se2923211474154528505it_int @ N @ A3 )
= zero_zero_int )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A3 ) ) ).
% take_bit_eq_0_iff
thf(fact_7267_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B3: real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7268_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B3: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7269_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B3: real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7270_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B3: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7271_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B3: nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7272_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B3: real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7273_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B3: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7274_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B3: real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B3 ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7275_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B3: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B3 ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7276_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B3: nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B3 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_7277_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_7278_binomial__strict__mono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_nat @ K @ K5 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% binomial_strict_mono
thf(fact_7279_binomial__strict__antimono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_nat @ K @ K5 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K5 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_7280_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_7281_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_7282_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_7283_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_7284_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > code_integer,A3: real > code_integer,B3: code_integer,Delta: code_integer] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7713935264441627589nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7713935264441627589nteger
@ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7285_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > code_integer,A3: nat > code_integer,B3: code_integer,Delta: code_integer] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7501900531339628137nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7286_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > code_integer,A3: int > code_integer,B3: code_integer,Delta: code_integer] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7873554091576472773nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7873554091576472773nteger
@ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7287_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > real,A3: real > real,B3: real,Delta: real] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ X @ I5 )
= one_one_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( times_times_real @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7288_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > real,A3: int > real,B3: real,Delta: real] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ X @ I5 )
= one_one_real )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( times_times_real @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7289_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > rat,A3: real > rat,B3: rat,Delta: rat] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups1300246762558778688al_rat
@ ^ [I3: real] : ( times_times_rat @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7290_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > rat,A3: nat > rat,B3: rat,Delta: rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7291_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > rat,A3: int > rat,B3: rat,Delta: rat] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups3906332499630173760nt_rat
@ ^ [I3: int] : ( times_times_rat @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7292_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > int,A3: real > int,B3: int,Delta: int] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups1932886352136224148al_int @ X @ I5 )
= one_one_int )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups1932886352136224148al_int
@ ^ [I3: real] : ( times_times_int @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7293_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > int,A3: nat > int,B3: int,Delta: int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups3539618377306564664at_int @ X @ I5 )
= one_one_int )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7294_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_7295_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_7296_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_7297_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_7298_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_7299_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_7300_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_7301_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_Suc_bit1
thf(fact_7302_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).
% take_bit_Suc_bit1
thf(fact_7303_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_bit1
thf(fact_7304_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_7305_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_7306_take__bit__Suc,axiom,
! [N: nat,A3: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A3 )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_7307_take__bit__Suc,axiom,
! [N: nat,A3: code_integer] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A3 )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_7308_take__bit__Suc,axiom,
! [N: nat,A3: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_7309_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_7310_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_7311_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_7312_stable__imp__take__bit__eq,axiom,
! [A3: nat,N: nat] :
( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( bit_se2925701944663578781it_nat @ N @ A3 )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
=> ( ( bit_se2925701944663578781it_nat @ N @ A3 )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_7313_stable__imp__take__bit__eq,axiom,
! [A3: code_integer,N: nat] :
( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A3 )
=> ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( bit_se1745604003318907178nteger @ N @ A3 )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
=> ( ( bit_se1745604003318907178nteger @ N @ A3 )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_7314_stable__imp__take__bit__eq,axiom,
! [A3: int,N: nat] :
( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( bit_se2923211474154528505it_int @ N @ A3 )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
=> ( ( bit_se2923211474154528505it_int @ N @ A3 )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_7315_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_7316_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_7317_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_7318_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_7319_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_7320_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_7321_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_numeral_bit1
thf(fact_7322_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).
% take_bit_numeral_bit1
thf(fact_7323_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_bit1
thf(fact_7324_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_7325_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_7326_Code__Numeral_Opositive__def,axiom,
code_positive = numera6620942414471956472nteger ).
% Code_Numeral.positive_def
thf(fact_7327_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_7328_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_7329_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri4939895301339042750nteger @ M )
= ( semiri4939895301339042750nteger @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_7330_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% abs_of_nat
thf(fact_7331_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_7332_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_7333_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( semiri4939895301339042750nteger @ N ) ) ).
% abs_of_nat
thf(fact_7334_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_7335_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_7336_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_7337_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_7338_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_less_mono
thf(fact_7339_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_7340_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_7341_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid6377331345833325938nteger @ ( semiri4939895301339042750nteger @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_7342_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4774559944035922753ze_int @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_7343_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4777050414544973029ze_nat @ ( semiri1316708129612266289at_nat @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_7344_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_7345_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_7346_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_7347_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_7348_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri4939895301339042750nteger @ M )
= zero_z3403309356797280102nteger )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_7349_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_7350_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_7351_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_7352_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_7353_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z3403309356797280102nteger
= ( semiri4939895301339042750nteger @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_7354_of__nat__0,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% of_nat_0
thf(fact_7355_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_7356_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_7357_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_7358_of__nat__0,axiom,
( ( semiri4939895301339042750nteger @ zero_zero_nat )
= zero_z3403309356797280102nteger ) ).
% of_nat_0
thf(fact_7359_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_7360_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_7361_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_7362_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_7363_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_7364_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_7365_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_7366_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_7367_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_7368_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_7369_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_7370_of__nat__numeral,axiom,
! [N: num] :
( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N ) )
= ( numera1916890842035813515d_enat @ N ) ) ).
% of_nat_numeral
thf(fact_7371_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_7372_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_7373_of__nat__numeral,axiom,
! [N: num] :
( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% of_nat_numeral
thf(fact_7374_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_7375_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_7376_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_7377_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_7378_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N ) )
= ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_add
thf(fact_7379_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_7380_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_7381_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_7382_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_7383_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mult
thf(fact_7384_exp__zero,axiom,
( ( exp_complex @ zero_zero_complex )
= one_one_complex ) ).
% exp_zero
thf(fact_7385_exp__zero,axiom,
( ( exp_real @ zero_zero_real )
= one_one_real ) ).
% exp_zero
thf(fact_7386_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_7387_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_7388_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_7389_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_7390_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_7391_of__nat__1,axiom,
( ( semiri4939895301339042750nteger @ one_one_nat )
= one_one_Code_integer ) ).
% of_nat_1
thf(fact_7392_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_7393_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_7394_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_7395_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_7396_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_7397_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_integer
= ( semiri4939895301339042750nteger @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_7398_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_7399_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_7400_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_7401_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_7402_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_7403_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri4939895301339042750nteger @ N )
= one_one_Code_integer )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_7404_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_7405_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_7406_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_7407_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_7408_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_7409_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_7410_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_7411_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_7412_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_7413_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_7414_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_7415_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_7416_of__nat__fact,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri1406184849735516958ct_int @ N ) ) ).
% of_nat_fact
thf(fact_7417_of__nat__fact,axiom,
! [N: nat] :
( ( semiri4939895301339042750nteger @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri3624122377584611663nteger @ N ) ) ).
% of_nat_fact
thf(fact_7418_of__nat__fact,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% of_nat_fact
thf(fact_7419_of__nat__fact,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri2265585572941072030t_real @ N ) ) ).
% of_nat_fact
thf(fact_7420_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% pred_numeral_inc
thf(fact_7421_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% of_nat_of_bool
thf(fact_7422_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% of_nat_of_bool
thf(fact_7423_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% of_nat_of_bool
thf(fact_7424_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_nat_of_bool
thf(fact_7425_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_7426_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_7427_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_7428_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_7429_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_7430_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_7431_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% of_nat_Suc
thf(fact_7432_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_7433_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_7434_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_7435_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ M ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).
% of_nat_Suc
thf(fact_7436_fact__Suc,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_Suc
thf(fact_7437_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_7438_fact__Suc,axiom,
! [N: nat] :
( ( semiri3624122377584611663nteger @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% fact_Suc
thf(fact_7439_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_7440_fact__Suc,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_Suc
thf(fact_7441_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_7442_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_7443_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_7444_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_less_exp_iff
thf(fact_7445_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_7446_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_7447_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_7448_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_7449_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_7450_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_7451_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_7452_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln_ln_real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_7453_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln_ln_real @ X ) )
= X )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% exp_ln_iff
thf(fact_7454_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_7455_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_7456_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_7457_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_7458_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_7459_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_7460_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_7461_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_7462_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_7463_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_7464_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_7465_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_7466_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_7467_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_7468_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_7469_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_7470_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_7471_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_7472_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_7473_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_7474_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_7475_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_7476_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_7477_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_7478_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_7479_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_7480_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_7481_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_7482_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_7483_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_7484_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B3: nat,W2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_7485_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_7486_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_7487_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_7488_sum__zero__power,axiom,
! [A4: set_nat,C: nat > complex] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A4 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A4 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_7489_sum__zero__power,axiom,
! [A4: set_nat,C: nat > rat] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A4 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A4 )
= zero_zero_rat ) ) ) ).
% sum_zero_power
thf(fact_7490_sum__zero__power,axiom,
! [A4: set_nat,C: nat > real] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A4 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A4 )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_7491_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_7492_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_7493_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_7494_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_7495_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_7496_sum__zero__power_H,axiom,
! [A4: set_nat,C: nat > complex,D: nat > complex] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= zero_zero_complex ) ) ) ).
% sum_zero_power'
thf(fact_7497_sum__zero__power_H,axiom,
! [A4: set_nat,C: nat > rat,D: nat > rat] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= zero_zero_rat ) ) ) ).
% sum_zero_power'
thf(fact_7498_sum__zero__power_H,axiom,
! [A4: set_nat,C: nat > real,D: nat > real] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= zero_zero_real ) ) ) ).
% sum_zero_power'
thf(fact_7499_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_7500_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_7501_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_7502_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_7503_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_7504_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_7505_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_7506_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_7507_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_7508_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_7509_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_7510_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_7511_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_7512_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_7513_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_7514_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_7515_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_7516_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_7517_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_7518_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_7519_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_7520_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_7521_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_7522_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_7523_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_7524_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_7525_real__arch__simple,axiom,
! [X: rat] :
? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% real_arch_simple
thf(fact_7526_reals__Archimedean2,axiom,
! [X: rat] :
? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% reals_Archimedean2
thf(fact_7527_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_7528_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7529_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7530_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7531_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7532_mult__of__nat__commute,axiom,
! [X: nat,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7533_exp__not__eq__zero,axiom,
! [X: real] :
( ( exp_real @ X )
!= zero_zero_real ) ).
% exp_not_eq_zero
thf(fact_7534_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_7535_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_7536_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s8582702949713902594nteger @ ( semiri4939895301339042750nteger @ X ) @ N )
= ( semiri4939895301339042750nteger @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_7537_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_7538_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_7539_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_7540_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_7541_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_7542_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_7543_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_7544_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_less_zero
thf(fact_7545_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_gt_zero
thf(fact_7546_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( exp_real @ X4 )
= Y ) ) ).
% exp_total
thf(fact_7547_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_7548_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_7549_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_le_zero
thf(fact_7550_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_ge_zero
thf(fact_7551_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_7552_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).
% of_nat_0_le_iff
thf(fact_7553_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_7554_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_7555_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_7556_sum__cong__Suc,axiom,
! [A4: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A4 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% sum_cong_Suc
thf(fact_7557_sum__cong__Suc,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A4 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A4 )
= ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).
% sum_cong_Suc
thf(fact_7558_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_7559_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_7560_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_7561_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_7562_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).
% of_nat_less_0_iff
thf(fact_7563_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_7564_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_7565_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_7566_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_7567_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_7568_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ N ) )
!= zero_z3403309356797280102nteger ) ).
% of_nat_neq_0
thf(fact_7569_div__mult2__eq_H,axiom,
! [A3: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A3 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7570_div__mult2__eq_H,axiom,
! [A3: int,M: nat,N: nat] :
( ( divide_divide_int @ A3 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7571_div__mult2__eq_H,axiom,
! [A3: code_integer,M: nat,N: nat] :
( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7572_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_7573_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_7574_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_7575_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_7576_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_7577_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_7578_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_7579_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_7580_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_7581_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_7582_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_7583_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).
% of_nat_mono
thf(fact_7584_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% of_nat_mono
thf(fact_7585_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_7586_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_7587_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_7588_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_7589_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N ) )
= ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_7590_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_7591_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_7592_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_7593_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_7594_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_7595_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_7596_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_7597_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_7598_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_7599_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_7600_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_7601_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_7602_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_7603_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mod
thf(fact_7604_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_7605_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z2: int] :
? [N3: nat] :
( Z2
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_7606_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_7607_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_7608_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_max
thf(fact_7609_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% of_nat_max
thf(fact_7610_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_max
thf(fact_7611_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).
% of_nat_max
thf(fact_7612_sum__subtractf__nat,axiom,
! [A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups977919841031483927at_nat
@ ^ [X3: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7613_sum__subtractf__nat,axiom,
! [A4: set_real,G: real > nat,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7614_sum__subtractf__nat,axiom,
! [A4: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( groups8294997508430121362at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7615_sum__subtractf__nat,axiom,
! [A4: set_int,G: int > nat,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7616_sum__subtractf__nat,axiom,
! [A4: set_nat,G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7617_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_7618_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_7619_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_7620_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_7621_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_less_as_int
thf(fact_7622_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_leq_as_int
thf(fact_7623_sum__eq__Suc0__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= ( suc @ zero_zero_nat ) )
& ! [Y3: int] :
( ( member_int @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_7624_sum__eq__Suc0__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= ( suc @ zero_zero_nat ) )
& ! [Y3: complex] :
( ( member_complex @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_7625_sum__eq__Suc0__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= ( suc @ zero_zero_nat ) )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_7626_sum__SucD,axiom,
! [F: nat > nat,A4: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ N ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_7627_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% exp_gt_one
thf(fact_7628_sum__eq__1__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= one_one_nat )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y3: int] :
( ( member_int @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_7629_sum__eq__1__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= one_one_nat )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y3: complex] :
( ( member_complex @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_7630_sum__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= one_one_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_7631_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_7632_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7633_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7634_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_7635_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_7636_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_7637_exp__minus__inverse,axiom,
! [X: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
= one_one_complex ) ).
% exp_minus_inverse
thf(fact_7638_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_7639_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_7640_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_7641_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_7642_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_7643_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N ) )
= ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% of_nat_diff
thf(fact_7644_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_7645_fact__le__power,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7646_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_7647_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_7648_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_7649_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_7650_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_7651_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_7652_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_7653_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_7654_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_7655_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z2: int] :
? [N3: nat] :
( Z2
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_7656_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_7657_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_7658_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_7659_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_7660_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_7661_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_7662_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_7663_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_7664_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_7665_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_7666_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z2: complex] :
( ( power_power_complex @ Z2 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_7667_sqrt2__less__2,axiom,
ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% sqrt2_less_2
thf(fact_7668_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_7669_sum__diff__nat,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_7670_sum__diff__nat,axiom,
! [B5: set_int,A4: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_7671_sum__diff__nat,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_7672_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_7673_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_7674_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_7675_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_7676_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_7677_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_7678_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_7679_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_7680_double__gauss__sum,axiom,
! [N: nat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% double_gauss_sum
thf(fact_7681_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_7682_double__gauss__sum,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum
thf(fact_7683_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_7684_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_7685_double__arith__series,axiom,
! [A3: complex,D: complex,N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7686_double__arith__series,axiom,
! [A3: rat,D: rat,N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7687_double__arith__series,axiom,
! [A3: extended_enat,D: extended_enat,N: nat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
@ ( groups7108830773950497114d_enat
@ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A3 @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A3 ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7688_double__arith__series,axiom,
! [A3: int,D: int,N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7689_double__arith__series,axiom,
! [A3: code_integer,D: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7690_double__arith__series,axiom,
! [A3: nat,D: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7691_double__arith__series,axiom,
! [A3: real,D: real,N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_7692_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_7693_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_7694_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_7695_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_7696_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ Y )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
& ( ( exp_real @ X4 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_7697_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_7698_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_7699_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_7700_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_7701_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_7702_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_7703_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_7704_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_7705_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_7706_mod__mult2__eq_H,axiom,
! [A3: nat,M: nat,N: nat] :
( ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7707_mod__mult2__eq_H,axiom,
! [A3: int,M: nat,N: nat] :
( ( modulo_modulo_int @ A3 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7708_mod__mult2__eq_H,axiom,
! [A3: code_integer,M: nat,N: nat] :
( ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A3 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7709_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_7710_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_7711_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_7712_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_7713_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_7714_pochhammer__rec_H,axiom,
! [Z: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7715_pochhammer__Suc,axiom,
! [A3: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A3 @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A3 @ N ) @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7716_pochhammer__Suc,axiom,
! [A3: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A3 @ N ) @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7717_pochhammer__Suc,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A3 @ N ) @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7718_pochhammer__Suc,axiom,
! [A3: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A3 @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A3 @ N ) @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7719_pochhammer__Suc,axiom,
! [A3: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A3 @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A3 @ N ) @ ( plus_p5714425477246183910nteger @ A3 @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7720_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_7721_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_7722_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_7723_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_7724_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_7725_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_7726_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_7727_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_7728_pochhammer__eq__0__iff,axiom,
! [A3: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A3 @ N )
= zero_zero_rat )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A3
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7729_pochhammer__eq__0__iff,axiom,
! [A3: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A3 @ N )
= zero_zero_real )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A3
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7730_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_7731_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_7732_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_7733_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_7734_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_7735_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_7736_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_7737_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_7738_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_7739_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_7740_pochhammer__product_H,axiom,
! [Z: code_integer,N: nat,M: nat] :
( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7741_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_7742_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_7743_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_7744_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% nat_less_real_le
thf(fact_7745_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_7746_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_7747_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_7748_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_7749_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_7750_int__ops_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B3 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B3 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_7751_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_7752_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_7753_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_7754_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_7755_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_7756_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_7757_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_7758_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_7759_gauss__sum,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7760_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_7761_arith__series,axiom,
! [A3: int,D: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7762_arith__series,axiom,
! [A3: code_integer,D: code_integer,N: nat] :
( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7763_arith__series,axiom,
! [A3: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7764_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_7765_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_7766_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7767_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_7768_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7769_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_7770_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_7771_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_7772_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_7773_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_7774_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_7775_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_7776_fact__numeral,axiom,
! [K: num] :
( ( semiri4449623510593786356d_enat @ ( numeral_numeral_nat @ K ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ K ) @ ( semiri4449623510593786356d_enat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7777_fact__numeral,axiom,
! [K: num] :
( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ K ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ K ) @ ( semiri3624122377584611663nteger @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7778_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_7779_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_7780_sum__atLeastAtMost__code,axiom,
! [F: nat > rat,A3: nat,B3: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A: nat] : ( plus_plus_rat @ ( F @ A ) )
@ A3
@ B3
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7781_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A3: nat,B3: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2581907887559384638at_int
@ ^ [A: nat] : ( plus_plus_int @ ( F @ A ) )
@ A3
@ B3
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_7782_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A3: nat,B3: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A: nat] : ( plus_plus_nat @ ( F @ A ) )
@ A3
@ B3
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7783_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A3: nat,B3: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo3111899725591712190t_real
@ ^ [A: nat] : ( plus_plus_real @ ( F @ A ) )
@ A3
@ B3
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_7784_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_7785_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sqrt_le_D
thf(fact_7786_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_7787_exp__le,axiom,
ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% exp_le
thf(fact_7788_nat__approx__posE,axiom,
! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_7789_nat__approx__posE,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_7790_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_7791_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_7792_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_7793_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_7794_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_7795_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_7796_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_7797_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_7798_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_7799_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7800_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7801_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7802_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7803_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_7804_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_7805_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 )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_7806_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_7807_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_7808_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_7809_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_7810_pochhammer__product,axiom,
! [M: nat,N: nat,Z: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s8582702949713902594nteger @ Z @ N )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7811_sum__count__set,axiom,
! [Xs: list_complex,X7: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X7 )
=> ( ( finite3207457112153483333omplex @ X7 )
=> ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X7 )
= ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_7812_sum__count__set,axiom,
! [Xs: list_VEBT_VEBT,X7: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X7 )
=> ( ( finite5795047828879050333T_VEBT @ X7 )
=> ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X7 )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_7813_sum__count__set,axiom,
! [Xs: list_o,X7: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X7 )
=> ( ( finite_finite_o @ X7 )
=> ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X7 )
= ( size_size_list_o @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_7814_sum__count__set,axiom,
! [Xs: list_int,X7: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X7 )
=> ( ( finite_finite_int @ X7 )
=> ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X7 )
= ( size_size_list_int @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_7815_sum__count__set,axiom,
! [Xs: list_nat,X7: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X7 )
=> ( ( finite_finite_nat @ X7 )
=> ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X7 )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_7816_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_7817_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_7818_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_7819_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_7820_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_7821_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_7822_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_7823_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_7824_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_7825_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_7826_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_7827_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A3: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A3 @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B3 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( 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_7828_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_7829_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_7830_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_7831_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_7832_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_7833_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_7834_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_7835_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_7836_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_7837_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_7838_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_7839_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_7840_fact__num__eq__if,axiom,
( semiri3624122377584611663nteger
= ( ^ [M6: nat] : ( if_Code_integer @ ( M6 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M6 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7841_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_7842_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_7843_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_7844_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_7845_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri3624122377584611663nteger @ N )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7846_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_7847_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_7848_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_7849_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_7850_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_7851_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_7852_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_7853_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_7854_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_7855_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_7856_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_7857_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_7858_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_7859_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_7860_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_7861_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_7862_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_7863_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_7864_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_7865_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_7866_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_7867_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_7868_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_7869_real__sqrt__ge__abs1,axiom,
! [X: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_7870_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_7871_pochhammer__minus_H,axiom,
! [B3: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B3 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7872_pochhammer__minus_H,axiom,
! [B3: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B3 @ ( 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 @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7873_pochhammer__minus_H,axiom,
! [B3: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B3 @ ( 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 @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7874_pochhammer__minus_H,axiom,
! [B3: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B3 @ ( 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 @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7875_pochhammer__minus_H,axiom,
! [B3: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B3 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B3 ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7876_pochhammer__minus,axiom,
! [B3: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B3 ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B3 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7877_pochhammer__minus,axiom,
! [B3: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B3 ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B3 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7878_pochhammer__minus,axiom,
! [B3: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B3 ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7879_pochhammer__minus,axiom,
! [B3: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B3 ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B3 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7880_pochhammer__minus,axiom,
! [B3: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B3 ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B3 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7881_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
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7882_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer )
= ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7883_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
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7884_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_7885_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_7886_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_7887_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_7888_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_7889_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_7890_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_7891_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_7892_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_7893_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_7894_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: 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 @ Y @ ( 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_7895_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_7896_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_7897_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
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_7898_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( 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_7899_fact__code,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7900_fact__code,axiom,
( semiri3624122377584611663nteger
= ( ^ [N3: nat] : ( semiri4939895301339042750nteger @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7901_fact__code,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7902_fact__code,axiom,
( semiri2265585572941072030t_real
= ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7903_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_7904_cos__x__y__le__one,axiom,
! [X: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% cos_x_y_le_one
thf(fact_7905_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_7906_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_7907_pochhammer__code,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A: complex,N3: nat] :
( if_complex @ ( N3 = zero_zero_nat ) @ one_one_complex
@ ( set_fo1517530859248394432omplex
@ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N3 @ one_one_nat )
@ one_one_complex ) ) ) ) ).
% pochhammer_code
thf(fact_7908_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A: rat,N3: nat] :
( if_rat @ ( N3 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N3 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_7909_pochhammer__code,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A: real,N3: nat] :
( if_real @ ( N3 = zero_zero_nat ) @ one_one_real
@ ( set_fo3111899725591712190t_real
@ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N3 @ one_one_nat )
@ one_one_real ) ) ) ) ).
% pochhammer_code
thf(fact_7910_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A: int,N3: nat] :
( if_int @ ( N3 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N3 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_7911_pochhammer__code,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A: code_integer,N3: nat] :
( if_Code_integer @ ( N3 = zero_zero_nat ) @ one_one_Code_integer
@ ( set_fo1084959871951514735nteger
@ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N3 @ one_one_nat )
@ one_one_Code_integer ) ) ) ) ).
% pochhammer_code
thf(fact_7912_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A: nat,N3: nat] :
( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N3 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_7913_arith__series__nat,axiom,
! [A3: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( 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 ) ) @ A3 ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_7914_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_7915_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( 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_7916_of__nat__code__if,axiom,
( semiri8010041392384452111omplex
= ( ^ [N3: nat] :
( if_complex @ ( N3 = zero_zero_nat ) @ zero_zero_complex
@ ( produc1917071388513777916omplex
@ ^ [M6: nat,Q5: nat] : ( if_complex @ ( Q5 = 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7917_of__nat__code__if,axiom,
( semiri681578069525770553at_rat
= ( ^ [N3: nat] :
( if_rat @ ( N3 = zero_zero_nat ) @ zero_zero_rat
@ ( produc6207742614233964070at_rat
@ ^ [M6: nat,Q5: nat] : ( if_rat @ ( Q5 = 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7918_of__nat__code__if,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N3: nat] :
( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
@ ( produc6842872674320459806at_nat
@ ^ [M6: nat,Q5: nat] : ( if_nat @ ( Q5 = 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7919_of__nat__code__if,axiom,
( semiri4216267220026989637d_enat
= ( ^ [N3: nat] :
( if_Extended_enat @ ( N3 = zero_zero_nat ) @ zero_z5237406670263579293d_enat
@ ( produc2676513652042109336d_enat
@ ^ [M6: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M6 ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M6 ) ) @ one_on7984719198319812577d_enat ) )
@ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7920_of__nat__code__if,axiom,
( semiri5074537144036343181t_real
= ( ^ [N3: nat] :
( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
@ ( produc1703576794950452218t_real
@ ^ [M6: nat,Q5: nat] : ( if_real @ ( Q5 = 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7921_of__nat__code__if,axiom,
( semiri1314217659103216013at_int
= ( ^ [N3: nat] :
( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int
@ ( produc6840382203811409530at_int
@ ^ [M6: nat,Q5: nat] : ( if_int @ ( Q5 = 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7922_of__nat__code__if,axiom,
( semiri4939895301339042750nteger
= ( ^ [N3: nat] :
( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger
@ ( produc1830744345554046123nteger
@ ^ [M6: nat,Q5: nat] : ( if_Code_integer @ ( Q5 = zero_zero_nat ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M6 ) ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M6 ) ) @ one_one_Code_integer ) )
@ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7923_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( topolo6980174941875973593q_real
@ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_7924_lemma__termdiff3,axiom,
! [H: real,Z: real,K4: real,N: nat] :
( ( H != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_7925_lemma__termdiff3,axiom,
! [H: complex,Z: complex,K4: real,N: nat] :
( ( H != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_7926_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_7927_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_7928_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_7929_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
@ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_7930_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_7931_prod_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [Uu3: nat] : one_one_nat
@ A4 )
= one_one_nat ) ).
% prod.neutral_const
thf(fact_7932_prod_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups705719431365010083at_int
@ ^ [Uu3: nat] : one_one_int
@ A4 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_7933_prod_Oneutral__const,axiom,
! [A4: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [Uu3: int] : one_one_int
@ A4 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_7934_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups129246275422532515t_real @ F @ A4 )
= zero_zero_real )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7935_prod__zero__iff,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups2316167850115554303t_real @ F @ A4 )
= zero_zero_real )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7936_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups766887009212190081x_real @ F @ A4 )
= zero_zero_real )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7937_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups73079841787564623at_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7938_prod__zero__iff,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1072433553688619179nt_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7939_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups225925009352817453ex_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7940_prod__zero__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1707563613775114915nt_nat @ F @ A4 )
= zero_zero_nat )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7941_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups861055069439313189ex_nat @ F @ A4 )
= zero_zero_nat )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7942_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups858564598930262913ex_int @ F @ A4 )
= zero_zero_int )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_int ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7943_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups708209901874060359at_nat @ F @ A4 )
= zero_zero_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7944_prod_Oempty,axiom,
! [G: real > complex] :
( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
= one_one_complex ) ).
% prod.empty
thf(fact_7945_prod_Oempty,axiom,
! [G: real > real] :
( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
= one_one_real ) ).
% prod.empty
thf(fact_7946_prod_Oempty,axiom,
! [G: real > rat] :
( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
= one_one_rat ) ).
% prod.empty
thf(fact_7947_prod_Oempty,axiom,
! [G: real > nat] :
( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
= one_one_nat ) ).
% prod.empty
thf(fact_7948_prod_Oempty,axiom,
! [G: real > int] :
( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
= one_one_int ) ).
% prod.empty
thf(fact_7949_prod_Oempty,axiom,
! [G: nat > complex] :
( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
= one_one_complex ) ).
% prod.empty
thf(fact_7950_prod_Oempty,axiom,
! [G: nat > real] :
( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
= one_one_real ) ).
% prod.empty
thf(fact_7951_prod_Oempty,axiom,
! [G: nat > rat] :
( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
= one_one_rat ) ).
% prod.empty
thf(fact_7952_prod_Oempty,axiom,
! [G: int > complex] :
( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
= one_one_complex ) ).
% prod.empty
thf(fact_7953_prod_Oempty,axiom,
! [G: int > real] :
( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
= one_one_real ) ).
% prod.empty
thf(fact_7954_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7955_prod_Oinfinite,axiom,
! [A4: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7956_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7957_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7958_prod_Oinfinite,axiom,
! [A4: set_int,G: int > real] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7959_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7960_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7961_prod_Oinfinite,axiom,
! [A4: set_int,G: int > rat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups1072433553688619179nt_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7962_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > rat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7963_prod_Oinfinite,axiom,
! [A4: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G @ A4 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_7964_dvd__prod__eqI,axiom,
! [A4: set_real,A3: real,B3: nat,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_nat @ B3 @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7965_dvd__prod__eqI,axiom,
! [A4: set_int,A3: int,B3: nat,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_nat @ B3 @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7966_dvd__prod__eqI,axiom,
! [A4: set_complex,A3: complex,B3: nat,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_nat @ B3 @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7967_dvd__prod__eqI,axiom,
! [A4: set_real,A3: real,B3: int,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_int @ B3 @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7968_dvd__prod__eqI,axiom,
! [A4: set_complex,A3: complex,B3: int,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_int @ B3 @ ( groups858564598930262913ex_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7969_dvd__prod__eqI,axiom,
! [A4: set_real,A3: real,B3: code_integer,F: real > code_integer] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_Code_integer @ B3 @ ( groups6225526099057966256nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7970_dvd__prod__eqI,axiom,
! [A4: set_nat,A3: nat,B3: code_integer,F: nat > code_integer] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_Code_integer @ B3 @ ( groups3455450783089532116nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7971_dvd__prod__eqI,axiom,
! [A4: set_int,A3: int,B3: code_integer,F: int > code_integer] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_Code_integer @ B3 @ ( groups3827104343326376752nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7972_dvd__prod__eqI,axiom,
! [A4: set_complex,A3: complex,B3: code_integer,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_Code_integer @ B3 @ ( groups8682486955453173170nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7973_dvd__prod__eqI,axiom,
! [A4: set_nat,A3: nat,B3: nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( B3
= ( F @ A3 ) )
=> ( dvd_dvd_nat @ B3 @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7974_dvd__prodI,axiom,
! [A4: set_real,A3: real,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7975_dvd__prodI,axiom,
! [A4: set_int,A3: int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7976_dvd__prodI,axiom,
! [A4: set_complex,A3: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7977_dvd__prodI,axiom,
! [A4: set_real,A3: real,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7978_dvd__prodI,axiom,
! [A4: set_complex,A3: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( groups858564598930262913ex_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7979_dvd__prodI,axiom,
! [A4: set_real,A3: real,F: real > code_integer] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( groups6225526099057966256nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7980_dvd__prodI,axiom,
! [A4: set_nat,A3: nat,F: nat > code_integer] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( groups3455450783089532116nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7981_dvd__prodI,axiom,
! [A4: set_int,A3: int,F: int > code_integer] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( groups3827104343326376752nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7982_dvd__prodI,axiom,
! [A4: set_complex,A3: complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( groups8682486955453173170nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7983_dvd__prodI,axiom,
! [A4: set_nat,A3: nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7984_norm__fact,axiom,
! [N: nat] :
( ( real_V1022390504157884413omplex @ ( semiri5044797733671781792omplex @ N ) )
= ( semiri2265585572941072030t_real @ N ) ) ).
% norm_fact
thf(fact_7985_norm__fact,axiom,
! [N: nat] :
( ( real_V7735802525324610683m_real @ ( semiri2265585572941072030t_real @ N ) )
= ( semiri2265585572941072030t_real @ N ) ) ).
% norm_fact
thf(fact_7986_prod_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7987_prod_Odelta_H,axiom,
! [S2: set_nat,A3: nat,B3: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7988_prod_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7989_prod_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7990_prod_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7991_prod_Odelta_H,axiom,
! [S2: set_nat,A3: nat,B3: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7992_prod_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7993_prod_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7994_prod_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7995_prod_Odelta_H,axiom,
! [S2: set_nat,A3: nat,B3: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7996_prod_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7997_prod_Odelta,axiom,
! [S2: set_nat,A3: nat,B3: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7998_prod_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7999_prod_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_8000_prod_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_8001_prod_Odelta,axiom,
! [S2: set_nat,A3: nat,B3: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_8002_prod_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_8003_prod_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_8004_prod_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_8005_prod_Odelta,axiom,
! [S2: set_nat,A3: nat,B3: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_8006_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_8007_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_8008_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_8009_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_8010_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_8011_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_8012_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_8013_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A4: set_real] :
( ( ( groups713298508707869441omplex @ G @ A4 )
!= one_one_complex )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8014_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A4: set_nat] :
( ( ( groups6464643781859351333omplex @ G @ A4 )
!= one_one_complex )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8015_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A4: set_int] :
( ( ( groups7440179247065528705omplex @ G @ A4 )
!= one_one_complex )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8016_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A4: set_real] :
( ( ( groups1681761925125756287l_real @ G @ A4 )
!= one_one_real )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8017_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A4: set_nat] :
( ( ( groups129246275422532515t_real @ G @ A4 )
!= one_one_real )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8018_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A4: set_int] :
( ( ( groups2316167850115554303t_real @ G @ A4 )
!= one_one_real )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8019_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A4: set_real] :
( ( ( groups4061424788464935467al_rat @ G @ A4 )
!= one_one_rat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8020_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A4: set_nat] :
( ( ( groups73079841787564623at_rat @ G @ A4 )
!= one_one_rat )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8021_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A4: set_int] :
( ( ( groups1072433553688619179nt_rat @ G @ A4 )
!= one_one_rat )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8022_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A4: set_real] :
( ( ( groups4696554848551431203al_nat @ G @ A4 )
!= one_one_nat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ( G @ A2 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8023_prod_Oneutral,axiom,
! [A4: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ A4 )
= one_one_nat ) ) ).
% prod.neutral
thf(fact_8024_prod_Oneutral,axiom,
! [A4: set_nat,G: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ A4 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_8025_prod_Oneutral,axiom,
! [A4: set_int,G: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G @ A4 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_8026_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > nat,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4696554848551431203al_nat
@ ^ [X3: real] :
( groups708209901874060359at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y3: nat] :
( groups4696554848551431203al_nat
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8027_prod_Oswap__restrict,axiom,
! [A4: set_int,B5: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [X3: int] :
( groups708209901874060359at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y3: nat] :
( groups1707563613775114915nt_nat
@ ^ [X3: int] : ( G @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8028_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups861055069439313189ex_nat
@ ^ [X3: complex] :
( groups708209901874060359at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y3: nat] :
( groups861055069439313189ex_nat
@ ^ [X3: complex] : ( G @ X3 @ Y3 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8029_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > int,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4694064378042380927al_int
@ ^ [X3: real] :
( groups705719431365010083at_int @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y3: nat] :
( groups4694064378042380927al_int
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8030_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > int,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups858564598930262913ex_int
@ ^ [X3: complex] :
( groups705719431365010083at_int @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y3: nat] :
( groups858564598930262913ex_int
@ ^ [X3: complex] : ( G @ X3 @ Y3 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8031_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_int,G: real > int > int,R: real > int > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups4694064378042380927al_int
@ ^ [X3: real] :
( groups1705073143266064639nt_int @ ( G @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups1705073143266064639nt_int
@ ^ [Y3: int] :
( groups4694064378042380927al_int
@ ^ [X3: real] : ( G @ X3 @ Y3 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8032_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups858564598930262913ex_int
@ ^ [X3: complex] :
( groups1705073143266064639nt_int @ ( G @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups1705073143266064639nt_int
@ ^ [Y3: int] :
( groups858564598930262913ex_int
@ ^ [X3: complex] : ( G @ X3 @ Y3 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8033_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_real,G: nat > real > nat,R: nat > real > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_real @ B5 )
=> ( ( groups708209901874060359at_nat
@ ^ [X3: nat] :
( groups4696554848551431203al_nat @ ( G @ X3 )
@ ( collect_real
@ ^ [Y3: real] :
( ( member_real @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups4696554848551431203al_nat
@ ^ [Y3: real] :
( groups708209901874060359at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8034_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_int,G: nat > int > nat,R: nat > int > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups708209901874060359at_nat
@ ^ [X3: nat] :
( groups1707563613775114915nt_nat @ ( G @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups1707563613775114915nt_nat
@ ^ [Y3: int] :
( groups708209901874060359at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8035_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_complex,G: nat > complex > nat,R: nat > complex > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups708209901874060359at_nat
@ ^ [X3: nat] :
( groups861055069439313189ex_nat @ ( G @ X3 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B5 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups861055069439313189ex_nat
@ ^ [Y3: complex] :
( groups708209901874060359at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_8036_mod__prod__eq,axiom,
! [F: nat > nat,A3: nat,A4: set_nat] :
( ( modulo_modulo_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ A3 ) ) ).
% mod_prod_eq
thf(fact_8037_mod__prod__eq,axiom,
! [F: nat > int,A3: int,A4: set_nat] :
( ( modulo_modulo_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A4 ) @ A3 ) ) ).
% mod_prod_eq
thf(fact_8038_mod__prod__eq,axiom,
! [F: int > int,A3: int,A4: set_int] :
( ( modulo_modulo_int
@ ( groups1705073143266064639nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ A3 ) ) ).
% mod_prod_eq
thf(fact_8039_prod__nonneg,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_8040_prod__nonneg,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_8041_prod__nonneg,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_8042_prod__mono,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8043_prod__mono,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8044_prod__mono,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8045_prod__mono,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8046_prod__mono,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8047_prod__mono,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8048_prod__mono,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8049_prod__mono,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8050_prod__mono,axiom,
! [A4: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
& ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8051_prod__mono,axiom,
! [A4: set_nat,F: nat > nat,G: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_8052_prod__pos,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_8053_prod__pos,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_8054_prod__pos,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_8055_prod__ge__1,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8056_prod__ge__1,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8057_prod__ge__1,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8058_prod__ge__1,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8059_prod__ge__1,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8060_prod__ge__1,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8061_prod__ge__1,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8062_prod__ge__1,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8063_prod__ge__1,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8064_prod__ge__1,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_8065_prod__zero,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) )
=> ( ( groups129246275422532515t_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_8066_prod__zero,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) )
=> ( ( groups2316167850115554303t_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_8067_prod__zero,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) )
=> ( ( groups766887009212190081x_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_8068_prod__zero,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) )
=> ( ( groups73079841787564623at_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_8069_prod__zero,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) )
=> ( ( groups1072433553688619179nt_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_8070_prod__zero,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) )
=> ( ( groups225925009352817453ex_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_8071_prod__zero,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_nat ) )
=> ( ( groups1707563613775114915nt_nat @ F @ A4 )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_8072_prod__zero,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_nat ) )
=> ( ( groups861055069439313189ex_nat @ F @ A4 )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_8073_prod__zero,axiom,
! [A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_int ) )
=> ( ( groups858564598930262913ex_int @ F @ A4 )
= zero_zero_int ) ) ) ).
% prod_zero
thf(fact_8074_prod__zero,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_nat ) )
=> ( ( groups708209901874060359at_nat @ F @ A4 )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_8075_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_8076_complex__mod__triangle__ineq2,axiom,
! [B3: complex,A3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B3 @ A3 ) ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ A3 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_8077_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > complex,P: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups713298508707869441omplex
@ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8078_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > complex,P: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups6464643781859351333omplex
@ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8079_prod_Ointer__filter,axiom,
! [A4: set_int,G: int > complex,P: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups7440179247065528705omplex
@ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8080_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > complex,P: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups3708469109370488835omplex
@ ^ [X3: complex] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8081_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups1681761925125756287l_real
@ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8082_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > real,P: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups129246275422532515t_real
@ ^ [X3: nat] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8083_prod_Ointer__filter,axiom,
! [A4: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups2316167850115554303t_real
@ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8084_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups766887009212190081x_real
@ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8085_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > rat,P: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups4061424788464935467al_rat @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups4061424788464935467al_rat
@ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8086_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > rat,P: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_8087_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_8088_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_8089_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_8090_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_8091_prod__le__1,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8092_prod__le__1,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8093_prod__le__1,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8094_prod__le__1,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8095_prod__le__1,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8096_prod__le__1,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8097_prod__le__1,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8098_prod__le__1,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8099_prod__le__1,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
& ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_8100_prod__le__1,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8101_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_nat,H: nat > complex,G: nat > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups6464643781859351333omplex @ H @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8102_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_int,H: int > complex,G: int > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups7440179247065528705omplex @ H @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8103_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_complex,H: complex > complex,G: complex > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3708469109370488835omplex @ H @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8104_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_nat,H: nat > real,G: nat > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups129246275422532515t_real @ H @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8105_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_int,H: int > real,G: int > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2316167850115554303t_real @ H @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8106_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_complex,H: complex > real,G: complex > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups766887009212190081x_real @ H @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8107_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_nat,H: nat > rat,G: nat > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups73079841787564623at_rat @ H @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8108_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_int,H: int > rat,G: int > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups1072433553688619179nt_rat @ H @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8109_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_complex,H: complex > rat,G: complex > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups225925009352817453ex_rat @ H @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8110_prod_Orelated,axiom,
! [R: nat > nat > $o,S2: set_int,H: int > nat,G: int > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times_nat @ X1 @ Y1 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups1707563613775114915nt_nat @ H @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_8111_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8112_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8113_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8114_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8115_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8116_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > code_integer] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8117_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8118_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8119_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_8120_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8121_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8122_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > int,G: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( dvd_dvd_int @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8123_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A4 )
=> ( dvd_dvd_int @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8124_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > code_integer,G: real > code_integer] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A4 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8125_prod__dvd__prod__subset2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8126_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > code_integer,G: complex > code_integer] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8127_prod__dvd__prod__subset2,axiom,
! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8128_prod__dvd__prod__subset2,axiom,
! [B5: set_int,A4: set_int,F: int > code_integer,G: int > code_integer] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8129_prod__dvd__prod__subset2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A2 ) @ ( G @ A2 ) ) )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_8130_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > complex,H: real > complex] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8131_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T2: set_int,G: real > complex,H: int > complex] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_int @ ( J @ A2 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8132_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T2: set_complex,G: real > complex,H: complex > complex] :
( ( finite_finite_real @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_complex @ ( J @ A2 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8133_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T2: set_real,G: int > complex,H: real > complex] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8134_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > complex,H: int > complex] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( member_int @ ( J @ A2 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8135_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T2: set_complex,G: int > complex,H: complex > complex] :
( ( finite_finite_int @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ ( minus_minus_set_int @ S2 @ S4 ) )
=> ( member_complex @ ( J @ A2 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8136_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T2: set_real,G: complex > complex,H: real > complex] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B2 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8137_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T2: set_int,G: complex > complex,H: int > complex] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( member_int @ ( J @ A2 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B2 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8138_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T2: set_complex,G: complex > complex,H: complex > complex] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
=> ( member_complex @ ( J @ A2 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B2 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8139_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( ( I @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S2 @ S4 ) )
=> ( member_real @ ( J @ A2 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S4 )
=> ( ( G @ A2 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_real ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S2 )
= ( groups1681761925125756287l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8140_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= one_one_complex ) ) ) )
= ( groups713298508707869441omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8141_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= one_one_complex ) ) ) )
= ( groups7440179247065528705omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8142_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X3: complex] :
( ( G @ X3 )
= one_one_complex ) ) ) )
= ( groups3708469109370488835omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8143_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= one_one_real ) ) ) )
= ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8144_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= one_one_real ) ) ) )
= ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8145_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X3: complex] :
( ( G @ X3 )
= one_one_real ) ) ) )
= ( groups766887009212190081x_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8146_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( groups4061424788464935467al_rat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= one_one_rat ) ) ) )
= ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8147_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( groups1072433553688619179nt_rat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= one_one_rat ) ) ) )
= ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8148_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X3: complex] :
( ( G @ X3 )
= one_one_rat ) ) ) )
= ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8149_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( groups4696554848551431203al_nat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X3: real] :
( ( G @ X3 )
= one_one_nat ) ) ) )
= ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8150_exp__sum,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( exp_real @ ( groups8778361861064173332t_real @ F @ I5 ) )
= ( groups2316167850115554303t_real
@ ^ [X3: int] : ( exp_real @ ( F @ X3 ) )
@ I5 ) ) ) ).
% exp_sum
thf(fact_8151_exp__sum,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( exp_real @ ( groups5808333547571424918x_real @ F @ I5 ) )
= ( groups766887009212190081x_real
@ ^ [X3: complex] : ( exp_real @ ( F @ X3 ) )
@ I5 ) ) ) ).
% exp_sum
thf(fact_8152_exp__sum,axiom,
! [I5: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( exp_complex @ ( groups7754918857620584856omplex @ F @ I5 ) )
= ( groups3708469109370488835omplex
@ ^ [X3: complex] : ( exp_complex @ ( F @ X3 ) )
@ I5 ) ) ) ).
% exp_sum
thf(fact_8153_exp__sum,axiom,
! [I5: set_nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( exp_real @ ( groups6591440286371151544t_real @ F @ I5 ) )
= ( groups129246275422532515t_real
@ ^ [X3: nat] : ( exp_real @ ( F @ X3 ) )
@ I5 ) ) ) ).
% exp_sum
thf(fact_8154_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_8155_norm__exp,axiom,
! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% norm_exp
thf(fact_8156_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_8157_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_8158_less__1__prod2,axiom,
! [I5: set_real,I: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8159_less__1__prod2,axiom,
! [I5: set_nat,I: nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8160_less__1__prod2,axiom,
! [I5: set_int,I: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8161_less__1__prod2,axiom,
! [I5: set_complex,I: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8162_less__1__prod2,axiom,
! [I5: set_real,I: real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8163_less__1__prod2,axiom,
! [I5: set_nat,I: nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8164_less__1__prod2,axiom,
! [I5: set_int,I: int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8165_less__1__prod2,axiom,
! [I5: set_complex,I: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8166_less__1__prod2,axiom,
! [I5: set_real,I: real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I @ I5 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8167_less__1__prod2,axiom,
! [I5: set_complex,I: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I @ I5 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8168_less__1__prod,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8169_less__1__prod,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8170_less__1__prod,axiom,
! [I5: set_nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8171_less__1__prod,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8172_less__1__prod,axiom,
! [I5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8173_less__1__prod,axiom,
! [I5: set_real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8174_less__1__prod,axiom,
! [I5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8175_less__1__prod,axiom,
! [I5: set_int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8176_less__1__prod,axiom,
! [I5: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8177_less__1__prod,axiom,
! [I5: set_real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8178_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8179_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > real] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8180_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8181_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ A4 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8182_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8183_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8184_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8185_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups1072433553688619179nt_rat @ G @ A4 )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8186_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G @ A4 )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8187_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups708209901874060359at_nat @ G @ A4 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8188_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > complex,H: real > complex] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups713298508707869441omplex @ G @ T2 )
= ( groups713298508707869441omplex @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8189_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > complex,H: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ T2 )
= ( groups3708469109370488835omplex @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8190_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ T2 )
= ( groups1681761925125756287l_real @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8191_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups766887009212190081x_real @ G @ T2 )
= ( groups766887009212190081x_real @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8192_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ T2 )
= ( groups4061424788464935467al_rat @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8193_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat,H: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ T2 )
= ( groups225925009352817453ex_rat @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8194_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ T2 )
= ( groups4696554848551431203al_nat @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8195_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat,H: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ T2 )
= ( groups861055069439313189ex_nat @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8196_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ T2 )
= ( groups4694064378042380927al_int @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8197_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int,H: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ T2 )
= ( groups858564598930262913ex_int @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8198_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > complex,G: real > complex] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8199_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_complex ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8200_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > real,G: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S2 )
= ( groups1681761925125756287l_real @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8201_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups766887009212190081x_real @ G @ S2 )
= ( groups766887009212190081x_real @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8202_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > rat,G: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ S2 )
= ( groups4061424788464935467al_rat @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8203_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ S2 )
= ( groups225925009352817453ex_rat @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8204_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > nat,G: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S2 )
= ( groups4696554848551431203al_nat @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8205_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8206_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > int,G: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ S2 )
= ( groups4694064378042380927al_int @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8207_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H @ X4 )
= one_one_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ S2 )
= ( groups858564598930262913ex_int @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8208_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ T2 )
= ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8209_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ T2 )
= ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8210_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ T2 )
= ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8211_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ T2 )
= ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8212_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ T2 )
= ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8213_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ T2 )
= ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8214_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ T2 )
= ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8215_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ T2 )
= ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8216_prod_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups7440179247065528705omplex @ G @ T2 )
= ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8217_prod_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ T2 )
= ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8218_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8219_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ S2 )
= ( groups766887009212190081x_real @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8220_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ S2 )
= ( groups225925009352817453ex_rat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8221_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8222_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ S2 )
= ( groups858564598930262913ex_int @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8223_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ S2 )
= ( groups6464643781859351333omplex @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8224_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ S2 )
= ( groups129246275422532515t_real @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8225_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ S2 )
= ( groups73079841787564623at_rat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8226_prod_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8227_prod_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ S2 )
= ( groups2316167850115554303t_real @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8228_prod_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > complex,H: real > complex] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ C2 )
= ( groups713298508707869441omplex @ H @ C2 ) )
=> ( ( groups713298508707869441omplex @ G @ A4 )
= ( groups713298508707869441omplex @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8229_prod_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H: complex > complex] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ C2 )
= ( groups3708469109370488835omplex @ H @ C2 ) )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= ( groups3708469109370488835omplex @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8230_prod_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ C2 )
= ( groups1681761925125756287l_real @ H @ C2 ) )
=> ( ( groups1681761925125756287l_real @ G @ A4 )
= ( groups1681761925125756287l_real @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8231_prod_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ C2 )
= ( groups766887009212190081x_real @ H @ C2 ) )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( groups766887009212190081x_real @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8232_prod_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ C2 )
= ( groups4061424788464935467al_rat @ H @ C2 ) )
=> ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( groups4061424788464935467al_rat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8233_prod_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H: complex > rat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ C2 )
= ( groups225925009352817453ex_rat @ H @ C2 ) )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( groups225925009352817453ex_rat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8234_prod_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ C2 )
= ( groups4696554848551431203al_nat @ H @ C2 ) )
=> ( ( groups4696554848551431203al_nat @ G @ A4 )
= ( groups4696554848551431203al_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8235_prod_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H: complex > nat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ C2 )
= ( groups861055069439313189ex_nat @ H @ C2 ) )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( groups861055069439313189ex_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8236_prod_Osame__carrierI,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ C2 )
= ( groups4694064378042380927al_int @ H @ C2 ) )
=> ( ( groups4694064378042380927al_int @ G @ A4 )
= ( groups4694064378042380927al_int @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8237_prod_Osame__carrierI,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H: complex > int] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ C2 )
= ( groups858564598930262913ex_int @ H @ C2 ) )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= ( groups858564598930262913ex_int @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8238_prod_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > complex,H: real > complex] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ A4 )
= ( groups713298508707869441omplex @ H @ B5 ) )
= ( ( groups713298508707869441omplex @ G @ C2 )
= ( groups713298508707869441omplex @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8239_prod_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H: complex > complex] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ A4 )
= ( groups3708469109370488835omplex @ H @ B5 ) )
= ( ( groups3708469109370488835omplex @ G @ C2 )
= ( groups3708469109370488835omplex @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8240_prod_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ A4 )
= ( groups1681761925125756287l_real @ H @ B5 ) )
= ( ( groups1681761925125756287l_real @ G @ C2 )
= ( groups1681761925125756287l_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8241_prod_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H: complex > real] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ A4 )
= ( groups766887009212190081x_real @ H @ B5 ) )
= ( ( groups766887009212190081x_real @ G @ C2 )
= ( groups766887009212190081x_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8242_prod_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > rat,H: real > rat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( groups4061424788464935467al_rat @ H @ B5 ) )
= ( ( groups4061424788464935467al_rat @ G @ C2 )
= ( groups4061424788464935467al_rat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8243_prod_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H: complex > rat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( groups225925009352817453ex_rat @ H @ B5 ) )
= ( ( groups225925009352817453ex_rat @ G @ C2 )
= ( groups225925009352817453ex_rat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8244_prod_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ A4 )
= ( groups4696554848551431203al_nat @ H @ B5 ) )
= ( ( groups4696554848551431203al_nat @ G @ C2 )
= ( groups4696554848551431203al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8245_prod_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H: complex > nat] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( groups861055069439313189ex_nat @ H @ B5 ) )
= ( ( groups861055069439313189ex_nat @ G @ C2 )
= ( groups861055069439313189ex_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8246_prod_Osame__carrier,axiom,
! [C2: set_real,A4: set_real,B5: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A4 @ C2 )
=> ( ( ord_less_eq_set_real @ B5 @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_int ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ A4 )
= ( groups4694064378042380927al_int @ H @ B5 ) )
= ( ( groups4694064378042380927al_int @ G @ C2 )
= ( groups4694064378042380927al_int @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8247_prod_Osame__carrier,axiom,
! [C2: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H: complex > int] :
( ( finite3207457112153483333omplex @ C2 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C2 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
=> ( ( G @ A2 )
= one_one_int ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B5 ) )
=> ( ( H @ B2 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ A4 )
= ( groups858564598930262913ex_int @ H @ B5 ) )
= ( ( groups858564598930262913ex_int @ G @ C2 )
= ( groups858564598930262913ex_int @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8248_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_8249_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_8250_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_8251_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_8252_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_8253_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_8254_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_8255_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_8256_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_8257_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_8258_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_8259_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_8260_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_8261_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_8262_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_8263_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_8264_fact__prod,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N3: nat] :
( semiri1314217659103216013at_int
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).
% fact_prod
thf(fact_8265_fact__prod,axiom,
( semiri3624122377584611663nteger
= ( ^ [N3: nat] :
( semiri4939895301339042750nteger
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).
% fact_prod
thf(fact_8266_fact__prod,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N3: nat] :
( semiri1316708129612266289at_nat
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).
% fact_prod
thf(fact_8267_fact__prod,axiom,
( semiri2265585572941072030t_real
= ( ^ [N3: nat] :
( semiri5074537144036343181t_real
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).
% fact_prod
thf(fact_8268_prod__atLeastAtMost__code,axiom,
! [F: nat > complex,A3: nat,B3: nat] :
( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo1517530859248394432omplex
@ ^ [A: nat] : ( times_times_complex @ ( F @ A ) )
@ A3
@ B3
@ one_one_complex ) ) ).
% prod_atLeastAtMost_code
thf(fact_8269_prod__atLeastAtMost__code,axiom,
! [F: nat > real,A3: nat,B3: nat] :
( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo3111899725591712190t_real
@ ^ [A: nat] : ( times_times_real @ ( F @ A ) )
@ A3
@ B3
@ one_one_real ) ) ).
% prod_atLeastAtMost_code
thf(fact_8270_prod__atLeastAtMost__code,axiom,
! [F: nat > rat,A3: nat,B3: nat] :
( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A: nat] : ( times_times_rat @ ( F @ A ) )
@ A3
@ B3
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_8271_prod__atLeastAtMost__code,axiom,
! [F: nat > nat,A3: nat,B3: nat] :
( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A: nat] : ( times_times_nat @ ( F @ A ) )
@ A3
@ B3
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_8272_prod__atLeastAtMost__code,axiom,
! [F: nat > int,A3: nat,B3: nat] :
( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2581907887559384638at_int
@ ^ [A: nat] : ( times_times_int @ ( F @ A ) )
@ A3
@ B3
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_8273_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8274_prod__mono__strict,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8275_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8276_prod__mono__strict,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8277_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8278_prod__mono__strict,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8279_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8280_prod__mono__strict,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8281_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8282_prod__mono__strict,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8283_even__prod__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8284_even__prod__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8285_even__prod__iff,axiom,
! [A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8286_even__prod__iff,axiom,
! [A4: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A4 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8287_even__prod__iff,axiom,
! [A4: set_int,F: int > code_integer] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3827104343326376752nteger @ F @ A4 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8288_even__prod__iff,axiom,
! [A4: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8289_even__prod__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8290_even__prod__iff,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8291_even__prod__iff,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X3 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8292_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8293_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8294_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8295_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8296_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
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_8297_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8298_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8299_prod__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8300_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A2 ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8301_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A2 ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8302_prod__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A2 ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8303_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A2 ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8304_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A2 ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8305_prod__mono2,axiom,
! [B5: set_int,A4: set_int,F: int > real] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8306_prod__mono2,axiom,
! [B5: set_int,A4: set_int,F: int > rat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A2: int] :
( ( member_int @ A2 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A2 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8307_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_8308_pochhammer__Suc__prod,axiom,
! [A3: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A3 @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8309_pochhammer__Suc__prod,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8310_pochhammer__Suc__prod,axiom,
! [A3: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A3 @ ( suc @ N ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A3 @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8311_pochhammer__Suc__prod,axiom,
! [A3: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8312_pochhammer__Suc__prod,axiom,
! [A3: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A3 @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8313_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A: rat,N3: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N3 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8314_pochhammer__prod__rev,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A: real,N3: nat] :
( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8315_pochhammer__prod__rev,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A: code_integer,N3: nat] :
( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N3 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8316_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A: nat,N3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N3 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8317_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A: int,N3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8318_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
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_8319_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_8320_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_8321_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_8322_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_8323_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_8324_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_8325_pochhammer__Suc__prod__rev,axiom,
! [A3: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A3 @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8326_pochhammer__Suc__prod__rev,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8327_pochhammer__Suc__prod__rev,axiom,
! [A3: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A3 @ ( suc @ N ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A3 @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8328_pochhammer__Suc__prod__rev,axiom,
! [A3: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8329_pochhammer__Suc__prod__rev,axiom,
! [A3: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A3 @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8330_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_8331_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_8332_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_8333_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_8334_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_8335_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_8336_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_8337_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_8338_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_8339_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_8340_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_8341_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_8342_suminf__zero,axiom,
( ( suminf_real
@ ^ [N3: nat] : zero_zero_real )
= zero_zero_real ) ).
% suminf_zero
thf(fact_8343_suminf__zero,axiom,
( ( suminf_nat
@ ^ [N3: nat] : zero_zero_nat )
= zero_zero_nat ) ).
% suminf_zero
thf(fact_8344_suminf__zero,axiom,
( ( suminf_int
@ ^ [N3: nat] : zero_zero_int )
= zero_zero_int ) ).
% suminf_zero
thf(fact_8345_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_8346_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_8347_prod__eq__1__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1707563613775114915nt_nat @ F @ A4 )
= one_one_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( ( F @ X3 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8348_prod__eq__1__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups861055069439313189ex_nat @ F @ A4 )
= one_one_nat )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A4 )
=> ( ( F @ X3 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8349_prod__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups708209901874060359at_nat @ F @ A4 )
= one_one_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( F @ X3 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8350_prod__pos__nat__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8351_prod__pos__nat__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8352_prod__pos__nat__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8353_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_8354_ln__prod,axiom,
! [I5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ I5 )
=> ( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups6036352826371341000t_real @ F @ I5 ) )
= ( groups4567486121110086003t_real
@ ^ [X3: product_prod_nat_nat] : ( ln_ln_real @ ( F @ X3 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_8355_ln__prod,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I5 ) )
= ( groups8097168146408367636l_real
@ ^ [X3: real] : ( ln_ln_real @ ( F @ X3 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_8356_ln__prod,axiom,
! [I5: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ I5 )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups3619160379726066777t_real @ F @ I5 ) )
= ( groups5107569545109728110t_real
@ ^ [X3: set_nat] : ( ln_ln_real @ ( F @ X3 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_8357_ln__prod,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I5 ) )
= ( groups8778361861064173332t_real
@ ^ [X3: int] : ( ln_ln_real @ ( F @ X3 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_8358_ln__prod,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I5 ) )
= ( groups5808333547571424918x_real
@ ^ [X3: complex] : ( ln_ln_real @ ( F @ X3 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_8359_ln__prod,axiom,
! [I5: set_nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I5 ) )
= ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( ln_ln_real @ ( F @ X3 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_8360_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_8361_norm__not__less__zero,axiom,
! [X: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_8362_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_8363_sum__norm__le,axiom,
! [S2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > complex,G: product_prod_nat_nat > real] :
( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6381953495645901045omplex @ F @ S2 ) ) @ ( groups4567486121110086003t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8364_sum__norm__le,axiom,
! [S2: set_real,F: real > complex,G: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8365_sum__norm__le,axiom,
! [S2: set_set_nat,F: set_nat > complex,G: set_nat > real] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S2 ) ) @ ( groups5107569545109728110t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8366_sum__norm__le,axiom,
! [S2: set_int,F: int > complex,G: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8367_sum__norm__le,axiom,
! [S2: set_nat,F: nat > complex,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8368_sum__norm__le,axiom,
! [S2: set_complex,F: complex > complex,G: complex > real] :
( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8369_sum__norm__le,axiom,
! [S2: set_nat,F: nat > real,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_8370_norm__sum,axiom,
! [F: nat > complex,A4: set_nat] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
@ A4 ) ) ).
% norm_sum
thf(fact_8371_norm__sum,axiom,
! [F: complex > complex,A4: set_complex] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A4 ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
@ A4 ) ) ).
% norm_sum
thf(fact_8372_norm__sum,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
@ A4 ) ) ).
% norm_sum
thf(fact_8373_nonzero__norm__divide,axiom,
! [B3: real,A3: real] :
( ( B3 != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A3 @ B3 ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ) ).
% nonzero_norm_divide
thf(fact_8374_nonzero__norm__divide,axiom,
! [B3: complex,A3: complex] :
( ( B3 != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ) ).
% nonzero_norm_divide
thf(fact_8375_power__eq__imp__eq__norm,axiom,
! [W2: real,N: nat,Z: real] :
( ( ( power_power_real @ W2 @ N )
= ( power_power_real @ Z @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W2 )
= ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_8376_power__eq__imp__eq__norm,axiom,
! [W2: complex,N: nat,Z: complex] :
( ( ( power_power_complex @ W2 @ N )
= ( power_power_complex @ Z @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W2 )
= ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_8377_norm__mult__less,axiom,
! [X: real,R2: real,Y: real,S3: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S3 ) ) ) ) ).
% norm_mult_less
thf(fact_8378_norm__mult__less,axiom,
! [X: complex,R2: real,Y: complex,S3: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S3 ) ) ) ) ).
% norm_mult_less
thf(fact_8379_norm__mult__ineq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_mult_ineq
thf(fact_8380_norm__mult__ineq,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_mult_ineq
thf(fact_8381_norm__triangle__lt,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_8382_norm__triangle__lt,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_8383_norm__add__less,axiom,
! [X: real,R2: real,Y: real,S3: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_add_less
thf(fact_8384_norm__add__less,axiom,
! [X: complex,R2: real,Y: complex,S3: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_add_less
thf(fact_8385_norm__triangle__mono,axiom,
! [A3: real,R2: real,B3: real,S3: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A3 ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ S3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A3 @ B3 ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_triangle_mono
thf(fact_8386_norm__triangle__mono,axiom,
! [A3: complex,R2: real,B3: complex,S3: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A3 ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A3 @ B3 ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_triangle_mono
thf(fact_8387_norm__triangle__ineq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_8388_norm__triangle__ineq,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_8389_norm__triangle__le,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_8390_norm__triangle__le,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_8391_norm__add__leD,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A3 @ B3 ) ) @ C )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A3 ) @ C ) ) ) ).
% norm_add_leD
thf(fact_8392_norm__add__leD,axiom,
! [A3: complex,B3: complex,C: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A3 @ B3 ) ) @ C )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A3 ) @ C ) ) ) ).
% norm_add_leD
thf(fact_8393_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_8394_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_8395_norm__diff__triangle__less,axiom,
! [X: real,Y: real,E1: real,Z: real,E22: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_8396_norm__diff__triangle__less,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_8397_norm__triangle__sub,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% norm_triangle_sub
thf(fact_8398_norm__triangle__sub,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% norm_triangle_sub
thf(fact_8399_norm__triangle__ineq4,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A3 @ B3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ).
% norm_triangle_ineq4
thf(fact_8400_norm__triangle__ineq4,axiom,
! [A3: complex,B3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A3 @ B3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ).
% norm_triangle_ineq4
thf(fact_8401_norm__diff__triangle__le,axiom,
! [X: real,Y: real,E1: real,Z: real,E22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ 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_8402_norm__diff__triangle__le,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_8403_norm__triangle__le__diff,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le_diff
thf(fact_8404_norm__triangle__le__diff,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le_diff
thf(fact_8405_norm__diff__ineq,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ).
% norm_diff_ineq
thf(fact_8406_norm__diff__ineq,axiom,
! [A3: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A3 @ B3 ) ) ) ).
% norm_diff_ineq
thf(fact_8407_norm__triangle__ineq2,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% norm_triangle_ineq2
thf(fact_8408_norm__triangle__ineq2,axiom,
! [A3: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A3 @ B3 ) ) ) ).
% norm_triangle_ineq2
thf(fact_8409_suminf__finite,axiom,
! [N6: set_nat,F: nat > int] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_int ) )
=> ( ( suminf_int @ F )
= ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_8410_suminf__finite,axiom,
! [N6: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_nat ) )
=> ( ( suminf_nat @ F )
= ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_8411_suminf__finite,axiom,
! [N6: set_nat,F: nat > real] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_real ) )
=> ( ( suminf_real @ F )
= ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_8412_power__eq__1__iff,axiom,
! [W2: real,N: nat] :
( ( ( power_power_real @ W2 @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_8413_power__eq__1__iff,axiom,
! [W2: complex,N: nat] :
( ( ( power_power_complex @ W2 @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_8414_norm__diff__triangle__ineq,axiom,
! [A3: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A3 @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_8415_norm__diff__triangle__ineq,axiom,
! [A3: complex,B3: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A3 @ B3 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A3 @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B3 @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_8416_norm__triangle__ineq3,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% norm_triangle_ineq3
thf(fact_8417_norm__triangle__ineq3,axiom,
! [A3: complex,B3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A3 @ B3 ) ) ) ).
% norm_triangle_ineq3
thf(fact_8418_norm__prod__diff,axiom,
! [I5: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > real,W2: product_prod_nat_nat > real] :
( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups6036352826371341000t_real @ Z @ I5 ) @ ( groups6036352826371341000t_real @ W2 @ I5 ) ) )
@ ( groups4567486121110086003t_real
@ ^ [I3: product_prod_nat_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8419_norm__prod__diff,axiom,
! [I5: set_real,Z: real > real,W2: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W2 @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8420_norm__prod__diff,axiom,
! [I5: set_set_nat,Z: set_nat > real,W2: set_nat > real] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups3619160379726066777t_real @ Z @ I5 ) @ ( groups3619160379726066777t_real @ W2 @ I5 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I3: set_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8421_norm__prod__diff,axiom,
! [I5: set_int,Z: int > real,W2: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W2 @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8422_norm__prod__diff,axiom,
! [I5: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > complex,W2: product_prod_nat_nat > complex] :
( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups8110221916422527690omplex @ Z @ I5 ) @ ( groups8110221916422527690omplex @ W2 @ I5 ) ) )
@ ( groups4567486121110086003t_real
@ ^ [I3: product_prod_nat_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8423_norm__prod__diff,axiom,
! [I5: set_real,Z: real > complex,W2: real > complex] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W2 @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8424_norm__prod__diff,axiom,
! [I5: set_set_nat,Z: set_nat > complex,W2: set_nat > complex] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups1092910753850256091omplex @ Z @ I5 ) @ ( groups1092910753850256091omplex @ W2 @ I5 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I3: set_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8425_norm__prod__diff,axiom,
! [I5: set_int,Z: int > complex,W2: int > complex] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W2 @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8426_norm__prod__diff,axiom,
! [I5: set_nat,Z: nat > real,W2: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W2 @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8427_norm__prod__diff,axiom,
! [I5: set_nat,Z: nat > complex,W2: nat > complex] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W2 @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8428_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_8429_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_8430_norm__power__diff,axiom,
! [Z: real,W2: real,M: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_8431_norm__power__diff,axiom,
! [Z: complex,W2: complex,M: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_8432_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_8433_lemma__termdiff2,axiom,
! [H: complex,Z: complex,N: nat] :
( ( H != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_complex @ H
@ ( groups2073611262835488442omplex
@ ^ [P5: nat] :
( groups2073611262835488442omplex
@ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ Q5 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
@ ( 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_8434_lemma__termdiff2,axiom,
! [H: rat,Z: rat,N: nat] :
( ( H != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_rat @ H
@ ( groups2906978787729119204at_rat
@ ^ [P5: nat] :
( groups2906978787729119204at_rat
@ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H ) @ Q5 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
@ ( 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_8435_lemma__termdiff2,axiom,
! [H: real,Z: real,N: nat] :
( ( H != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_real @ H
@ ( groups6591440286371151544t_real
@ ^ [P5: nat] :
( groups6591440286371151544t_real
@ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
@ ( 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_8436_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_8437_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_8438_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_8439_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_8440_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] :
( if_Code_integer
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) )
@ zero_z3403309356797280102nteger )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8441_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_8442_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_8443_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_8444_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_8445_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( if_Code_integer @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) @ zero_z3403309356797280102nteger )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8446_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_8447_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_8448_lessThan__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( set_ord_lessThan_int @ X )
= ( set_ord_lessThan_int @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_8449_lessThan__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( set_or5984915006950818249n_real @ X )
= ( set_or5984915006950818249n_real @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_8450_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_8451_atMost__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( set_ord_atMost_int @ X )
= ( set_ord_atMost_int @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_8452_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_8453_lessThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
= ( ord_less_rat @ I @ K ) ) ).
% lessThan_iff
thf(fact_8454_lessThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I @ K ) ) ).
% lessThan_iff
thf(fact_8455_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_8456_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_8457_lessThan__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I @ K ) ) ).
% lessThan_iff
thf(fact_8458_atMost__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I @ K ) ) ).
% atMost_iff
thf(fact_8459_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_8460_atMost__iff,axiom,
! [I: set_int,K: set_int] :
( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
= ( ord_less_eq_set_int @ I @ K ) ) ).
% atMost_iff
thf(fact_8461_atMost__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
= ( ord_less_eq_rat @ I @ K ) ) ).
% atMost_iff
thf(fact_8462_atMost__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
= ( ord_less_eq_num @ I @ K ) ) ).
% atMost_iff
thf(fact_8463_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_8464_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_8465_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_8466_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_8467_summable__single,axiom,
! [I: nat,F: nat > real] :
( summable_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% summable_single
thf(fact_8468_summable__single,axiom,
! [I: nat,F: nat > nat] :
( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% summable_single
thf(fact_8469_summable__single,axiom,
! [I: nat,F: nat > int] :
( summable_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% summable_single
thf(fact_8470_summable__zero,axiom,
( summable_real
@ ^ [N3: nat] : zero_zero_real ) ).
% summable_zero
thf(fact_8471_summable__zero,axiom,
( summable_nat
@ ^ [N3: nat] : zero_zero_nat ) ).
% summable_zero
thf(fact_8472_summable__zero,axiom,
( summable_int
@ ^ [N3: nat] : zero_zero_int ) ).
% summable_zero
thf(fact_8473_lessThan__subset__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8474_lessThan__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8475_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8476_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8477_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8478_atMost__subset__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8479_atMost__subset__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8480_atMost__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8481_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8482_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8483_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_8484_summable__cmult__iff,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_cmult_iff
thf(fact_8485_summable__divide__iff,axiom,
! [F: nat > real,C: real] :
( ( summable_real
@ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_8486_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( summable_real
@ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% summable_If_finite_set
thf(fact_8487_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% summable_If_finite_set
thf(fact_8488_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( summable_int
@ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% summable_If_finite_set
thf(fact_8489_summable__If__finite,axiom,
! [P: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( summable_real
@ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% summable_If_finite
thf(fact_8490_summable__If__finite,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% summable_If_finite
thf(fact_8491_summable__If__finite,axiom,
! [P: nat > $o,F: nat > int] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( summable_int
@ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% summable_If_finite
thf(fact_8492_Icc__subset__Iic__iff,axiom,
! [L: set_int,H: set_int,H2: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H ) @ ( set_or58775011639299419et_int @ H2 ) )
= ( ~ ( ord_less_eq_set_int @ L @ H )
| ( ord_less_eq_set_int @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8493_Icc__subset__Iic__iff,axiom,
! [L: rat,H: rat,H2: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atMost_rat @ H2 ) )
= ( ~ ( ord_less_eq_rat @ L @ H )
| ( ord_less_eq_rat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8494_Icc__subset__Iic__iff,axiom,
! [L: num,H: num,H2: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H ) @ ( set_ord_atMost_num @ H2 ) )
= ( ~ ( ord_less_eq_num @ L @ H )
| ( ord_less_eq_num @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8495_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8496_Icc__subset__Iic__iff,axiom,
! [L: int,H: int,H2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H2 ) )
= ( ~ ( ord_less_eq_int @ L @ H )
| ( ord_less_eq_int @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8497_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8498_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_8499_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_8500_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_8501_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_8502_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_8503_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_8504_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_8505_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_8506_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_8507_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_8508_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_8509_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_8510_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_8511_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_8512_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_8513_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_8514_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_8515_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_8516_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_8517_Iic__subset__Iio__iff,axiom,
! [A3: rat,B3: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A3 ) @ ( set_ord_lessThan_rat @ B3 ) )
= ( ord_less_rat @ A3 @ B3 ) ) ).
% Iic_subset_Iio_iff
thf(fact_8518_Iic__subset__Iio__iff,axiom,
! [A3: num,B3: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A3 ) @ ( set_ord_lessThan_num @ B3 ) )
= ( ord_less_num @ A3 @ B3 ) ) ).
% Iic_subset_Iio_iff
thf(fact_8519_Iic__subset__Iio__iff,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A3 ) @ ( set_ord_lessThan_nat @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% Iic_subset_Iio_iff
thf(fact_8520_Iic__subset__Iio__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A3 ) @ ( set_ord_lessThan_int @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% Iic_subset_Iio_iff
thf(fact_8521_Iic__subset__Iio__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A3 ) @ ( set_or5984915006950818249n_real @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% Iic_subset_Iio_iff
thf(fact_8522_infinite__Iio,axiom,
! [A3: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A3 ) ) ).
% infinite_Iio
thf(fact_8523_infinite__Iio,axiom,
! [A3: real] :
~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A3 ) ) ).
% infinite_Iio
thf(fact_8524_lessThan__non__empty,axiom,
! [X: int] :
( ( set_ord_lessThan_int @ X )
!= bot_bot_set_int ) ).
% lessThan_non_empty
thf(fact_8525_lessThan__non__empty,axiom,
! [X: real] :
( ( set_or5984915006950818249n_real @ X )
!= bot_bot_set_real ) ).
% lessThan_non_empty
thf(fact_8526_not__empty__eq__Iic__eq__empty,axiom,
! [H: real] :
( bot_bot_set_real
!= ( set_ord_atMost_real @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8527_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8528_not__empty__eq__Iic__eq__empty,axiom,
! [H: int] :
( bot_bot_set_int
!= ( set_ord_atMost_int @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8529_infinite__Iic,axiom,
! [A3: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A3 ) ) ).
% infinite_Iic
thf(fact_8530_not__Iic__eq__Icc,axiom,
! [H2: int,L: int,H: int] :
( ( set_ord_atMost_int @ H2 )
!= ( set_or1266510415728281911st_int @ L @ H ) ) ).
% not_Iic_eq_Icc
thf(fact_8531_not__Iic__eq__Icc,axiom,
! [H2: real,L: real,H: real] :
( ( set_ord_atMost_real @ H2 )
!= ( set_or1222579329274155063t_real @ L @ H ) ) ).
% not_Iic_eq_Icc
thf(fact_8532_summable__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N8 @ N2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test
thf(fact_8533_summable__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N8 @ N2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable_real @ G )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test
thf(fact_8534_summable__comparison__test_H,axiom,
! [G: nat > real,N6: nat,F: nat > real] :
( ( summable_real @ G )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test'
thf(fact_8535_summable__comparison__test_H,axiom,
! [G: nat > real,N6: nat,F: nat > complex] :
( ( summable_real @ G )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test'
thf(fact_8536_summable__const__iff,axiom,
! [C: real] :
( ( summable_real
@ ^ [Uu3: nat] : C )
= ( C = zero_zero_real ) ) ).
% summable_const_iff
thf(fact_8537_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_8538_suminf__le__const,axiom,
! [F: nat > int,X: int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_8539_suminf__le__const,axiom,
! [F: nat > nat,X: nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_8540_suminf__le__const,axiom,
! [F: nat > real,X: real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_8541_lessThan__def,axiom,
( set_or890127255671739683et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8542_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8543_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8544_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8545_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8546_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8547_bounded__imp__summable,axiom,
! [A3: nat > int,B5: int] :
( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A3 @ ( set_ord_atMost_nat @ N2 ) ) @ B5 )
=> ( summable_int @ A3 ) ) ) ).
% bounded_imp_summable
thf(fact_8548_bounded__imp__summable,axiom,
! [A3: nat > nat,B5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A3 @ ( set_ord_atMost_nat @ N2 ) ) @ B5 )
=> ( summable_nat @ A3 ) ) ) ).
% bounded_imp_summable
thf(fact_8549_bounded__imp__summable,axiom,
! [A3: nat > real,B5: real] :
( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A3 @ ( set_ord_atMost_nat @ N2 ) ) @ B5 )
=> ( summable_real @ A3 ) ) ) ).
% bounded_imp_summable
thf(fact_8550_summableI__nonneg__bounded,axiom,
! [F: nat > int,X: int] :
( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
=> ( summable_int @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_8551_summableI__nonneg__bounded,axiom,
! [F: nat > nat,X: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
=> ( summable_nat @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_8552_summableI__nonneg__bounded,axiom,
! [F: nat > real,X: real] :
( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
=> ( summable_real @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_8553_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8554_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8555_atMost__def,axiom,
( set_or58775011639299419et_int
= ( ^ [U2: set_int] :
( collect_set_int
@ ^ [X3: set_int] : ( ord_less_eq_set_int @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8556_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8557_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X3: num] : ( ord_less_eq_num @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8558_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8559_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8560_sum_Onested__swap_H,axiom,
! [A3: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A3 @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8561_sum_Onested__swap_H,axiom,
! [A3: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A3 @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8562_prod_Onested__swap_H,axiom,
! [A3: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A3 @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8563_prod_Onested__swap_H,axiom,
! [A3: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A3 @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8564_powser__insidea,axiom,
! [F: nat > real,X: real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_8565_powser__insidea,axiom,
! [F: nat > complex,X: complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_real
@ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_8566_suminf__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% suminf_le
thf(fact_8567_suminf__le,axiom,
! [F: nat > nat,G: nat > nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% suminf_le
thf(fact_8568_suminf__le,axiom,
! [F: nat > int,G: nat > int] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% suminf_le
thf(fact_8569_summable__finite,axiom,
! [N6: set_nat,F: nat > real] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_real ) )
=> ( summable_real @ F ) ) ) ).
% summable_finite
thf(fact_8570_summable__finite,axiom,
! [N6: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_nat ) )
=> ( summable_nat @ F ) ) ) ).
% summable_finite
thf(fact_8571_summable__finite,axiom,
! [N6: set_nat,F: nat > int] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_int ) )
=> ( summable_int @ F ) ) ) ).
% summable_finite
thf(fact_8572_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_8573_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_8574_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_8575_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_8576_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_8577_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_8578_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_8579_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_8580_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_8581_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_8582_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_8583_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_8584_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_8585_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_8586_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_8587_summable__mult__D,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
=> ( ( C != zero_zero_real )
=> ( summable_real @ F ) ) ) ).
% summable_mult_D
thf(fact_8588_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_8589_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_8590_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_8591_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
% pi_gt_zero
thf(fact_8592_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
% pi_not_less_zero
thf(fact_8593_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
% pi_ge_zero
thf(fact_8594_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_8595_sum__less__suminf,axiom,
! [F: nat > int,N: nat] :
( ( summable_int @ F )
=> ( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_8596_sum__less__suminf,axiom,
! [F: nat > nat,N: nat] :
( ( summable_nat @ F )
=> ( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_8597_sum__less__suminf,axiom,
! [F: nat > real,N: nat] :
( ( summable_real @ F )
=> ( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_8598_not__Iic__le__Icc,axiom,
! [H: int,L2: int,H2: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_8599_not__Iic__le__Icc,axiom,
! [H: real,L2: real,H2: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_8600_polyfun__linear__factor__root,axiom,
! [C: nat > complex,A3: complex,N: nat] :
( ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ~ ! [B2: nat > complex] :
~ ! [Z5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ Z5 @ A3 )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B2 @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8601_polyfun__linear__factor__root,axiom,
! [C: nat > rat,A3: rat,N: nat] :
( ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat )
=> ~ ! [B2: nat > rat] :
~ ! [Z5: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ Z5 @ A3 )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B2 @ I3 ) @ ( power_power_rat @ Z5 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8602_polyfun__linear__factor__root,axiom,
! [C: nat > int,A3: int,N: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int )
=> ~ ! [B2: nat > int] :
~ ! [Z5: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( minus_minus_int @ Z5 @ A3 )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B2 @ I3 ) @ ( power_power_int @ Z5 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8603_polyfun__linear__factor__root,axiom,
! [C: nat > real,A3: real,N: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ~ ! [B2: nat > real] :
~ ! [Z5: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( minus_minus_real @ Z5 @ A3 )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B2 @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8604_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_8605_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_8606_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_8607_finite__nat__bounded,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_8608_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_8609_prod_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8610_prod_Otriangle__reindex,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8611_sum__less__suminf2,axiom,
! [F: nat > int,N: nat,I: nat] :
( ( summable_int @ F )
=> ( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
=> ( ( ord_less_eq_nat @ N @ I )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_8612_sum__less__suminf2,axiom,
! [F: nat > nat,N: nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
=> ( ( ord_less_eq_nat @ N @ I )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_8613_sum__less__suminf2,axiom,
! [F: nat > real,N: nat,I: nat] :
( ( summable_real @ F )
=> ( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
=> ( ( ord_less_eq_nat @ N @ I )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_8614_suminf__eq__zero__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ( ( suminf_real @ F )
= zero_zero_real )
= ( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_real ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8615_suminf__eq__zero__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ( ( suminf_nat @ F )
= zero_zero_nat )
= ( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_nat ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8616_suminf__eq__zero__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ( ( suminf_int @ F )
= zero_zero_int )
= ( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_int ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8617_suminf__nonneg,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8618_suminf__nonneg,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8619_suminf__nonneg,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8620_suminf__pos,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_pos
thf(fact_8621_suminf__pos,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_pos
thf(fact_8622_suminf__pos,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_pos
thf(fact_8623_summable__0__powser,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% summable_0_powser
thf(fact_8624_summable__0__powser,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% summable_0_powser
thf(fact_8625_summable__zero__power_H,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% summable_zero_power'
thf(fact_8626_summable__zero__power_H,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% summable_zero_power'
thf(fact_8627_summable__zero__power_H,axiom,
! [F: nat > int] :
( summable_int
@ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).
% summable_zero_power'
thf(fact_8628_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
=> ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% powser_split_head(3)
thf(fact_8629_powser__split__head_I3_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% powser_split_head(3)
thf(fact_8630_summable__powser__split__head,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
= ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% summable_powser_split_head
thf(fact_8631_summable__powser__split__head,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
= ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% summable_powser_split_head
thf(fact_8632_summable__norm__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N8 @ N2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_8633_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N8 @ N2 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_8634_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_8635_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_8636_summable__rabs,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% summable_rabs
thf(fact_8637_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_8638_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_8639_sum__diff__distrib,axiom,
! [Q: int > nat,P: int > nat,N: int] :
( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
=> ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_ord_lessThan_int @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_8640_sum__diff__distrib,axiom,
! [Q: real > nat,P: real > nat,N: real] :
( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
=> ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_8641_sum__diff__distrib,axiom,
! [Q: nat > nat,P: nat > nat,N: nat] :
( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_8642_suminf__pos2,axiom,
! [F: nat > real,I: nat] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8643_suminf__pos2,axiom,
! [F: nat > nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8644_suminf__pos2,axiom,
! [F: nat > int,I: nat] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8645_suminf__pos__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
= ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8646_suminf__pos__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
= ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8647_suminf__pos__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
= ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8648_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_8649_polyfun__diff__alt,axiom,
! [N: nat,A3: nat > complex,X: complex,Y: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A3 @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A3 @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A3 @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ 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_8650_polyfun__diff__alt,axiom,
! [N: nat,A3: nat > rat,X: rat,Y: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A3 @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ 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_8651_polyfun__diff__alt,axiom,
! [N: nat,A3: nat > int,X: int,Y: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( power_power_int @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A3 @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ 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_8652_polyfun__diff__alt,axiom,
! [N: nat,A3: nat > real,X: real,Y: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A3 @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A3 @ I3 ) @ ( power_power_real @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A3 @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ 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_8653_powser__inside,axiom,
! [F: nat > real,X: real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ).
% powser_inside
thf(fact_8654_powser__inside,axiom,
! [F: nat > complex,X: complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ).
% powser_inside
thf(fact_8655_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_8656_summable__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% summable_geometric
thf(fact_8657_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_8658_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_8659_suminf__split__head,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_8660_pi__less__4,axiom,
ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% pi_less_4
thf(fact_8661_pi__ge__two,axiom,
ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% pi_ge_two
thf(fact_8662_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_8663_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_8664_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_8665_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_8666_sum__pos__lt__pair,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_8667_sum__telescope,axiom,
! [F: nat > rat,I: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8668_sum__telescope,axiom,
! [F: nat > int,I: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8669_sum__telescope,axiom,
! [F: nat > real,I: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8670_polyfun__eq__coeffs,axiom,
! [C: nat > complex,N: nat,D: nat > complex] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8671_polyfun__eq__coeffs,axiom,
! [C: nat > real,N: nat,D: nat > real] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8672_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_8673_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_8674_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_8675_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_8676_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_8677_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_8678_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_8679_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_8680_sum__lessThan__telescope,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8681_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8682_sum__lessThan__telescope,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8683_sum__lessThan__telescope_H,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8684_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8685_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8686_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_8687_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_8688_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_8689_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_8690_summable__norm,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) ) ).
% summable_norm
thf(fact_8691_summable__norm,axiom,
! [F: nat > complex] :
( ( summable_real
@ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
@ ( suminf_real
@ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).
% summable_norm
thf(fact_8692_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_8693_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_8694_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_8695_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_8696_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_8697_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_8698_polyfun__diff,axiom,
! [N: nat,A3: nat > complex,X: complex,Y: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A3 @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A3 @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A3 @ I3 ) @ ( power_power_complex @ Y @ ( 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_8699_polyfun__diff,axiom,
! [N: nat,A3: nat > rat,X: rat,Y: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( power_power_rat @ Y @ ( 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_8700_polyfun__diff,axiom,
! [N: nat,A3: nat > int,X: int,Y: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( power_power_int @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( power_power_int @ Y @ ( 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_8701_polyfun__diff,axiom,
! [N: nat,A3: nat > real,X: real,Y: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A3 @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A3 @ I3 ) @ ( power_power_real @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A3 @ I3 ) @ ( power_power_real @ Y @ ( 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_8702_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_8703_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_8704_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_8705_sum__le__suminf,axiom,
! [F: nat > int,I5: set_nat] :
( ( summable_int @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_8706_sum__le__suminf,axiom,
! [F: nat > nat,I5: set_nat] :
( ( summable_nat @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_8707_sum__le__suminf,axiom,
! [F: nat > real,I5: set_nat] :
( ( summable_real @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_8708_polyfun__eq__0,axiom,
! [C: nat > complex,N: nat] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ 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_8709_polyfun__eq__0,axiom,
! [C: nat > real,N: nat] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ 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_8710_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > complex,N: nat,K: nat] :
( ! [W: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W @ 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_8711_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N: nat,K: nat] :
( ! [W: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W @ 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_8712_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_8713_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_8714_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_8715_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_8716_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_8717_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_8718_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_8719_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_8720_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_8721_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_8722_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_8723_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_8724_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_8725_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_8726_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_8727_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_8728_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_8729_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_8730_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_8731_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8732_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8733_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_8734_powser__split__head_I1_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
=> ( ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
= ( plus_plus_complex @ ( F @ zero_zero_nat )
@ ( times_times_complex
@ ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
@ Z ) ) ) ) ).
% powser_split_head(1)
thf(fact_8735_powser__split__head_I1_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
=> ( ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
= ( plus_plus_real @ ( F @ zero_zero_nat )
@ ( times_times_real
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
@ Z ) ) ) ) ).
% powser_split_head(1)
thf(fact_8736_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
@ Z )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_8737_powser__split__head_I2_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
@ Z )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_8738_summable__partial__sum__bound,axiom,
! [F: nat > complex,E2: real] :
( ( summable_complex @ F )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N9: nat] :
~ ! [M2: nat] :
( ( ord_less_eq_nat @ N9 @ M2 )
=> ! [N7: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E2 ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_8739_summable__partial__sum__bound,axiom,
! [F: nat > real,E2: real] :
( ( summable_real @ F )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [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 ) ) ) @ E2 ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_8740_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_8741_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_8742_summable__power__series,axiom,
! [F: nat > real,Z: real] :
( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
=> ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ( 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_8743_Abel__lemma,axiom,
! [R2: real,R0: real,A3: nat > complex,M7: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_real @ R2 @ R0 )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A3 @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M7 )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A3 @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_8744_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_8745_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_8746_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_8747_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_8748_polyfun__finite__roots,axiom,
! [C: nat > complex,N: nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ 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_8749_polyfun__finite__roots,axiom,
! [C: nat > real,N: nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ 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_8750_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_8751_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_8752_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_8753_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_8754_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_8755_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_8756_diff__power__eq__sum,axiom,
! [X: complex,N: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8757_diff__power__eq__sum,axiom,
! [X: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8758_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8759_diff__power__eq__sum,axiom,
! [X: real,N: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8760_power__diff__sumr2,axiom,
! [X: complex,N: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8761_power__diff__sumr2,axiom,
! [X: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8762_power__diff__sumr2,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8763_power__diff__sumr2,axiom,
! [X: real,N: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8764_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_8765_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_8766_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_8767_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_8768_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_8769_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_8770_summable__ratio__test,axiom,
! [C: real,N6: nat,F: nat > real] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_ratio_test
thf(fact_8771_summable__ratio__test,axiom,
! [C: real,N6: nat,F: nat > complex] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_ratio_test
thf(fact_8772_arctan__ubound,axiom,
! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arctan_ubound
thf(fact_8773_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > code_integer,K4: code_integer,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_le3102999989581377725nteger @ ( F @ P7 ) @ K4 ) )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ K4 )
=> ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8774_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > rat,K4: rat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_rat @ ( F @ P7 ) @ K4 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ K4 )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8775_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > int,K4: int,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_int @ ( F @ P7 ) @ K4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K4 )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8776_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > nat,K4: nat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_nat @ ( F @ P7 ) @ K4 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8777_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > real,K4: real,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_real @ ( F @ P7 ) @ K4 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ K4 )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8778_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_8779_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_8780_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_8781_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_8782_polynomial__product,axiom,
! [M: nat,A3: nat > complex,N: nat,B3: nat > complex,X: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A3 @ I2 )
= zero_zero_complex ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B3 @ J2 )
= zero_zero_complex ) )
=> ( ( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A3 @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( times_times_complex @ ( B3 @ J3 ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [R5: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( A3 @ K3 ) @ ( B3 @ ( 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_8783_polynomial__product,axiom,
! [M: nat,A3: nat > rat,N: nat,B3: nat > rat,X: rat] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A3 @ I2 )
= zero_zero_rat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B3 @ J2 )
= zero_zero_rat ) )
=> ( ( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A3 @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( times_times_rat @ ( B3 @ 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 @ ( A3 @ K3 ) @ ( B3 @ ( 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_8784_polynomial__product,axiom,
! [M: nat,A3: nat > int,N: nat,B3: nat > int,X: int] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A3 @ I2 )
= zero_zero_int ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B3 @ J2 )
= zero_zero_int ) )
=> ( ( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A3 @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( B3 @ 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 @ ( A3 @ K3 ) @ ( B3 @ ( 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_8785_polynomial__product,axiom,
! [M: nat,A3: nat > real,N: nat,B3: nat > real,X: real] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A3 @ I2 )
= zero_zero_real ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B3 @ J2 )
= zero_zero_real ) )
=> ( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A3 @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( B3 @ 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 @ ( A3 @ K3 ) @ ( B3 @ ( 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_8786_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_8787_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_8788_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_8789_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_8790_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_8791_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_8792_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_8793_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_8794_polyfun__eq__const,axiom,
! [C: nat > complex,N: nat,K: complex] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C @ X3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_8795_polyfun__eq__const,axiom,
! [C: nat > real,N: nat,K: real] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C @ X3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_8796_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_8797_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > literal > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_literal ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_literal ) ) ) ) ).
% Maclaurin_zero
thf(fact_8798_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_8799_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_8800_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_8801_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_8802_polynomial__product__nat,axiom,
! [M: nat,A3: nat > nat,N: nat,B3: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A3 @ I2 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B3 @ J2 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A3 @ I3 ) @ ( power_power_nat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( B3 @ 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 @ ( A3 @ K3 ) @ ( B3 @ ( 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_8803_arctan__lbound,axiom,
! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_8804_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arctan_bounded
thf(fact_8805_Maclaurin__lemma,axiom,
! [H: real,F: real > real,J: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H )
=> ? [B8: real] :
( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_8806_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > rat,H: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8807_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > int,H: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8808_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8809_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8810_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > complex,H: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8811_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8812_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > rat,H: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8813_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8814_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > int,H: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8815_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_8816_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_8817_root__polyfun,axiom,
! [N: nat,Z: complex,A3: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_complex @ Z @ N )
= A3 )
= ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A3 ) @ ( 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_8818_root__polyfun,axiom,
! [N: nat,Z: int,A3: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_int @ Z @ N )
= A3 )
= ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A3 ) @ ( 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_8819_root__polyfun,axiom,
! [N: nat,Z: code_integer,A3: code_integer] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_8256067586552552935nteger @ Z @ N )
= A3 )
= ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A3 ) @ ( 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_8820_root__polyfun,axiom,
! [N: nat,Z: rat,A3: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_rat @ Z @ N )
= A3 )
= ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A3 ) @ ( 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_8821_root__polyfun,axiom,
! [N: nat,Z: real,A3: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_real @ Z @ N )
= A3 )
= ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A3 ) @ ( 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_8822_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_8823_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_8824_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_8825_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_8826_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_8827_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_8828_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_8829_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_8830_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_8831_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_8832_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_8833_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_8834_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_8835_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_8836_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_8837_polyfun__extremal__lemma,axiom,
! [E2: real,C: nat > complex,N: nat] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M8: real] :
! [Z5: complex] :
( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z5 ) )
=> ( ord_less_eq_real
@ ( real_V1022390504157884413omplex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
@ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8838_polyfun__extremal__lemma,axiom,
! [E2: real,C: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M8: real] :
! [Z5: real] :
( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z5 ) )
=> ( ord_less_eq_real
@ ( real_V7735802525324610683m_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
@ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8839_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
& ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_8840_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_real @ T5 @ 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 @ T5 @ ( 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_8841_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ 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 @ T5 @ ( 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_8842_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 @ ( 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_8843_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_8844_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_8845_sin__zero,axiom,
( ( sin_real @ zero_zero_real )
= zero_zero_real ) ).
% sin_zero
thf(fact_8846_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_8847_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_8848_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% sin_le_one
thf(fact_8849_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_8850_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_8851_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_8852_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_8853_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_8854_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_8855_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
thf(fact_8856_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
thf(fact_8857_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_8858_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_8859_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_8860_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_8861_sin__inj__pi,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( sin_real @ X )
= ( sin_real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_8862_sin__mono__le__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_8863_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_8864_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_8865_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_8866_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_8867_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_8868_sin__mono__less__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_8869_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X4 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
& ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ Y5 )
= Y ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% sin_total
thf(fact_8870_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_8871_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_8872_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( 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 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_8873_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_real @ T5 @ 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 @ T5 @ ( 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_8874_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ? [T5: real] :
( ( ord_less_real @ X @ T5 )
& ( ord_less_real @ T5 @ 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 @ T5 @ ( 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_8875_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 @ ( 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_8876_gbinomial__partial__row__sum,axiom,
! [A3: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A3 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ ( 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 @ A3 @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8877_gbinomial__partial__row__sum,axiom,
! [A3: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A3 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A3 @ ( 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 @ A3 @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8878_gbinomial__partial__row__sum,axiom,
! [A3: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A3 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A3 @ ( 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 @ A3 @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8879_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_8880_cos__zero,axiom,
( ( cos_complex @ zero_zero_complex )
= one_one_complex ) ).
% cos_zero
thf(fact_8881_cos__zero,axiom,
( ( cos_real @ zero_zero_real )
= one_one_real ) ).
% cos_zero
thf(fact_8882_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
= zero_zero_real ) ).
% gbinomial_0(2)
thf(fact_8883_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
= zero_zero_rat ) ).
% gbinomial_0(2)
thf(fact_8884_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_8885_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_8886_gbinomial__0_I1_J,axiom,
! [A3: complex] :
( ( gbinomial_complex @ A3 @ zero_zero_nat )
= one_one_complex ) ).
% gbinomial_0(1)
thf(fact_8887_gbinomial__0_I1_J,axiom,
! [A3: real] :
( ( gbinomial_real @ A3 @ zero_zero_nat )
= one_one_real ) ).
% gbinomial_0(1)
thf(fact_8888_gbinomial__0_I1_J,axiom,
! [A3: rat] :
( ( gbinomial_rat @ A3 @ zero_zero_nat )
= one_one_rat ) ).
% gbinomial_0(1)
thf(fact_8889_gbinomial__0_I1_J,axiom,
! [A3: nat] :
( ( gbinomial_nat @ A3 @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_8890_gbinomial__0_I1_J,axiom,
! [A3: int] :
( ( gbinomial_int @ A3 @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_8891_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_8892_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_8893_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_8894_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_8895_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_8896_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_8897_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% cos_le_one
thf(fact_8898_cos__one__sin__zero,axiom,
! [X: complex] :
( ( ( cos_complex @ X )
= one_one_complex )
=> ( ( sin_complex @ X )
= zero_zero_complex ) ) ).
% cos_one_sin_zero
thf(fact_8899_cos__one__sin__zero,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
=> ( ( sin_real @ X )
= zero_zero_real ) ) ).
% cos_one_sin_zero
thf(fact_8900_gbinomial__Suc__Suc,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) )
= ( plus_plus_complex @ ( gbinomial_complex @ A3 @ K ) @ ( gbinomial_complex @ A3 @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8901_gbinomial__Suc__Suc,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) )
= ( plus_plus_real @ ( gbinomial_real @ A3 @ K ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8902_gbinomial__Suc__Suc,axiom,
! [A3: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ A3 @ K ) @ ( gbinomial_rat @ A3 @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8903_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_8904_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_8905_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ( cos_real @ X )
= ( cos_real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_8906_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_8907_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_8908_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_8909_sin__zero__norm__cos__one,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
= one_one_real ) ) ).
% sin_zero_norm_cos_one
thf(fact_8910_sin__zero__norm__cos__one,axiom,
! [X: complex] :
( ( ( sin_complex @ X )
= zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
= one_one_real ) ) ).
% sin_zero_norm_cos_one
thf(fact_8911_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_8912_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_8913_gbinomial__addition__formula,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ A3 @ ( suc @ K ) )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8914_gbinomial__addition__formula,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ A3 @ ( suc @ K ) )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8915_gbinomial__addition__formula,axiom,
! [A3: rat,K: nat] :
( ( gbinomial_rat @ A3 @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8916_gbinomial__absorb__comp,axiom,
! [A3: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A3 @ K ) )
= ( times_times_complex @ A3 @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8917_gbinomial__absorb__comp,axiom,
! [A3: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A3 @ K ) )
= ( times_times_rat @ A3 @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8918_gbinomial__absorb__comp,axiom,
! [A3: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A3 @ K ) )
= ( times_times_real @ A3 @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8919_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A3: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A3 )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A3 @ K ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_8920_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A3: rat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A3 )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A3 @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A3 @ K ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_8921_gbinomial__mult__1,axiom,
! [A3: rat,K: nat] :
( ( times_times_rat @ A3 @ ( gbinomial_rat @ A3 @ K ) )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A3 @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_8922_gbinomial__mult__1,axiom,
! [A3: real,K: nat] :
( ( times_times_real @ A3 @ ( gbinomial_real @ A3 @ K ) )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A3 @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_8923_gbinomial__mult__1_H,axiom,
! [A3: rat,K: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A3 @ K ) @ A3 )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A3 @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_8924_gbinomial__mult__1_H,axiom,
! [A3: real,K: nat] :
( ( times_times_real @ ( gbinomial_real @ A3 @ K ) @ A3 )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A3 @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_8925_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_8926_cos__two__less__zero,axiom,
ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_less_zero
thf(fact_8927_cos__is__zero,axiom,
? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X4 )
= zero_zero_real )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
& ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ Y5 )
= zero_zero_real ) )
=> ( Y5 = X4 ) ) ) ).
% cos_is_zero
thf(fact_8928_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_8929_Suc__times__gbinomial,axiom,
! [K: nat,A3: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) ) )
= ( times_times_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( gbinomial_complex @ A3 @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8930_Suc__times__gbinomial,axiom,
! [K: nat,A3: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( suc @ K ) ) )
= ( times_times_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( gbinomial_rat @ A3 @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8931_Suc__times__gbinomial,axiom,
! [K: nat,A3: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) ) )
= ( times_times_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( gbinomial_real @ A3 @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8932_gbinomial__absorption,axiom,
! [K: nat,A3: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A3 @ ( suc @ K ) ) )
= ( times_times_complex @ A3 @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8933_gbinomial__absorption,axiom,
! [K: nat,A3: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A3 @ ( suc @ K ) ) )
= ( times_times_rat @ A3 @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8934_gbinomial__absorption,axiom,
! [K: nat,A3: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A3 @ ( suc @ K ) ) )
= ( times_times_real @ A3 @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8935_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_8936_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ pi )
& ( ( cos_real @ X4 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
& ( ord_less_eq_real @ Y5 @ pi )
& ( ( cos_real @ Y5 )
= Y ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% cos_total
thf(fact_8937_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A3: rat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_rat @ ( gbinomial_rat @ A3 @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A3 @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_8938_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A3: real] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_real @ ( gbinomial_real @ A3 @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
= ( times_times_real @ ( gbinomial_real @ A3 @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_8939_sincos__principal__value,axiom,
! [X: real] :
? [Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
& ( ord_less_eq_real @ Y4 @ pi )
& ( ( sin_real @ Y4 )
= ( sin_real @ X ) )
& ( ( cos_real @ Y4 )
= ( cos_real @ X ) ) ) ).
% sincos_principal_value
thf(fact_8940_gbinomial__parallel__sum,axiom,
! [A3: complex,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8941_gbinomial__parallel__sum,axiom,
! [A3: rat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8942_gbinomial__parallel__sum,axiom,
! [A3: real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8943_gbinomial__rec,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( gbinomial_complex @ A3 @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8944_gbinomial__rec,axiom,
! [A3: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A3 @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8945_gbinomial__rec,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( gbinomial_real @ A3 @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8946_gbinomial__factors,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A3 @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A3 @ K ) ) ) ).
% gbinomial_factors
thf(fact_8947_gbinomial__factors,axiom,
! [A3: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A3 @ K ) ) ) ).
% gbinomial_factors
thf(fact_8948_gbinomial__factors,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A3 @ K ) ) ) ).
% gbinomial_factors
thf(fact_8949_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_8950_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_8951_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_8952_gbinomial__negated__upper,axiom,
( gbinomial_complex
= ( ^ [A: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8953_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A: 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 ) @ A ) @ one_one_rat ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8954_gbinomial__negated__upper,axiom,
( gbinomial_real
= ( ^ [A: 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 ) @ A ) @ one_one_real ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8955_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% sin_cos_le1
thf(fact_8956_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_8957_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_8958_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_8959_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_8960_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_8961_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_8962_gbinomial__minus,axiom,
! [A3: complex,K: nat] :
( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A3 ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8963_gbinomial__minus,axiom,
! [A3: rat,K: nat] :
( ( gbinomial_rat @ ( uminus_uminus_rat @ A3 ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8964_gbinomial__minus,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ ( uminus_uminus_real @ A3 ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8965_gbinomial__reduce__nat,axiom,
! [K: nat,A3: complex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_complex @ A3 @ K )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8966_gbinomial__reduce__nat,axiom,
! [K: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_real @ A3 @ K )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8967_gbinomial__reduce__nat,axiom,
! [K: nat,A3: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A3 @ K )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8968_gbinomial__pochhammer,axiom,
( gbinomial_complex
= ( ^ [A: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8969_gbinomial__pochhammer,axiom,
( gbinomial_rat
= ( ^ [A: 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 @ A ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8970_gbinomial__pochhammer,axiom,
( gbinomial_real
= ( ^ [A: 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 @ A ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8971_gbinomial__pochhammer_H,axiom,
( gbinomial_complex
= ( ^ [A: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8972_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8973_gbinomial__pochhammer_H,axiom,
( gbinomial_real
= ( ^ [A: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8974_cos__double__cos,axiom,
! [W2: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% cos_double_cos
thf(fact_8975_cos__double__cos,axiom,
! [W2: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% cos_double_cos
thf(fact_8976_gbinomial__sum__lower__neg,axiom,
! [A3: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A3 @ 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 @ A3 @ one_one_complex ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8977_gbinomial__sum__lower__neg,axiom,
! [A3: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A3 @ 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 @ A3 @ one_one_rat ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8978_gbinomial__sum__lower__neg,axiom,
! [A3: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A3 @ 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 @ A3 @ one_one_real ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8979_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_8980_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_8981_cos__double__sin,axiom,
! [W2: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_8982_cos__double__sin,axiom,
! [W2: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_8983_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8984_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8985_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8986_gbinomial__Suc,axiom,
! [A3: rat,K: nat] :
( ( gbinomial_rat @ A3 @ ( suc @ K ) )
= ( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8987_gbinomial__Suc,axiom,
! [A3: code_integer,K: nat] :
( ( gbinom8545251970709558553nteger @ A3 @ ( suc @ K ) )
= ( divide6298287555418463151nteger
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( minus_8373710615458151222nteger @ A3 @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri3624122377584611663nteger @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8988_gbinomial__Suc,axiom,
! [A3: real,K: nat] :
( ( gbinomial_real @ A3 @ ( suc @ K ) )
= ( divide_divide_real
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8989_gbinomial__Suc,axiom,
! [A3: nat,K: nat] :
( ( gbinomial_nat @ A3 @ ( suc @ K ) )
= ( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( minus_minus_nat @ A3 @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8990_gbinomial__Suc,axiom,
! [A3: int,K: nat] :
( ( gbinomial_int @ A3 @ ( suc @ K ) )
= ( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( minus_minus_int @ A3 @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8991_gbinomial__absorption_H,axiom,
! [K: nat,A3: complex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_complex @ A3 @ K )
= ( times_times_complex @ ( divide1717551699836669952omplex @ A3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A3 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8992_gbinomial__absorption_H,axiom,
! [K: nat,A3: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A3 @ K )
= ( times_times_rat @ ( divide_divide_rat @ A3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A3 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8993_gbinomial__absorption_H,axiom,
! [K: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_real @ A3 @ K )
= ( times_times_real @ ( divide_divide_real @ A3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A3 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8994_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A3: complex,X: complex,Y: complex] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A3 ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A3 ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8995_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A3: rat,X: rat,Y: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A3 ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A3 ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8996_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A3: real,X: real,Y: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A3 ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A3 ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8997_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ pi )
& ( X
= ( cos_real @ T5 ) )
& ( Y
= ( sin_real @ T5 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_8998_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_8999_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_9000_gbinomial__code,axiom,
( gbinomial_complex
= ( ^ [A: complex,K3: nat] :
( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
@ ( divide1717551699836669952omplex
@ ( set_fo1517530859248394432omplex
@ ^ [L3: nat] : ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ L3 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_complex )
@ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_9001_gbinomial__code,axiom,
( gbinomial_rat
= ( ^ [A: rat,K3: nat] :
( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
@ ( divide_divide_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [L3: nat] : ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ L3 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_rat )
@ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_9002_gbinomial__code,axiom,
( gbinomial_real
= ( ^ [A: real,K3: nat] :
( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
@ ( divide_divide_real
@ ( set_fo3111899725591712190t_real
@ ^ [L3: nat] : ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ L3 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_real )
@ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_9003_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( 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 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_9004_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_9005_gchoose__row__sum__weighted,axiom,
! [R2: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_9006_gchoose__row__sum__weighted,axiom,
! [R2: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_9007_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( X
= ( cos_real @ T5 ) )
& ( Y
= ( sin_real @ T5 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_9008_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
& ( X
= ( cos_real @ T5 ) )
& ( Y
= ( sin_real @ T5 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_9009_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_9010_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_9011_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_9012_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ~ ! [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
=> ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X
= ( cos_real @ T5 ) )
=> ( Y
!= ( sin_real @ T5 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_9013_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_9014_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_9015_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_9016_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_9017_ceiling__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_9018_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V1022390504157884413omplex @ Z )
= one_one_real )
=> ~ ! [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
=> ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_9019_tan__zero,axiom,
( ( tan_real @ zero_zero_real )
= zero_zero_real ) ).
% tan_zero
thf(fact_9020_zero__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A3 @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_9021_log__less__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A3 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_9022_one__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ ( log @ A3 @ X ) )
= ( ord_less_real @ A3 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_9023_log__less__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A3 @ X ) @ one_one_real )
= ( ord_less_real @ X @ A3 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_9024_log__less__cancel__iff,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_9025_log__eq__one,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ A3 @ A3 )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_9026_log__le__cancel__iff,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_9027_log__le__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A3 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A3 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_9028_one__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A3 @ X ) )
= ( ord_less_eq_real @ A3 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_9029_log__le__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A3 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_9030_zero__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A3 @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_9031_log__pow__cancel,axiom,
! [A3: real,B3: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ A3 @ ( power_power_real @ A3 @ B3 ) )
= ( semiri5074537144036343181t_real @ B3 ) ) ) ) ).
% log_pow_cancel
thf(fact_9032_log__base__change,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ B3 @ X )
= ( divide_divide_real @ ( log @ A3 @ X ) @ ( log @ A3 @ B3 ) ) ) ) ) ).
% log_base_change
thf(fact_9033_less__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B3 @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_9034_log__of__power__eq,axiom,
! [M: nat,B3: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B3 @ N ) )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_9035_log__mult,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A3 @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_9036_log__divide,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A3 @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_9037_le__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B3 @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_9038_log__base__pow,axiom,
! [A3: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( log @ ( power_power_real @ A3 @ N ) @ X )
= ( divide_divide_real @ ( log @ A3 @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log_base_pow
thf(fact_9039_log__nat__power,axiom,
! [X: real,B3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B3 @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X ) ) ) ) ).
% log_nat_power
thf(fact_9040_log__of__power__less,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_9041_log__eq__div__ln__mult__log,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( B3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A3 @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( ln_ln_real @ A3 ) ) @ ( log @ B3 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_9042_log__of__power__le,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_9043_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% lemma_tan_total
thf(fact_9044_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_9045_lemma__tan__total1,axiom,
! [Y: real] :
? [X4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_9046_tan__mono__lt__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_9047_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( 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 @ Y @ X )
= ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_9048_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_9049_tan__total,axiom,
! [Y: real] :
? [X4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
& ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ Y5 )
= Y ) )
=> ( Y5 = X4 ) ) ) ).
% tan_total
thf(fact_9050_add__tan__eq,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_9051_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_9052_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_9053_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_9054_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_9055_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_9056_tan__mono__le__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_9057_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_9058_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_9059_arctan__unique,axiom,
! [X: real,Y: 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 )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_9060_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_9061_arctan,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_9062_tan__add,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_9063_tan__add,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
!= zero_zero_real )
=> ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_9064_tan__diff,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_9065_tan__diff,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
!= zero_zero_real )
=> ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
= ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_9066_lemma__tan__add1,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
= ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_9067_lemma__tan__add1,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
= ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_9068_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_9069_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_9070_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_9071_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_9072_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_9073_tan__half,axiom,
( tan_complex
= ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).
% tan_half
thf(fact_9074_tan__half,axiom,
( tan_real
= ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).
% tan_half
thf(fact_9075_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_9076_ceiling__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
=> ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_9077_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
@ ( sin_real @ X ) ) ).
% sin_paired
thf(fact_9078_ceiling__log__eq__powr__iff,axiom,
! [X: real,B3: real,K: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ( archim7802044766580827645g_real @ ( log @ B3 @ X ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
= ( ( ord_less_real @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ X )
& ( ord_less_eq_real @ X @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_9079_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_9080_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( sin_real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_9081_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_9082_powr__0,axiom,
! [Z: real] :
( ( powr_real @ zero_zero_real @ Z )
= zero_zero_real ) ).
% powr_0
thf(fact_9083_powr__eq__0__iff,axiom,
! [W2: real,Z: real] :
( ( ( powr_real @ W2 @ Z )
= zero_zero_real )
= ( W2 = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_9084_powr__one__eq__one,axiom,
! [A3: real] :
( ( powr_real @ one_one_real @ A3 )
= one_one_real ) ).
% powr_one_eq_one
thf(fact_9085_sums__zero,axiom,
( sums_real
@ ^ [N3: nat] : zero_zero_real
@ zero_zero_real ) ).
% sums_zero
thf(fact_9086_sums__zero,axiom,
( sums_nat
@ ^ [N3: nat] : zero_zero_nat
@ zero_zero_nat ) ).
% sums_zero
thf(fact_9087_sums__zero,axiom,
( sums_int
@ ^ [N3: nat] : zero_zero_int
@ zero_zero_int ) ).
% sums_zero
thf(fact_9088_powr__zero__eq__one,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_9089_powr__gt__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A3 ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_9090_powr__nonneg__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ ( powr_real @ A3 @ X ) @ zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_9091_powr__less__cancel__iff,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% powr_less_cancel_iff
thf(fact_9092_powr__eq__one__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ( powr_real @ A3 @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_9093_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_9094_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_9095_powr__le__cancel__iff,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% powr_le_cancel_iff
thf(fact_9096_powr__log__cancel,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ A3 @ ( log @ A3 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_9097_log__powr__cancel,axiom,
! [A3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ A3 @ ( powr_real @ A3 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_9098_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_9099_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_9100_powser__sums__zero__iff,axiom,
! [A3: nat > complex,X: complex] :
( ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( A3 @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
@ X )
= ( ( A3 @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_9101_powser__sums__zero__iff,axiom,
! [A3: nat > real,X: real] :
( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( A3 @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
@ X )
= ( ( A3 @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_9102_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_9103_sums__0,axiom,
! [F: nat > real] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_real )
=> ( sums_real @ F @ zero_zero_real ) ) ).
% sums_0
thf(fact_9104_sums__0,axiom,
! [F: nat > nat] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_nat )
=> ( sums_nat @ F @ zero_zero_nat ) ) ).
% sums_0
thf(fact_9105_sums__0,axiom,
! [F: nat > int] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_int )
=> ( sums_int @ F @ zero_zero_int ) ) ).
% sums_0
thf(fact_9106_sums__le,axiom,
! [F: nat > real,G: nat > real,S3: real,T: real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( sums_real @ F @ S3 )
=> ( ( sums_real @ G @ T )
=> ( ord_less_eq_real @ S3 @ T ) ) ) ) ).
% sums_le
thf(fact_9107_sums__le,axiom,
! [F: nat > nat,G: nat > nat,S3: nat,T: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( sums_nat @ F @ S3 )
=> ( ( sums_nat @ G @ T )
=> ( ord_less_eq_nat @ S3 @ T ) ) ) ) ).
% sums_le
thf(fact_9108_sums__le,axiom,
! [F: nat > int,G: nat > int,S3: int,T: int] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( sums_int @ F @ S3 )
=> ( ( sums_int @ G @ T )
=> ( ord_less_eq_int @ S3 @ T ) ) ) ) ).
% sums_le
thf(fact_9109_sums__single,axiom,
! [I: nat,F: nat > real] :
( sums_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
@ ( F @ I ) ) ).
% sums_single
thf(fact_9110_sums__single,axiom,
! [I: nat,F: nat > nat] :
( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( F @ I ) ) ).
% sums_single
thf(fact_9111_sums__single,axiom,
! [I: nat,F: nat > int] :
( sums_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
@ ( F @ I ) ) ).
% sums_single
thf(fact_9112_powr__non__neg,axiom,
! [A3: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A3 @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_9113_powr__less__mono2__neg,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A3 ) @ ( powr_real @ X @ A3 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_9114_powr__mono2,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y @ A3 ) ) ) ) ) ).
% powr_mono2
thf(fact_9115_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_9116_powr__less__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B3 ) ) ) ) ).
% powr_less_mono
thf(fact_9117_powr__less__cancel,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B3 ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% powr_less_cancel
thf(fact_9118_powr__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B3 ) ) ) ) ).
% powr_mono
thf(fact_9119_sums__mult__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
@ ( times_times_real @ C @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_9120_sums__mult2__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
@ ( times_times_real @ D @ C ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_9121_powr__less__mono2,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y @ A3 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_9122_powr__mono2_H,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A3 ) @ ( powr_real @ X @ A3 ) ) ) ) ) ).
% powr_mono2'
thf(fact_9123_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_9124_powr__inj,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ( powr_real @ A3 @ X )
= ( powr_real @ A3 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_9125_powr__le1,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_9126_powr__mono__both,axiom,
! [A3: real,B3: real,X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y @ B3 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_9127_ge__one__powr__ge__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A3 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_9128_powr__divide,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A3 )
= ( divide_divide_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y @ A3 ) ) ) ) ) ).
% powr_divide
thf(fact_9129_powr__mult,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( times_times_real @ X @ Y ) @ A3 )
= ( times_times_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_9130_sums__mult__D,axiom,
! [C: real,F: nat > real,A3: real] :
( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
@ A3 )
=> ( ( C != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A3 @ C ) ) ) ) ).
% sums_mult_D
thf(fact_9131_sums__Suc__imp,axiom,
! [F: nat > real,S3: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ S3 )
=> ( sums_real @ F @ S3 ) ) ) ).
% sums_Suc_imp
thf(fact_9132_sums__Suc,axiom,
! [F: nat > real,L: real] :
( ( sums_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ L )
=> ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_9133_sums__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( sums_nat
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ L )
=> ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_9134_sums__Suc,axiom,
! [F: nat > int,L: int] :
( ( sums_int
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ L )
=> ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_9135_sums__Suc__iff,axiom,
! [F: nat > real,S3: real] :
( ( sums_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ S3 )
= ( sums_real @ F @ ( plus_plus_real @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc_iff
thf(fact_9136_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > real,S3: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( F @ I2 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S3 )
= ( sums_real @ F @ S3 ) ) ) ).
% sums_zero_iff_shift
thf(fact_9137_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( sums_int
@ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( groups3539618377306564664at_int @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_9138_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_9139_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( sums_real
@ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( groups6591440286371151544t_real @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_9140_sums__If__finite,axiom,
! [P: nat > $o,F: nat > int] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( sums_int
@ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% sums_If_finite
thf(fact_9141_sums__If__finite,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% sums_If_finite
thf(fact_9142_sums__If__finite,axiom,
! [P: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( sums_real
@ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% sums_If_finite
thf(fact_9143_sums__finite,axiom,
! [N6: set_nat,F: nat > int] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_int ) )
=> ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_9144_sums__finite,axiom,
! [N6: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_nat ) )
=> ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_9145_sums__finite,axiom,
! [N6: set_nat,F: nat > real] :
( ( finite_finite_nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member_nat @ N2 @ N6 )
=> ( ( F @ N2 )
= zero_zero_real ) )
=> ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_9146_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_9147_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
& ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_9148_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_9149_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_9150_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_9151_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_9152_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_9153_powser__sums__if,axiom,
! [M: nat,Z: complex] :
( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N3 ) )
@ ( power_power_complex @ Z @ M ) ) ).
% powser_sums_if
thf(fact_9154_powser__sums__if,axiom,
! [M: nat,Z: real] :
( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N3 ) )
@ ( power_power_real @ Z @ M ) ) ).
% powser_sums_if
thf(fact_9155_powser__sums__if,axiom,
! [M: nat,Z: int] :
( sums_int
@ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N3 ) )
@ ( power_power_int @ Z @ M ) ) ).
% powser_sums_if
thf(fact_9156_powser__sums__zero,axiom,
! [A3: nat > complex] :
( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( A3 @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
@ ( A3 @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_9157_powser__sums__zero,axiom,
! [A3: nat > real] :
( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( A3 @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
@ ( A3 @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_9158_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_9159_powr__less__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( powr_real @ B3 @ Y ) @ X )
= ( ord_less_real @ Y @ ( log @ B3 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_9160_less__powr__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( powr_real @ B3 @ Y ) )
= ( ord_less_real @ ( log @ B3 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_9161_log__less__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ B3 @ X ) @ Y )
= ( ord_less_real @ X @ ( powr_real @ B3 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_9162_less__log__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ ( log @ B3 @ X ) )
= ( ord_less_real @ ( powr_real @ B3 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_9163_sums__If__finite__set_H,axiom,
! [G: nat > real,S2: real,A4: set_nat,S4: real,F: nat > real] :
( ( sums_real @ G @ S2 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( S4
= ( plus_plus_real @ S2
@ ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
@ A4 ) ) )
=> ( sums_real
@ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A4 ) @ ( F @ N3 ) @ ( G @ N3 ) )
@ S4 ) ) ) ) ).
% sums_If_finite_set'
thf(fact_9164_powr__minus__divide,axiom,
! [X: real,A3: real] :
( ( powr_real @ X @ ( uminus_uminus_real @ A3 ) )
= ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A3 ) ) ) ).
% powr_minus_divide
thf(fact_9165_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_9166_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_9167_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_9168_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_9169_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_9170_powr__neg__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X ) ) ) ).
% powr_neg_one
thf(fact_9171_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_9172_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
= ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_9173_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_9174_le__log__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( log @ B3 @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B3 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_9175_log__le__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B3 @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( powr_real @ B3 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_9176_le__powr__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B3 @ Y ) )
= ( ord_less_eq_real @ ( log @ B3 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_9177_powr__le__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B3 @ Y ) @ X )
= ( ord_less_eq_real @ Y @ ( log @ B3 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_9178_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_9179_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_9180_ln__powr__bound,axiom,
! [X: real,A3: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A3 ) @ A3 ) ) ) ) ).
% ln_powr_bound
thf(fact_9181_ln__powr__bound2,axiom,
! [X: real,A3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A3 ) @ ( times_times_real @ ( powr_real @ A3 @ A3 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_9182_log__add__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( B3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ ( log @ B3 @ X ) @ Y )
= ( log @ B3 @ ( times_times_real @ X @ ( powr_real @ B3 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_9183_add__log__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( B3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ Y @ ( log @ B3 @ X ) )
= ( log @ B3 @ ( times_times_real @ ( powr_real @ B3 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_9184_minus__log__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( B3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ Y @ ( log @ B3 @ X ) )
= ( log @ B3 @ ( divide_divide_real @ ( powr_real @ B3 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_9185_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_9186_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_9187_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ( sin_real @ ( arccos @ X ) )
!= zero_zero_real ) ) ) ).
% sin_arccos_nonzero
thf(fact_9188_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_9189_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_9190_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X ) )
!= zero_zero_real ) ) ) ).
% cos_arcsin_nonzero
thf(fact_9191_power__half__series,axiom,
( sums_real
@ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
@ one_one_real ) ).
% power_half_series
thf(fact_9192_log__minus__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( B3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ ( log @ B3 @ X ) @ Y )
= ( log @ B3 @ ( times_times_real @ X @ ( powr_real @ B3 @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_9193_arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
& ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_9194_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_9195_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( sums_real
@ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_9196_sums__if,axiom,
! [G: nat > real,X: real,F: nat > real,Y: real] :
( ( sums_real @ G @ X )
=> ( ( sums_real @ F @ Y )
=> ( sums_real
@ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ X @ Y ) ) ) ) ).
% sums_if
thf(fact_9197_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_9198_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_9199_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_9200_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_9201_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_9202_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_9203_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_9204_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_9205_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_9206_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
& ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_9207_arcsin__le__iff,axiom,
! [X: real,Y: 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 ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_9208_le__arcsin__iff,axiom,
! [X: real,Y: 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 ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_9209_floor__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_9210_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_9211_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_9212_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_9213_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_9214_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_9215_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_9216_floor__divide__real__eq__div,axiom,
! [B3: int,A3: real] :
( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A3 @ ( ring_1_of_int_real @ B3 ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A3 ) @ B3 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_9217_floor__log__eq__powr__iff,axiom,
! [X: real,B3: real,K: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B3 )
=> ( ( ( archim6058952711729229775r_real @ ( log @ B3 @ X ) )
= K )
= ( ( ord_less_eq_real @ ( powr_real @ B3 @ ( ring_1_of_int_real @ K ) ) @ X )
& ( ord_less_real @ X @ ( powr_real @ B3 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_9218_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_9219_floor__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
=> ( ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_9220_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_9221_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_9222_inverse__powr,axiom,
! [Y: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A3 )
= ( inverse_inverse_real @ ( powr_real @ Y @ A3 ) ) ) ) ).
% inverse_powr
thf(fact_9223_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_9224_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_9225_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_9226_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_9227_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% ln_inverse
thf(fact_9228_log__inverse,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A3 @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log @ A3 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_9229_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_9230_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_9231_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_9232_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_9233_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_9234_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_9235_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z2: int] :
( ( ord_less_eq_int @ D5 @ Z7 )
& ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_9236_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z2: int] :
( ( ord_less_eq_int @ D5 @ Z2 )
& ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_9237_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_9238_sinh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% sinh_real_le_iff
thf(fact_9239_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% sinh_real_neg_iff
thf(fact_9240_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% sinh_real_pos_iff
thf(fact_9241_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_9242_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_9243_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_9244_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_9245_cosh__real__pos,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% cosh_real_pos
thf(fact_9246_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% cosh_real_nonneg
thf(fact_9247_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_9248_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_9249_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% cosh_real_ge_1
thf(fact_9250_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_9251_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_9252_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_9253_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_9254_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_9255_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_9256_arctan__def,axiom,
( arctan
= ( ^ [Y3: real] :
( the_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 )
= Y3 ) ) ) ) ) ).
% arctan_def
thf(fact_9257_arcsin__def,axiom,
( arcsin
= ( ^ [Y3: real] :
( the_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 )
= Y3 ) ) ) ) ) ).
% arcsin_def
thf(fact_9258_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= 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 @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L )
@ ( 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_9259_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_9260_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_9261_mask__integer_Oabs__eq,axiom,
( bit_se2119862282449309892nteger
= ( ^ [X3: nat] : ( code_integer_of_int @ ( bit_se2000444600071755411sk_int @ X3 ) ) ) ) ).
% mask_integer.abs_eq
thf(fact_9262_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_9263_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_9264_sgn__mod,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ L @ K )
=> ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
= ( sgn_sgn_int @ L ) ) ) ) ).
% sgn_mod
thf(fact_9265_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_9266_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ln_ln_real @ X )
= ( the_real
@ ^ [X3: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_9267_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_9268_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_9269_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_9270_arccos__def,axiom,
( arccos
= ( ^ [Y3: real] :
( the_real
@ ^ [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ pi )
& ( ( cos_real @ X3 )
= Y3 ) ) ) ) ) ).
% arccos_def
thf(fact_9271_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
=> ( ( K
= ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_9272_mask__nat__def,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% mask_nat_def
thf(fact_9273_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_9274_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 ) ) )
| ? [L3: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L3 )
& ( A33
= ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
& ( L3 != zero_zero_int )
& ( K3
= ( times_times_int @ Q5 @ L3 ) ) )
| ? [R5: int,L3: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L3 )
& ( A33
= ( product_Pair_int_int @ Q5 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L3 ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
& ( K3
= ( plus_plus_int @ ( times_times_int @ Q5 @ L3 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_9275_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_9276_pi__half,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( the_real
@ ^ [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 ) ) ) ) ).
% pi_half
thf(fact_9277_pi__def,axiom,
( pi
= ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( the_real
@ ^ [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 ) ) ) ) ) ).
% pi_def
thf(fact_9278_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= 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 @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( 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_9279_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K3: int,L3: int] :
( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L3 ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_9280_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
= ( power_power_real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_9281_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K3: int,L3: int] :
( if_int @ ( L3 = zero_zero_int ) @ K3
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L3 ) )
@ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L3 )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L3 )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_9282_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_9283_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_9284_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
= one_one_int ) ).
% abs_division_segment
thf(fact_9285_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_9286_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_9287_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_9288_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_9289_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_9290_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_9291_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_9292_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_9293_nat__ceiling__le__eq,axiom,
! [X: real,A3: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A3 )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_9294_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_9295_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_9296_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_nat @ ( nat2 @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_9297_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_9298_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_9299_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_9300_division__segment__integer_Oabs__eq,axiom,
! [X: int] :
( ( euclid6289375185220004616nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( euclid3395696857347342551nt_int @ X ) ) ) ).
% division_segment_integer.abs_eq
thf(fact_9301_division__segment__nat__def,axiom,
( euclid3398187327856392827nt_nat
= ( ^ [N3: nat] : one_one_nat ) ) ).
% division_segment_nat_def
thf(fact_9302_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_9303_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_9304_eq__nat__nat__iff,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z8 ) )
= ( Z = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_9305_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_9306_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_9307_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_9308_sgn__integer_Oabs__eq,axiom,
! [X: int] :
( ( sgn_sgn_Code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( sgn_sgn_int @ X ) ) ) ).
% sgn_integer.abs_eq
thf(fact_9309_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ( euclid3395696857347342551nt_int @ K )
= ( sgn_sgn_int @ K ) ) ) ).
% division_segment_eq_sgn
thf(fact_9310_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_9311_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_9312_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_9313_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_9314_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_9315_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_9316_sgn__real__def,axiom,
( sgn_sgn_real
= ( ^ [A: real] : ( if_real @ ( A = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_real_def
thf(fact_9317_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_9318_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_9319_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_9320_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_9321_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_9322_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_9323_nat__add__distrib,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z8 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_9324_nat__mult__distrib,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z8 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_9325_Suc__as__int,axiom,
( suc
= ( ^ [A: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_9326_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_9327_nat__diff__distrib,axiom,
! [Z8: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( ord_less_eq_int @ Z8 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z8 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_9328_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_9329_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_9330_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_9331_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_9332_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_9333_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
= ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_9334_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_9335_le__nat__floor,axiom,
! [X: nat,A3: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A3 )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_9336_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_9337_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_9338_sgn__integer__code,axiom,
( sgn_sgn_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% sgn_integer_code
thf(fact_9339_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_9340_sgn__power__injE,axiom,
! [A3: real,N: nat,X: real,B3: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A3 ) @ ( power_power_real @ ( abs_abs_real @ A3 ) @ N ) )
= X )
=> ( ( X
= ( times_times_real @ ( sgn_sgn_real @ B3 ) @ ( power_power_real @ ( abs_abs_real @ B3 ) @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ).
% sgn_power_injE
thf(fact_9341_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_9342_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_9343_nat__mult__distrib__neg,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z8 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_9344_nat__abs__int__diff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
= ( minus_minus_nat @ B3 @ A3 ) ) )
& ( ~ ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_9345_division__segment__int__def,axiom,
( euclid3395696857347342551nt_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% division_segment_int_def
thf(fact_9346_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_9347_diff__nat__eq__if,axiom,
! [Z8: int,Z: int] :
( ( ( ord_less_int @ Z8 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_9348_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_9349_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_9350_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_9351_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_9352_set__encode__insert,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_9353_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_9354_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_9355_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_9356_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_9357_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_9358_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9359_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9360_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_9361_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_9362_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_9363_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_9364_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_9365_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_9366_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_9367_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_9368_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_9369_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_9370_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_9371_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_9372_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_9373_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less_nat @ N @ M )
& ( bit_se1146084159140164899it_int @ K @ N ) )
| ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_9374_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( bit_se1146084159140164899it_int @ K @ M2 )
= ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_9375_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_9376_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = 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 @ N3 @ ( 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_9377_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_9378_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_9379_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_9380_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_9381_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
!= ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% xor_negative_int_iff
thf(fact_9382_xor__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( bit_se3222712562003087583nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se6526347334894502574or_int @ Xa2 @ X ) ) ) ).
% xor_integer.abs_eq
thf(fact_9383_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_9384_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_9385_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_9386_bit__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_se9216721137139052372nteger @ ( code_integer_of_int @ X ) )
= ( bit_se1146084159140164899it_int @ X ) ) ).
% bit_integer.abs_eq
thf(fact_9387_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_9388_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_9389_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_9390_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_9391_XOR__upper,axiom,
! [X: int,N: nat,Y: 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 @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_9392_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X3: real] :
( the_int
@ ^ [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 )
& ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_9393_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_9394_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L3: 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 @ L3 @ ( 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 ) ) @ L3 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_9395_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y )
=> ( ( ( ( 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 ) ) ) )
=> ( Y
= ( 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 ) ) ) )
=> ( Y
= ( 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_9396_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
=> ( ( ( ( member_int @ K @ ( 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 ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member_int @ K @ ( 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 ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_9397_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
| ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_9398_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% and_negative_int_iff
thf(fact_9399_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_9400_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_9401_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_9402_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_9403_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_9404_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_9405_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_9406_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_9407_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y )
=> ( ( 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 ) ) ) )
=> ( Y
= ( 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 ) ) ) )
=> ( Y
= ( 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_9408_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X3: rat] :
( the_int
@ ^ [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 )
& ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_9409_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_9410_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_9411_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_9412_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_9413_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_9414_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_9415_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_9416_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_9417_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_9418_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_9419_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_9420_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_9421_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_9422_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_9423_sgn__rat__def,axiom,
( sgn_sgn_rat
= ( ^ [A: rat] : ( if_rat @ ( A = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_rat_def
thf(fact_9424_abs__rat__def,axiom,
( abs_abs_rat
= ( ^ [A: rat] : ( if_rat @ ( ord_less_rat @ A @ zero_zero_rat ) @ ( uminus_uminus_rat @ A ) @ A ) ) ) ).
% abs_rat_def
thf(fact_9425_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S: rat] :
( ( ord_less_rat @ zero_zero_rat @ S )
=> ! [T5: rat] :
( ( ord_less_rat @ zero_zero_rat @ T5 )
=> ( R2
!= ( plus_plus_rat @ S @ T5 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_9426_push__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se7788150548672797655nteger @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se545348938243370406it_int @ Xa2 @ X ) ) ) ).
% push_bit_integer.abs_eq
thf(fact_9427_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% less_eq_rat_def
thf(fact_9428_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M6: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
thf(fact_9429_and__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( bit_se3949692690581998587nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se725231765392027082nd_int @ Xa2 @ X ) ) ) ).
% and_integer.abs_eq
thf(fact_9430_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M6: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_9431_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_9432_bit__push__bit__iff__nat,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_9433_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M6: nat,N3: nat] :
( if_nat
@ ( ( M6 = zero_zero_nat )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_9434_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_9435_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_9436_normalize__negative,axiom,
! [Q4: int,P6: int] :
( ( ord_less_int @ Q4 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P6 @ Q4 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_9437_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( 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_9438_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_9439_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_9440_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_9441_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
| ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% or_negative_int_iff
thf(fact_9442_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_9443_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_9444_or__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( bit_se1080825931792720795nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se1409905431419307370or_int @ Xa2 @ X ) ) ) ).
% or_integer.abs_eq
thf(fact_9445_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_9446_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_9447_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_9448_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_9449_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_9450_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_9451_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_9452_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_9453_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_9454_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_9455_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_9456_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_fact
thf(fact_9457_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P6: int,Q4: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P6 @ Q4 ) )
=> ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_9458_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_9459_OR__upper,axiom,
! [X: int,N: nat,Y: 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 @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_9460_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_9461_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_9462_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B3: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A3 @ I2 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A3 @ I3 ) @ ( B3 @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_9463_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M6: nat] :
( ( ord_less_eq_nat @ M9 @ M6 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ M9 @ N3 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M6 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_9464_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_9465_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_9466_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9467_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_9468_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_9469_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L3 ) @ ( modulo364778990260209775nteger @ K3 @ L3 ) ) ) ) ).
% divmod_integer_def
thf(fact_9470_integer__of__num__def,axiom,
code_integer_of_num = numera6620942414471956472nteger ).
% integer_of_num_def
thf(fact_9471_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one )
= one_one_Code_integer ) ).
% integer_of_num_triv(1)
thf(fact_9472_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_9473_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% integer_of_num_triv(2)
thf(fact_9474_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_9475_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_9476_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_9477_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_9478_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_9479_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_9480_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_9481_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_9482_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_9483_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_9484_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_9485_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_9486_card__less__Suc2,axiom,
! [M7: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_9487_card__less__Suc,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_9488_card__less,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_9489_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_9490_subset__card__intvl__is__intvl,axiom,
! [A4: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) )
=> ( A4
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_9491_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_9492_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_9493_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N @ M4 ) )
@ M ) ) ).
% max_Suc1
thf(fact_9494_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_9495_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_9496_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_9497_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_9498_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_9499_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_9500_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_9501_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( 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 @ L3 @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( 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 @ L3 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_9502_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_9503_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_9504_finite__enumerate,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [R3: nat > nat] :
( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
& ! [N7: nat] :
( ( ord_less_nat @ N7 @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( R3 @ N7 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_9505_binomial__def,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] :
( finite_card_set_nat
@ ( collect_set_nat
@ ^ [K6: set_nat] :
( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
& ( ( finite_card_nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_9506_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K3 )
= ( sgn_sgn_Code_integer @ L3 ) )
@ ( code_divmod_abs @ K3 @ L3 )
@ ( 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 @ L3 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_9507_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_9508_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_9509_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_9510_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_9511_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_9512_drop__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se3928097537394005634nteger @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se8568078237143864401it_int @ Xa2 @ X ) ) ) ).
% drop_bit_integer.abs_eq
thf(fact_9513_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_9514_Inf__nat__def1,axiom,
! [K4: set_nat] :
( ( K4 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% Inf_nat_def1
thf(fact_9515_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_9516_euclidean__size__int__def,axiom,
( euclid4774559944035922753ze_int
= ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% euclidean_size_int_def
thf(fact_9517_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% Code_Numeral.negative_def
thf(fact_9518_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_9519_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_9520_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_9521_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_9522_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_9523_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_9524_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_9525_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_9526_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_9527_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_9528_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_9529_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_9530_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_9531_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_9532_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_9533_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_9534_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_9535_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_9536_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_9537_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_9538_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_9539_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_9540_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_9541_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_9542_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_9543_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_9544_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9545_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_9546_real__root__strict__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_9547_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_9548_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_9549_real__root__strict__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_9550_real__root__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_9551_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_9552_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_9553_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_9554_real__root__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_9555_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_9556_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_9557_ln__root,axiom,
! [N: nat,B3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ln_ln_real @ ( root @ N @ B3 ) )
= ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_9558_log__root,axiom,
! [N: nat,A3: real,B3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( log @ B3 @ ( root @ N @ A3 ) )
= ( divide_divide_real @ ( log @ B3 @ A3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_9559_log__base__root,axiom,
! [N: nat,B3: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( log @ ( root @ N @ B3 ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_9560_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 )
=> ! [Y3: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
= X )
=> ( P @ Y3 ) ) ) ) ) ).
% split_root
thf(fact_9561_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_9562_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_9563_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_9564_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_9565_nat__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( semiri4939895301339042750nteger @ N ) )
= N ) ).
% nat_of_integer_of_nat
thf(fact_9566_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_9567_of__nat__of__integer,axiom,
! [K: code_integer] :
( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
= ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).
% of_nat_of_integer
thf(fact_9568_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_nat ) ).
% nat_of_integer_code_post(1)
thf(fact_9569_nat__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
= ( nat2 @ X ) ) ).
% nat_of_integer.abs_eq
thf(fact_9570_nat__of__integer__code__post_I2_J,axiom,
( ( code_nat_of_integer @ one_one_Code_integer )
= one_one_nat ) ).
% nat_of_integer_code_post(2)
thf(fact_9571_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L3: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_9572_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9573_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
@ ^ [L3: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_9574_Gcd__remove0__nat,axiom,
! [M7: set_nat] :
( ( finite_finite_nat @ M7 )
=> ( ( gcd_Gcd_nat @ M7 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_9575_int__of__integer__sub,axiom,
! [K: num,L: num] :
( ( code_int_of_integer @ ( neg_nu5755505904847501662nteger @ K @ L ) )
= ( neg_numeral_sub_int @ K @ L ) ) ).
% int_of_integer_sub
thf(fact_9576_integer__of__int__int__of__integer,axiom,
! [K: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
= K ) ).
% integer_of_int_int_of_integer
thf(fact_9577_int__of__integer__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
= K ) ).
% int_of_integer_integer_of_int
thf(fact_9578_int__of__integer__inverse,axiom,
! [X: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
= X ) ).
% int_of_integer_inverse
thf(fact_9579_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_max_Code_integer @ K @ L ) )
= ( ord_max_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_max
thf(fact_9580_of__int__integer__of,axiom,
! [K: code_integer] :
( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
= K ) ).
% of_int_integer_of
thf(fact_9581_int__of__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
= K ) ).
% int_of_integer_of_int
thf(fact_9582_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_int ) ).
% zero_integer.rep_eq
thf(fact_9583_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% int_of_integer_numeral
thf(fact_9584_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_9585_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus1351360451143612070nteger @ X ) )
= ( uminus_uminus_int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_9586_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_9587_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ one_one_Code_integer )
= one_one_int ) ).
% one_integer.rep_eq
thf(fact_9588_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa2 ) )
= ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% minus_integer.rep_eq
thf(fact_9589_abs__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( abs_abs_Code_integer @ X ) )
= ( abs_abs_int @ ( code_int_of_integer @ X ) ) ) ).
% abs_integer.rep_eq
thf(fact_9590_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa2 ) )
= ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% divide_integer.rep_eq
thf(fact_9591_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X @ Xa2 ) )
= ( modulo_modulo_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% modulo_integer.rep_eq
thf(fact_9592_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_9593_sgn__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( sgn_sgn_Code_integer @ X ) )
= ( sgn_sgn_int @ ( code_int_of_integer @ X ) ) ) ).
% sgn_integer.rep_eq
thf(fact_9594_division__segment__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( euclid6289375185220004616nteger @ X ) )
= ( euclid3395696857347342551nt_int @ ( code_int_of_integer @ X ) ) ) ).
% division_segment_integer.rep_eq
thf(fact_9595_euclidean__size__integer_Orep__eq,axiom,
( euclid6377331345833325938nteger
= ( ^ [X3: code_integer] : ( euclid4774559944035922753ze_int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% euclidean_size_integer.rep_eq
thf(fact_9596_int__of__integer__inject,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( code_int_of_integer @ X )
= ( code_int_of_integer @ Y ) )
= ( X = Y ) ) ).
% int_of_integer_inject
thf(fact_9597_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( ( code_int_of_integer @ K )
= ( code_int_of_integer @ L ) )
=> ( K = L ) ) ).
% integer_eqI
thf(fact_9598_integer__eq__iff,axiom,
( ( ^ [Y6: code_integer,Z4: code_integer] : Y6 = Z4 )
= ( ^ [K3: code_integer,L3: code_integer] :
( ( code_int_of_integer @ K3 )
= ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_eq_iff
thf(fact_9599_Gcd__nat__eq__one,axiom,
! [N6: set_nat] :
( ( member_nat @ one_one_nat @ N6 )
=> ( ( gcd_Gcd_nat @ N6 )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_9600_less__integer_Orep__eq,axiom,
( ord_le6747313008572928689nteger
= ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_9601_integer__less__iff,axiom,
( ord_le6747313008572928689nteger
= ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_iff
thf(fact_9602_less__eq__integer_Orep__eq,axiom,
( ord_le3102999989581377725nteger
= ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_9603_integer__less__eq__iff,axiom,
( ord_le3102999989581377725nteger
= ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_9604_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X3: code_integer] : ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_9605_take__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se1745604003318907178nteger @ X @ Xa2 ) )
= ( bit_se2923211474154528505it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% take_bit_integer.rep_eq
thf(fact_9606_and__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se3949692690581998587nteger @ X @ Xa2 ) )
= ( bit_se725231765392027082nd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% and_integer.rep_eq
thf(fact_9607_bit__integer_Orep__eq,axiom,
( bit_se9216721137139052372nteger
= ( ^ [X3: code_integer] : ( bit_se1146084159140164899it_int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% bit_integer.rep_eq
thf(fact_9608_or__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se1080825931792720795nteger @ X @ Xa2 ) )
= ( bit_se1409905431419307370or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% or_integer.rep_eq
thf(fact_9609_xor__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se3222712562003087583nteger @ X @ Xa2 ) )
= ( bit_se6526347334894502574or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% xor_integer.rep_eq
thf(fact_9610_drop__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se3928097537394005634nteger @ X @ Xa2 ) )
= ( bit_se8568078237143864401it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% drop_bit_integer.rep_eq
thf(fact_9611_push__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se7788150548672797655nteger @ X @ Xa2 ) )
= ( bit_se545348938243370406it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% push_bit_integer.rep_eq
thf(fact_9612_mask__integer_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( bit_se2119862282449309892nteger @ X ) )
= ( bit_se2000444600071755411sk_int @ X ) ) ).
% mask_integer.rep_eq
thf(fact_9613_unset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa2 ) )
= ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% unset_bit_integer.rep_eq
thf(fact_9614_set__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se2793503036327961859nteger @ X @ Xa2 ) )
= ( bit_se7879613467334960850it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% set_bit_integer.rep_eq
thf(fact_9615_flip__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se1345352211410354436nteger @ X @ Xa2 ) )
= ( bit_se2159334234014336723it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% flip_bit_integer.rep_eq
thf(fact_9616_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_9617_Gcd__int__greater__eq__0,axiom,
! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_9618_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_9619_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_9620_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_9621_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_9622_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_9623_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_9624_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M6: nat,N3: nat] :
( N3
= ( suc @ M6 ) ) ) ) ) ).
% pred_nat_def
thf(fact_9625_num__of__nat__numeral__eq,axiom,
! [Q4: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q4 ) )
= Q4 ) ).
% num_of_nat_numeral_eq
thf(fact_9626_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_9627_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_9628_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_9629_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_9630_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_9631_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_9632_upto_Opelims,axiom,
! [X: int,Xa2: int,Y: list_int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_9633_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_9634_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_9635_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_9636_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_9637_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_9638_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_9639_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_9640_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_9641_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_9642_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_9643_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_9644_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_9645_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_9646_upto_Oelims,axiom,
! [X: int,Xa2: int,Y: list_int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9647_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_9648_num__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
= ( num_of_nat @ ( nat2 @ X ) ) ) ).
% num_of_integer.abs_eq
thf(fact_9649_num__of__integer_Orep__eq,axiom,
( code_num_of_integer
= ( ^ [X3: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ) ).
% num_of_integer.rep_eq
thf(fact_9650_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L3: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_9651_rat__floor__lemma,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A3 @ B3 ) ) @ ( fract @ A3 @ B3 ) )
& ( ord_less_rat @ ( fract @ A3 @ B3 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ) ).
% rat_floor_lemma
thf(fact_9652_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_9653_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_9654_less__rat,axiom,
! [B3: int,D: int,A3: int,C: int] :
( ( B3 != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A3 @ B3 ) @ ( fract @ C @ D ) )
= ( ord_less_int @ ( times_times_int @ ( times_times_int @ A3 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% less_rat
thf(fact_9655_le__rat,axiom,
! [B3: int,D: int,A3: int,C: int] :
( ( B3 != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A3 @ B3 ) @ ( fract @ C @ D ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A3 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% le_rat
thf(fact_9656_Rat__induct__pos,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( P @ ( fract @ A2 @ B2 ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct_pos
thf(fact_9657_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_9658_zero__less__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A3 @ B3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ) ).
% zero_less_Fract_iff
thf(fact_9659_Fract__less__zero__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_rat @ ( fract @ A3 @ B3 ) @ zero_zero_rat )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% Fract_less_zero_iff
thf(fact_9660_Fract__less__one__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_rat @ ( fract @ A3 @ B3 ) @ one_one_rat )
= ( ord_less_int @ A3 @ B3 ) ) ) ).
% Fract_less_one_iff
thf(fact_9661_one__less__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A3 @ B3 ) )
= ( ord_less_int @ B3 @ A3 ) ) ) ).
% one_less_Fract_iff
thf(fact_9662_Fract__le__zero__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_rat @ ( fract @ A3 @ B3 ) @ zero_zero_rat )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ) ).
% Fract_le_zero_iff
thf(fact_9663_zero__le__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A3 @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).
% zero_le_Fract_iff
thf(fact_9664_one__le__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A3 @ B3 ) )
= ( ord_less_eq_int @ B3 @ A3 ) ) ) ).
% one_le_Fract_iff
thf(fact_9665_Fract__le__one__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_rat @ ( fract @ A3 @ B3 ) @ one_one_rat )
= ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% Fract_le_one_iff
thf(fact_9666_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9667_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y: nat,X: nat] :
( ( ( ord_less_nat @ C @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_9668_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_9669_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_9670_zero__notin__Suc__image,axiom,
! [A4: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_9671_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_9672_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_9673_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_9674_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_9675_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_9676_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_9677_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9678_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_9679_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_9680_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_9681_finite__int__iff__bounded__le,axiom,
( finite_finite_int
= ( ^ [S5: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_9682_finite__int__iff__bounded,axiom,
( finite_finite_int
= ( ^ [S5: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_9683_image__int__atLeastAtMost,axiom,
! [A3: nat,B3: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_9684_image__int__atLeastLessThan,axiom,
! [A3: nat,B3: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A3 @ B3 ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_9685_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_9686_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9687_infinite__int__iff__infinite__nat__abs,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S2 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_9688_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_9689_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_9690_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_9691_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_9692_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_9693_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_9694_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_9695_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9696_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_9697_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_9698_suminf__eq__SUP__real,axiom,
! [X7: nat > real] :
( ( summable_real @ X7 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X7 @ I2 ) )
=> ( ( suminf_real @ X7 )
= ( comple1385675409528146559p_real
@ ( image_nat_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ X7 @ ( set_ord_lessThan_nat @ I3 ) )
@ top_top_set_nat ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_9699_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_9700_range__mult,axiom,
! [A3: real] :
( ( ( A3 = zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A3 ) @ top_top_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
& ( ( A3 != zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A3 ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% range_mult
thf(fact_9701_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_9702_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ( code_integer_of_int @ X )
= ( code_integer_of_int @ Y ) )
= ( X = Y ) ) ) ) ).
% integer_of_int_inject
thf(fact_9703_integer__of__int__induct,axiom,
! [P: code_integer > $o,X: code_integer] :
( ! [Y4: int] :
( ( member_int @ Y4 @ top_top_set_int )
=> ( P @ ( code_integer_of_int @ Y4 ) ) )
=> ( P @ X ) ) ).
% integer_of_int_induct
thf(fact_9704_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y4: int] :
( ( X
= ( code_integer_of_int @ Y4 ) )
=> ~ ( member_int @ Y4 @ top_top_set_int ) ) ).
% integer_of_int_cases
thf(fact_9705_int__of__integer__induct,axiom,
! [Y: int,P: int > $o] :
( ( member_int @ Y @ top_top_set_int )
=> ( ! [X4: code_integer] : ( P @ ( code_int_of_integer @ X4 ) )
=> ( P @ Y ) ) ) ).
% int_of_integer_induct
thf(fact_9706_int__of__integer__cases,axiom,
! [Y: int] :
( ( member_int @ Y @ top_top_set_int )
=> ~ ! [X4: code_integer] :
( Y
!= ( code_int_of_integer @ X4 ) ) ) ).
% int_of_integer_cases
thf(fact_9707_int__of__integer,axiom,
! [X: code_integer] : ( member_int @ ( code_int_of_integer @ X ) @ top_top_set_int ) ).
% int_of_integer
thf(fact_9708_integer__of__int__inverse,axiom,
! [Y: int] :
( ( member_int @ Y @ top_top_set_int )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
= Y ) ) ).
% integer_of_int_inverse
thf(fact_9709_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_9710_root__def,axiom,
( root
= ( ^ [N3: nat,X3: real] :
( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N3 ) )
@ X3 ) ) ) ) ).
% root_def
thf(fact_9711_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_9712_DERIV__isconst3,axiom,
! [A3: real,B3: real,X: real,Y: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ( F @ X )
= ( F @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_9713_DERIV__neg__dec__left,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_9714_DERIV__pos__inc__left,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_9715_DERIV__neg__dec__right,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_9716_DERIV__pos__inc__right,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_9717_DERIV__pos__imp__increasing,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_9718_DERIV__neg__imp__decreasing,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F @ B3 ) @ ( F @ A3 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_9719_DERIV__nonneg__imp__nondecreasing,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_9720_DERIV__nonpos__imp__nonincreasing,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
=> ( ord_less_eq_real @ ( F @ B3 ) @ ( F @ A3 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_9721_deriv__nonneg__imp__mono,axiom,
! [A3: real,B3: real,G: real > real,G2: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A3 @ B3 ) )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A3 @ B3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( G @ A3 ) @ ( G @ B3 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_9722_has__real__derivative__neg__dec__left,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_9723_has__real__derivative__pos__inc__left,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_9724_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_9725_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_9726_MVT2,axiom,
! [A3: real,B3: real,F: real > real,F5: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B3 )
=> ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A3 @ Z3 )
& ( ord_less_real @ Z3 @ B3 )
& ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) )
= ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ ( F5 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_9727_DERIV__local__const,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
=> ( ( F @ X )
= ( F @ Y4 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_const
thf(fact_9728_DERIV__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_ln
thf(fact_9729_DERIV__local__min,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_min
thf(fact_9730_DERIV__local__max,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_max
thf(fact_9731_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_ln_divide
thf(fact_9732_DERIV__pow,axiom,
! [N: nat,X: real,S3: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( power_power_real @ X3 @ N )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ S3 ) ) ).
% DERIV_pow
thf(fact_9733_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
@ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ 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_9734_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_9735_DERIV__series_H,axiom,
! [F: real > nat > real,F5: real > nat > real,X0: real,A3: real,B3: real,L5: nat > real] :
( ! [N2: nat] :
( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( F @ X3 @ N2 )
@ ( F5 @ X0 @ N2 )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( summable_real @ ( F @ X4 ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( ( summable_real @ ( F5 @ X0 ) )
=> ( ( summable_real @ L5 )
=> ( ! [N2: nat,X4: real,Y4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
@ ( suminf_real @ ( F5 @ X0 ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_9736_DERIV__log,axiom,
! [X: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( log @ B3 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B3 ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_log
thf(fact_9737_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
@ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ 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_9738_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
@ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
@ ( 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_9739_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_9740_DERIV__real__sqrt__generic,axiom,
! [X: real,D6: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D6
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D6
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_9741_arcosh__real__has__field__derivative,axiom,
! [X: real,A4: 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 @ A4 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_9742_artanh__real__has__field__derivative,axiom,
! [X: real,A4: 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 @ A4 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_9743_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X4 @ N3 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X3: real] :
( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X3 @ ( suc @ N3 ) ) ) )
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_9744_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_9745_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_9746_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_9747_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_9748_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_9749_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_9750_Maclaurin,axiom,
! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_real @ T5 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_9751_Maclaurin2,axiom,
! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_9752_Maclaurin__minus,axiom,
! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ H @ T5 )
& ( ord_less_eq_real @ T5 @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_real @ H @ T5 )
& ( ord_less_real @ T5 @ zero_zero_real )
& ( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_9753_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 )
=> ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
& ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_9754_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_9755_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A3: real,B3: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ A3 @ T5 )
& ( ord_less_eq_real @ T5 @ B3 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A3 @ C )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ? [T5: real] :
( ( ord_less_real @ A3 @ T5 )
& ( ord_less_real @ T5 @ C )
& ( ( F @ A3 )
= ( 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 @ A3 @ C ) @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_9756_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A3: real,B3: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ A3 @ T5 )
& ( ord_less_eq_real @ T5 @ B3 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A3 @ C )
=> ( ( ord_less_real @ C @ B3 )
=> ? [T5: real] :
( ( ord_less_real @ C @ T5 )
& ( ord_less_real @ T5 @ B3 )
& ( ( F @ B3 )
= ( 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 @ B3 @ C ) @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_9757_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A3: real,B3: real,C: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ A3 @ T5 )
& ( ord_less_eq_real @ T5 @ B3 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A3 @ C )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ( ord_less_eq_real @ A3 @ X )
=> ( ( ord_less_eq_real @ X @ B3 )
=> ( ( X != C )
=> ? [T5: real] :
( ( ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ X @ T5 )
& ( ord_less_real @ T5 @ C ) ) )
& ( ~ ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ C @ T5 )
& ( ord_less_real @ T5 @ 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 @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_9758_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K: nat,B5: real] :
( ! [M3: nat,T5: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M2: nat,T6: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H ) )
=> ( 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 @ B5 @ ( 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 ) @ T6 )
@ ( 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 @ T6 @ P5 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_9759_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X10: 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 @ X10 @ ( 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_9760_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D6: 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 )
=> ( D6
= ( 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 )
=> ( D6
= ( 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 )
=> ( D6
= ( 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 ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_9761_LIM__fun__less__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X5: real] :
( ( ( X5 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
=> ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_9762_LIM__fun__not__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
=> ( ( L != zero_zero_real )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X5: real] :
( ( ( X5 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
=> ( ( F @ X5 )
!= zero_zero_real ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_9763_LIM__fun__gt__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X5: real] :
( ( ( X5 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
=> ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_9764_summable__Leibniz_I2_J,axiom,
! [A3: nat > real] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ ( A3 @ zero_zero_nat ) )
=> ! [N7: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ 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 ) @ ( A3 @ 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_9765_summable__Leibniz_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A3 )
=> ( ( ord_less_real @ ( A3 @ 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 ) @ ( A3 @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ 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 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_9766_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ 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_9767_trivial__limit__sequentially,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_sequentially
thf(fact_9768_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_9769_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_9770_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_9771_monoseq__convergent,axiom,
! [X7: nat > real,B5: real] :
( ( topolo6980174941875973593q_real @ X7 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X7 @ I2 ) ) @ B5 )
=> ~ ! [L6: real] :
~ ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% monoseq_convergent
thf(fact_9772_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( filterlim_nat_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
@ ( 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_9773_LIMSEQ__inverse__zero,axiom,
! [X7: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N8 @ N2 )
=> ( ord_less_real @ R3 @ ( X7 @ N2 ) ) )
=> ( filterlim_nat_real
@ ^ [N3: nat] : ( inverse_inverse_real @ ( X7 @ N3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_9774_LIMSEQ__root__const,axiom,
! [C: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( filterlim_nat_real
@ ^ [N3: nat] : ( root @ N3 @ C )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ) ).
% LIMSEQ_root_const
thf(fact_9775_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim_nat_real
@ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_9776_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_9777_increasing__LIMSEQ,axiom,
! [F: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N7: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N7 ) @ E ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_9778_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_9779_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N3: nat] : ( divide_divide_real @ A3 @ ( power_power_real @ X @ N3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_9780_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_9781_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_9782_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_9783_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9784_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9785_summable,axiom,
! [A3: nat > real] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A3 @ N3 ) ) ) ) ) ) ).
% summable
thf(fact_9786_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( filterlim_nat_real
@ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% zeroseq_arctan_series
thf(fact_9787_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ 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 ) @ ( A3 @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_9788_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim_nat_real
@ ^ [N3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_9789_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ( filterlim_nat_real @ A3 @ ( 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 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
@ L4 )
& ( filterlim_nat_real
@ ^ [N3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
@ ( 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 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
& ( filterlim_nat_real
@ ^ [N3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_9790_summable__Leibniz_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A3 )
=> ( filterlim_nat_real
@ ^ [N3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(5)
thf(fact_9791_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim_nat_real
@ ^ [N3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A3 @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_9792_DERIV__neg__imp__decreasing__at__top,axiom,
! [B3: real,F: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ B3 @ X4 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y5 @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
=> ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_9793_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: real > real,F4: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
@ at_top_real
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_9794_at__top__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% at_top_le_at_infinity
thf(fact_9795_at__bot__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% at_bot_le_at_infinity
thf(fact_9796_DERIV__pos__imp__increasing__at__bot,axiom,
! [B3: real,F: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y5 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
=> ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_9797_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: real > real,F4: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
@ at_bot_real
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_9798_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_9799_eventually__sequentiallyI,axiom,
! [C: nat,P: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq_nat @ C @ X4 )
=> ( P @ X4 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_9800_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N5: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ( P @ N3 ) ) ) ) ).
% eventually_sequentially
thf(fact_9801_le__sequentially,axiom,
! [F4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
= ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_9802_eventually__at__left__real,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( eventually_real
@ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B3 @ A3 ) )
@ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) ) ) ).
% eventually_at_left_real
thf(fact_9803_Bseq__eq__bounded,axiom,
! [F: nat > real,A3: real,B3: real] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A3 @ B3 ) )
=> ( bfun_nat_real @ F @ at_top_nat ) ) ).
% Bseq_eq_bounded
thf(fact_9804_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_9805_eventually__at__right__real,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( eventually_real
@ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
@ ( topolo2177554685111907308n_real @ A3 @ ( set_or5849166863359141190n_real @ A3 ) ) ) ) ).
% eventually_at_right_real
thf(fact_9806_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_9807_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_9808_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_9809_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_9810_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_9811_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_9812_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_9813_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_9814_decseq__bounded,axiom,
! [X7: nat > real,B5: real] :
( ( order_9091379641038594480t_real @ X7 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X7 @ I2 ) )
=> ( bfun_nat_real @ X7 @ at_top_nat ) ) ) ).
% decseq_bounded
thf(fact_9815_decseq__convergent,axiom,
! [X7: nat > real,B5: real] :
( ( order_9091379641038594480t_real @ X7 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X7 @ I2 ) )
=> ~ ! [L6: real] :
( ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
=> ~ ! [I4: nat] : ( ord_less_eq_real @ L6 @ ( X7 @ I4 ) ) ) ) ) ).
% decseq_convergent
thf(fact_9816_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_9817_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_9818_Gcd__eq__Max,axiom,
! [M7: set_nat] :
( ( finite_finite_nat @ M7 )
=> ( ( M7 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( gcd_Gcd_nat @ M7 )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M6: nat] :
( collect_nat
@ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M6 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_9819_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_9820_card__le__Suc__Max,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_9821_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% Sup_nat_def
thf(fact_9822_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N3: nat] :
( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N3 ) @ M6 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_9823_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( 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 @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_9824_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( 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 @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_9825_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_VEBT_valid @ X3 @ ( 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 @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_9826_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( 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 @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_9827_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa2 = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( 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 @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_9828_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 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( 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 @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_9829_Sup__int__def,axiom,
( complete_Sup_Sup_int
= ( ^ [X8: set_int] :
( the_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X8 )
& ! [Y3: int] :
( ( member_int @ Y3 @ X8 )
=> ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_9830_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 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( 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 @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
@ ( 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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
=> ( ( ord_less_nat @ Mi3 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_9831_uniformity__real__def,axiom,
( topolo1511823702728130853y_real
= ( comple2936214249959783750l_real
@ ( image_2178119161166701260l_real
@ ^ [E3: real] :
( princi6114159922880469582l_real
@ ( collec3799799289383736868l_real
@ ( produc5414030515140494994real_o
@ ^ [X3: real,Y3: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y3 ) @ E3 ) ) ) )
@ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% uniformity_real_def
thf(fact_9832_uniformity__complex__def,axiom,
( topolo896644834953643431omplex
= ( comple8358262395181532106omplex
@ ( image_5971271580939081552omplex
@ ^ [E3: real] :
( princi3496590319149328850omplex
@ ( collec8663557070575231912omplex
@ ( produc6771430404735790350plex_o
@ ^ [X3: complex,Y3: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y3 ) @ E3 ) ) ) )
@ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% uniformity_complex_def
thf(fact_9833_isCont__Lb__Ub,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
=> ? [L6: real,M8: real] :
( ! [X5: real] :
( ( ( ord_less_eq_real @ A3 @ X5 )
& ( ord_less_eq_real @ X5 @ B3 ) )
=> ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
& ( ord_less_eq_real @ ( F @ X5 ) @ M8 ) ) )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ L6 @ Y5 )
& ( ord_less_eq_real @ Y5 @ M8 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ( F @ X4 )
= Y5 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_9834_isCont__inverse__function2,axiom,
! [A3: real,X: real,B3: real,G: real > real,F: real > real] :
( ( ord_less_real @ A3 @ X )
=> ( ( ord_less_real @ X @ B3 )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A3 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B3 )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A3 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B3 )
=> ( 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_9835_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% isCont_arcosh
thf(fact_9836_DERIV__inverse__function,axiom,
! [F: real > real,D6: real,G: real > real,X: real,A3: real,B3: real] :
( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
=> ( ( D6 != zero_zero_real )
=> ( ( ord_less_real @ A3 @ X )
=> ( ( ord_less_real @ X @ B3 )
=> ( ! [Y4: real] :
( ( ord_less_real @ A3 @ Y4 )
=> ( ( ord_less_real @ Y4 @ B3 )
=> ( ( F @ ( G @ Y4 ) )
= Y4 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_9837_isCont__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_9838_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_9839_LIM__less__bound,axiom,
! [B3: real,X: real,F: real > real] :
( ( ord_less_real @ B3 @ X )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B3 @ X ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_9840_isCont__artanh,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% isCont_artanh
thf(fact_9841_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_9842_GMVT_H,axiom,
! [A3: real,B3: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A3 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B3 )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A3 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B3 )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A3 @ Z3 )
=> ( ( ord_less_real @ Z3 @ B3 )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A3 @ Z3 )
=> ( ( ord_less_real @ Z3 @ B3 )
=> ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ? [C3: real] :
( ( ord_less_real @ A3 @ C3 )
& ( ord_less_real @ C3 @ B3 )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) ) @ ( G2 @ C3 ) )
= ( times_times_real @ ( minus_minus_real @ ( G @ B3 ) @ ( G @ A3 ) ) @ ( F5 @ C3 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_9843_bdd__above__nat,axiom,
condit2214826472909112428ve_nat = finite_finite_nat ).
% bdd_above_nat
thf(fact_9844_GMVT,axiom,
! [A3: real,B3: real,F: real > real,G: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A3 @ X4 )
& ( ord_less_real @ X4 @ B3 ) )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A3 @ X4 )
& ( ord_less_real @ X4 @ B3 ) )
=> ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ? [G_c: real,F_c: real,C3: real] :
( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
& ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
& ( ord_less_real @ A3 @ C3 )
& ( ord_less_real @ C3 @ B3 )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) ) @ G_c )
= ( times_times_real @ ( minus_minus_real @ ( G @ B3 ) @ ( G @ A3 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_9845_MVT,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less_real @ A3 @ Z3 )
& ( ord_less_real @ Z3 @ B3 )
& ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
& ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) )
= ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_9846_continuous__on__arcosh_H,axiom,
! [A4: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A4 @ F )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( topolo5044208981011980120l_real @ A4
@ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_9847_eventually__prod__sequentially,axiom,
! [P: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N5: nat] :
! [M6: nat] :
( ( ord_less_eq_nat @ N5 @ M6 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ( P @ ( product_Pair_nat_nat @ N3 @ M6 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_9848_continuous__image__closed__interval,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ? [C3: real,D4: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A3 @ B3 ) )
= ( set_or1222579329274155063t_real @ C3 @ D4 ) )
& ( ord_less_eq_real @ C3 @ D4 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_9849_continuous__on__arcosh,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( set_ord_atLeast_real @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A4 @ arcosh_real ) ) ).
% continuous_on_arcosh
thf(fact_9850_Rolle__deriv,axiom,
! [A3: real,B3: real,F: real > real,F5: real > real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ A3 )
= ( F @ B3 ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A3 @ Z3 )
& ( ord_less_real @ Z3 @ B3 )
& ( ( F5 @ Z3 )
= ( ^ [V3: real] : zero_zero_real ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_9851_mvt,axiom,
! [A3: real,B3: real,F: real > real,F5: real > real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less_real @ A3 @ Xi )
=> ( ( ord_less_real @ Xi @ B3 )
=> ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) )
!= ( F5 @ Xi @ ( minus_minus_real @ B3 @ A3 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_9852_DERIV__pos__imp__increasing__open,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ord_less_real @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_9853_DERIV__neg__imp__decreasing__open,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ord_less_real @ ( F @ B3 ) @ ( F @ A3 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_9854_DERIV__isconst__end,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ( ( F @ B3 )
= ( F @ A3 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_9855_continuous__on__artanh,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A4 @ artanh_real ) ) ).
% continuous_on_artanh
thf(fact_9856_DERIV__isconst2,axiom,
! [A3: real,B3: real,F: real > real,X: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ( ( ord_less_eq_real @ A3 @ X )
=> ( ( ord_less_eq_real @ X @ B3 )
=> ( ( F @ X )
= ( F @ A3 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_9857_Rolle,axiom,
! [A3: real,B3: real,F: real > real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ A3 )
= ( F @ B3 ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A3 @ X4 )
=> ( ( ord_less_real @ X4 @ B3 )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A3 @ Z3 )
& ( ord_less_real @ Z3 @ B3 )
& ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% Rolle
thf(fact_9858_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_9859_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_9860_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9861_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_9862_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_9863_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_9864_incseq__bounded,axiom,
! [X7: nat > real,B5: real] :
( ( order_mono_nat_real @ X7 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ ( X7 @ I2 ) @ B5 )
=> ( bfun_nat_real @ X7 @ at_top_nat ) ) ) ).
% incseq_bounded
thf(fact_9865_incseq__convergent,axiom,
! [X7: nat > real,B5: real] :
( ( order_mono_nat_real @ X7 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ ( X7 @ I2 ) @ B5 )
=> ~ ! [L6: real] :
( ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
=> ~ ! [I4: nat] : ( ord_less_eq_real @ ( X7 @ I4 ) @ L6 ) ) ) ) ).
% incseq_convergent
thf(fact_9866_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_9867_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M6: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_9868_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_9869_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F: nat > real,G: nat > nat] :
( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ( order_mono_nat_real @ F )
=> ( ( order_5726023648592871131at_nat @ G )
=> ( ( bfun_nat_real
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ at_top_nat )
= ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
thf(fact_9870_infinite__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ? [R3: nat > nat] :
( ( order_5726023648592871131at_nat @ R3 )
& ! [N7: nat] : ( member_nat @ ( R3 @ N7 ) @ S2 ) ) ) ).
% infinite_enumerate
thf(fact_9871_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_9872_pos__deriv__imp__strict__mono,axiom,
! [F: real > real,F5: real > real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X4 ) )
=> ( order_7092887310737990675l_real @ F ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_9873_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_real_real
@ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_9874_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_9875_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_9876_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_9877_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_9878_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_9879_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_9880_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_9881_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q4 ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_9882_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q4 )
= ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_9883_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N @ M4 ) )
@ M ) ) ).
% min_Suc1
thf(fact_9884_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_9885_log__inj,axiom,
! [B3: real] :
( ( ord_less_real @ one_one_real @ B3 )
=> ( inj_on_real_real @ ( log @ B3 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% log_inj
thf(fact_9886_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_min_Code_integer @ K @ L ) )
= ( ord_min_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_min
thf(fact_9887_inj__on__set__encode,axiom,
inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% inj_on_set_encode
thf(fact_9888_inj__Suc,axiom,
! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_9889_inj__on__diff__nat,axiom,
! [N6: set_nat,K: nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ K @ N2 ) )
=> ( inj_on_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_9890_summable__reindex,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% summable_reindex
thf(fact_9891_suminf__reindex__mono,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_9892_suminf__reindex,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ! [X4: nat] :
( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( ( F @ X4 )
= zero_zero_real ) )
=> ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
= ( suminf_real @ F ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_9893_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_9894_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_9895_Rats__eq__int__div__nat,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu3: real] :
? [I3: int,N3: nat] :
( ( Uu3
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( N3 != zero_zero_nat ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_9896_Rats__no__top__le,axiom,
! [X: real] :
? [X4: real] :
( ( member_real @ X4 @ field_5140801741446780682s_real )
& ( ord_less_eq_real @ X @ X4 ) ) ).
% Rats_no_top_le
thf(fact_9897_Rats__no__bot__less,axiom,
! [X: real] :
? [X4: real] :
( ( member_real @ X4 @ field_5140801741446780682s_real )
& ( ord_less_real @ X4 @ X ) ) ).
% Rats_no_bot_less
thf(fact_9898_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [X4: real] :
( ( member_real @ X4 @ field_5140801741446780682s_real )
& ( ord_less_real @ X @ X4 )
& ( ord_less_real @ X4 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_9899_positive__rat,axiom,
! [A3: int,B3: int] :
( ( positive @ ( fract @ A3 @ B3 ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ).
% positive_rat
thf(fact_9900_less__rat__def,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y3: rat] : ( positive @ ( minus_minus_rat @ Y3 @ X3 ) ) ) ) ).
% less_rat_def
thf(fact_9901_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_9902_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_9903_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_9904_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_9905_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_9906_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_9907_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_9908_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_9909_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q4 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_9910_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N3: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M6 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_9911_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_9912_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N3: nat,M6: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M6 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_9913_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N3: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M6 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_9914_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_9915_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_9916_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_9917_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_9918_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_9919_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_9920_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_9921_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_9922_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_9923_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_9924_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_9925_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_9926_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_9927_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_9928_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_9929_bij__betw__Suc,axiom,
! [M7: set_nat,N6: set_nat] :
( ( bij_betw_nat_nat @ suc @ M7 @ N6 )
= ( ( image_nat_nat @ suc @ M7 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_9930_Arg__def,axiom,
( arg
= ( ^ [Z2: complex] :
( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
@ ( fChoice_real
@ ^ [A: real] :
( ( ( sgn_sgn_complex @ Z2 )
= ( cis @ A ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A )
& ( ord_less_eq_real @ A @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_9931_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_9932_card__length__sum__list__rec,axiom,
! [M: nat,N6: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( groups4561878855575611511st_nat @ L3 )
= N6 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L3 )
= N6 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_9933_card__length__sum__list,axiom,
! [M: nat,N6: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( groups4561878855575611511st_nat @ L3 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M ) @ one_one_nat ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_9934_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_9935_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_9936_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_9937_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_9938_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_9939_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
= ( divide6298287555418463151nteger @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_9940_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
= ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_9941_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_9942_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( 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_9943_bezw_Osimps,axiom,
( bezw
= ( ^ [X3: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_9944_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_9945_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( 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_9946_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
= ( modulo364778990260209775nteger @ K @ L ) ) ).
% snd_divmod_integer
thf(fact_9947_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L ) )
= ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).
% snd_divmod_abs
thf(fact_9948_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_9949_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_9950_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_9951_gcd__pos__int,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
= ( ( M != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
% gcd_pos_int
thf(fact_9952_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_ge_0_int
thf(fact_9953_gcd__le2__int,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A3 @ B3 ) @ B3 ) ) ).
% gcd_le2_int
thf(fact_9954_gcd__le1__int,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A3 @ B3 ) @ A3 ) ) ).
% gcd_le1_int
thf(fact_9955_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_9956_gcd__unique__int,axiom,
! [D: int,A3: int,B3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ D @ A3 )
& ( dvd_dvd_int @ D @ B3 )
& ! [E3: int] :
( ( ( dvd_dvd_int @ E3 @ A3 )
& ( dvd_dvd_int @ E3 @ B3 ) )
=> ( dvd_dvd_int @ E3 @ D ) ) )
= ( D
= ( gcd_gcd_int @ A3 @ B3 ) ) ) ).
% gcd_unique_int
thf(fact_9957_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_9958_gcd__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( gcd_gcd_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
& ( B3 = zero_zero_nat ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_9959_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% gcd_nat.left_neutral
thf(fact_9960_gcd__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B3: nat] :
( ( zero_zero_nat
= ( gcd_gcd_nat @ A3 @ B3 ) )
= ( ( A3 = zero_zero_nat )
& ( B3 = zero_zero_nat ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_9961_gcd__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% gcd_nat.right_neutral
thf(fact_9962_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ zero_zero_nat )
= X ) ).
% gcd_0_nat
thf(fact_9963_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_9964_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
% gcd_1_nat
thf(fact_9965_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_9966_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_9967_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa2 )
= Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_9968_gcd__nat_Osimps,axiom,
( gcd_gcd_nat
= ( ^ [X3: nat,Y3: nat] : ( if_nat @ ( Y3 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_9969_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y != zero_zero_nat )
=> ( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_9970_Gcd__in,axiom,
! [A4: set_nat] :
( ! [A2: nat,B2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ A4 )
=> ( member_nat @ ( gcd_gcd_nat @ A2 @ B2 ) @ A4 ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( member_nat @ ( gcd_Gcd_nat @ A4 ) @ A4 ) ) ) ).
% Gcd_in
thf(fact_9971_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_9972_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_9973_gcd__le1__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A3 ) ) ).
% gcd_le1_nat
thf(fact_9974_gcd__le2__nat,axiom,
! [B3: nat,A3: nat] :
( ( B3 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ B3 ) ) ).
% gcd_le2_nat
thf(fact_9975_bezout__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ? [X4: nat,Y4: nat] :
( ( times_times_nat @ A3 @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).
% bezout_nat
thf(fact_9976_bezout__gcd__nat_H,axiom,
! [B3: nat,A3: nat] :
? [X4: nat,Y4: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( times_times_nat @ A3 @ X4 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A3 @ X4 ) @ ( times_times_nat @ B3 @ Y4 ) )
= ( gcd_gcd_nat @ A3 @ B3 ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ Y4 ) @ ( times_times_nat @ B3 @ X4 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B3 @ X4 ) @ ( times_times_nat @ A3 @ Y4 ) )
= ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_9977_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
@ ^ [M6: nat,N3: nat] :
( ( dvd_dvd_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9978_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
@ ^ [D5: nat] :
( ( dvd_dvd_nat @ D5 @ M )
& ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_9979_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( 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_9980_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y3: nat,Z2: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y3 @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9981_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X3: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y3: nat,Z2: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y3 @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9982_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
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_9983_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
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
@ Xa2
@ X ) ) ).
% less_int.abs_eq
thf(fact_9984_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9985_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9986_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9987_Gcd__nat__set__eq__fold,axiom,
! [Xs: list_nat] :
( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
= ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_9988_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) )
= ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_9989_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_9990_wf__less,axiom,
wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% wf_less
thf(fact_9991_cauchy__def,axiom,
( cauchy
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [M6: nat] :
( ( ord_less_eq_nat @ K3 @ M6 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M6 ) @ ( X8 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_9992_cauchyI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K7: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ K7 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ K7 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X7 ) ) ).
% cauchyI
thf(fact_9993_cauchy__imp__bounded,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ? [B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ B2 )
& ! [N7: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N7 ) ) @ B2 ) ) ) ).
% cauchy_imp_bounded
thf(fact_9994_cauchyD,axiom,
! [X7: nat > rat,R2: rat] :
( ( cauchy @ X7 )
=> ( ( 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 @ ( X7 @ M2 ) @ ( X7 @ N7 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_9995_le__Real,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( plus_plus_rat @ ( Y7 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_9996_cauchy__not__vanishes,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [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 @ ( X7 @ N7 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_9997_vanishes__mult__bounded,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ? [A8: rat] :
( ( ord_less_rat @ zero_zero_rat @ A8 )
& ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N2 ) ) @ A8 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N3: nat] : ( times_times_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_9998_vanishes__def,axiom,
( vanishes
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_9999_vanishesI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K7: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K7 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N2 ) ) @ R3 ) ) )
=> ( vanishes @ X7 ) ) ).
% vanishesI
thf(fact_10000_vanishesD,axiom,
! [X7: nat > rat,R2: rat] :
( ( vanishes @ X7 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N7 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_10001_cauchy__not__vanishes__cases,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [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 @ ( X7 @ N7 ) ) ) )
| ! [N7: nat] :
( ( ord_less_eq_nat @ K2 @ N7 )
=> ( ord_less_rat @ B2 @ ( X7 @ N7 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_10002_not__positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( ~ ( positive2 @ ( real2 @ X7 ) ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_eq_rat @ ( X7 @ N3 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_10003_positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( positive2 @ ( real2 @ X7 ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ R5 @ ( X7 @ N3 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_10004_less__real__def,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] : ( positive2 @ ( minus_minus_real @ Y3 @ X3 ) ) ) ) ).
% less_real_def
thf(fact_10005_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X3: real] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N3 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_10006_le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_10007_enumerate__Ex,axiom,
! [S2: set_nat,S3: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( member_nat @ S3 @ S2 )
=> ? [N2: nat] :
( ( infini8530281810654367211te_nat @ S2 @ N2 )
= S3 ) ) ) ).
% enumerate_Ex
thf(fact_10008_strict__mono__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S2 ) ) ) ).
% strict_mono_enumerate
thf(fact_10009_range__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat )
= S2 ) ) ).
% range_enumerate
thf(fact_10010_finite__le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_10011_bij__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat @ S2 ) ) ).
% bij_enumerate
thf(fact_10012_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_10013_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_10014_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_10015_Sup__real__def,axiom,
( comple1385675409528146559p_real
= ( ^ [X8: set_real] :
( ord_Least_real
@ ^ [Z2: real] :
! [X3: real] :
( ( member_real @ X3 @ X8 )
=> ( ord_less_eq_real @ X3 @ Z2 ) ) ) ) ) ).
% Sup_real_def
thf(fact_10016_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P5: rat,Q5: rat] :
( produc4947309494688390418_int_o
@ ^ [A: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A @ D5 ) @ ( times_times_int @ C4 @ B ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_10017_quotient__of__denom__pos,axiom,
! [R2: rat,P6: int,Q4: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ P6 @ Q4 ) )
=> ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_10018_quotient__of__denom__pos_H,axiom,
! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_10019_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P5: rat,Q5: rat] :
( produc4947309494688390418_int_o
@ ^ [A: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B: int,D5: int] : ( ord_less_int @ ( times_times_int @ A @ D5 ) @ ( times_times_int @ C4 @ B ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_10020_quotient__of__def,axiom,
( quotient_of
= ( ^ [X3: rat] :
( the_Pr4378521158711661632nt_int
@ ^ [Pair: product_prod_int_int] :
( ( X3
= ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_10021_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_10022_coprime__abs__left__iff,axiom,
! [K: int,L: int] :
( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
= ( algebr932160517623751201me_int @ K @ L ) ) ).
% coprime_abs_left_iff
thf(fact_10023_coprime__abs__right__iff,axiom,
! [K: int,L: int] :
( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
= ( algebr932160517623751201me_int @ K @ L ) ) ).
% coprime_abs_right_iff
thf(fact_10024_normalize__stable,axiom,
! [Q4: int,P6: int] :
( ( ord_less_int @ zero_zero_int @ Q4 )
=> ( ( algebr932160517623751201me_int @ P6 @ Q4 )
=> ( ( normalize @ ( product_Pair_int_int @ P6 @ Q4 ) )
= ( product_Pair_int_int @ P6 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_10025_coprime__common__divisor__int,axiom,
! [A3: int,B3: int,X: int] :
( ( algebr932160517623751201me_int @ A3 @ B3 )
=> ( ( dvd_dvd_int @ X @ A3 )
=> ( ( dvd_dvd_int @ X @ B3 )
=> ( ( abs_abs_int @ X )
= one_one_int ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_10026_Rat__cases,axiom,
! [Q4: rat] :
~ ! [A2: int,B2: int] :
( ( Q4
= ( fract @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ~ ( algebr932160517623751201me_int @ A2 @ B2 ) ) ) ).
% Rat_cases
thf(fact_10027_Rat__induct,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( algebr932160517623751201me_int @ A2 @ B2 )
=> ( P @ ( fract @ A2 @ B2 ) ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct
thf(fact_10028_Rat__cases__nonzero,axiom,
! [Q4: rat] :
( ! [A2: int,B2: int] :
( ( Q4
= ( fract @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( A2 != zero_zero_int )
=> ~ ( algebr932160517623751201me_int @ A2 @ B2 ) ) ) )
=> ( Q4 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
thf(fact_10029_quotient__of__unique,axiom,
! [R2: rat] :
? [X4: product_prod_int_int] :
( ( R2
= ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
& ! [Y5: product_prod_int_int] :
( ( ( R2
= ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
=> ( Y5 = X4 ) ) ) ).
% quotient_of_unique
thf(fact_10030_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( algebr934650988132801477me_nat @ M @ N ) ) ).
% coprime_int_iff
thf(fact_10031_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
= ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_10032_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_10033_coprime__common__divisor__nat,axiom,
! [A3: nat,B3: nat,X: nat] :
( ( algebr934650988132801477me_nat @ A3 @ B3 )
=> ( ( dvd_dvd_nat @ X @ A3 )
=> ( ( dvd_dvd_nat @ X @ B3 )
=> ( X = one_one_nat ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_10034_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_10035_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_10036_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% coprime_Suc_0_right
thf(fact_10037_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% coprime_Suc_0_left
thf(fact_10038_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_10039_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_10040_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member_real @ X @ field_5140801741446780682s_real )
=> ~ ! [M3: nat,N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( ( abs_abs_real @ X )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
=> ~ ( algebr934650988132801477me_nat @ M3 @ N2 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_10041_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_10042_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_10043_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_10044_not__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_ri7632146776885996613nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_ri7919022796975470100ot_int @ X ) ) ) ).
% not_integer.abs_eq
thf(fact_10045_not__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( bit_ri7632146776885996613nteger @ X ) )
= ( bit_ri7919022796975470100ot_int @ ( code_int_of_integer @ X ) ) ) ).
% not_integer.rep_eq
thf(fact_10046_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] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ R5 @ ( X @ N3 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_10047_Real_Opositive_Orsp,axiom,
( bNF_re728719798268516973at_o_o @ realrel
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_10048_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_10049_sqr_Osimps_I1_J,axiom,
( ( sqr @ one )
= one ) ).
% sqr.simps(1)
thf(fact_10050_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_10051_sqr__conv__mult,axiom,
( sqr
= ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).
% sqr_conv_mult
thf(fact_10052_less__eq__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ord_less_eq_nat
@ ord_less_eq_nat ) ).
% less_eq_natural.rsp
thf(fact_10053_less__eq__integer_Orsp,axiom,
( bNF_re3403563459893282935_int_o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re5089333283451836215nt_o_o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ord_less_eq_int
@ ord_less_eq_int ) ).
% less_eq_integer.rsp
thf(fact_10054_num__of__integer_Orsp,axiom,
( bNF_re7626690874201225453um_num
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ( comp_nat_num_int @ num_of_nat @ nat2 )
@ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_10055_sgn__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ sgn_sgn_int
@ sgn_sgn_int ) ).
% sgn_integer.rsp
thf(fact_10056_less__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ord_less_nat
@ ord_less_nat ) ).
% less_natural.rsp
thf(fact_10057_less__integer_Orsp,axiom,
( bNF_re3403563459893282935_int_o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re5089333283451836215nt_o_o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ord_less_int
@ ord_less_int ) ).
% less_integer.rsp
thf(fact_10058_integer__of__natural_Orsp,axiom,
( bNF_re6650684261131312217nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ semiri1314217659103216013at_int
@ semiri1314217659103216013at_int ) ).
% integer_of_natural.rsp
thf(fact_10059_drop__bit__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se8570568707652914677it_nat
@ bit_se8570568707652914677it_nat ) ).
% drop_bit_natural.rsp
thf(fact_10060_drop__bit__integer_Orsp,axiom,
( bNF_re4785983289428654063nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se8568078237143864401it_int
@ bit_se8568078237143864401it_int ) ).
% drop_bit_integer.rsp
thf(fact_10061_dup_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_10062_and__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se725231765392027082nd_int
@ bit_se725231765392027082nd_int ) ).
% and_integer.rsp
thf(fact_10063_and__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se727722235901077358nd_nat
@ bit_se727722235901077358nd_nat ) ).
% and_natural.rsp
thf(fact_10064_push__bit__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se547839408752420682it_nat
@ bit_se547839408752420682it_nat ) ).
% push_bit_natural.rsp
thf(fact_10065_push__bit__integer_Orsp,axiom,
( bNF_re4785983289428654063nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se545348938243370406it_int
@ bit_se545348938243370406it_int ) ).
% push_bit_integer.rsp
thf(fact_10066_abs__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ abs_abs_int
@ abs_abs_int ) ).
% abs_integer.rsp
thf(fact_10067_flip__bit__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se2161824704523386999it_nat
@ bit_se2161824704523386999it_nat ) ).
% flip_bit_natural.rsp
thf(fact_10068_flip__bit__integer_Orsp,axiom,
( bNF_re4785983289428654063nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se2159334234014336723it_int
@ bit_se2159334234014336723it_int ) ).
% flip_bit_integer.rsp
thf(fact_10069_set__bit__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se7882103937844011126it_nat
@ bit_se7882103937844011126it_nat ) ).
% set_bit_natural.rsp
thf(fact_10070_set__bit__integer_Orsp,axiom,
( bNF_re4785983289428654063nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se7879613467334960850it_int
@ bit_se7879613467334960850it_int ) ).
% set_bit_integer.rsp
thf(fact_10071_plus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ plus_plus_nat
@ plus_plus_nat ) ).
% plus_natural.rsp
thf(fact_10072_plus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ plus_plus_int
@ plus_plus_int ) ).
% plus_integer.rsp
thf(fact_10073_uminus__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ uminus_uminus_int
@ uminus_uminus_int ) ).
% uminus_integer.rsp
thf(fact_10074_bit__integer_Orsp,axiom,
( bNF_re3376528473927230327_nat_o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ bit_se1146084159140164899it_int
@ bit_se1146084159140164899it_int ) ).
% bit_integer.rsp
thf(fact_10075_bit__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ bit_se1148574629649215175it_nat
@ bit_se1148574629649215175it_nat ) ).
% bit_natural.rsp
thf(fact_10076_xor__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se6526347334894502574or_int
@ bit_se6526347334894502574or_int ) ).
% xor_integer.rsp
thf(fact_10077_xor__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se6528837805403552850or_nat
@ bit_se6528837805403552850or_nat ) ).
% xor_natural.rsp
thf(fact_10078_division__segment__natural_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ euclid3398187327856392827nt_nat
@ euclid3398187327856392827nt_nat ) ).
% division_segment_natural.rsp
thf(fact_10079_division__segment__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ euclid3395696857347342551nt_int
@ euclid3395696857347342551nt_int ) ).
% division_segment_integer.rsp
thf(fact_10080_take__bit__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se2925701944663578781it_nat
@ bit_se2925701944663578781it_nat ) ).
% take_bit_natural.rsp
thf(fact_10081_take__bit__integer_Orsp,axiom,
( bNF_re4785983289428654063nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se2923211474154528505it_int
@ bit_se2923211474154528505it_int ) ).
% take_bit_integer.rsp
thf(fact_10082_unset__bit__integer_Orsp,axiom,
( bNF_re4785983289428654063nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se4203085406695923979it_int
@ bit_se4203085406695923979it_int ) ).
% unset_bit_integer.rsp
thf(fact_10083_Suc_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_10084_divide__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ divide_divide_int
@ divide_divide_int ) ).
% divide_integer.rsp
thf(fact_10085_divide__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ divide_divide_nat
@ divide_divide_nat ) ).
% divide_natural.rsp
thf(fact_10086_modulo__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ modulo_modulo_nat
@ modulo_modulo_nat ) ).
% modulo_natural.rsp
thf(fact_10087_modulo__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ modulo_modulo_int
@ modulo_modulo_int ) ).
% modulo_integer.rsp
thf(fact_10088_natural__of__integer_Orsp,axiom,
( bNF_re3715656647883201625at_nat
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_10089_mask__natural_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ bit_se2002935070580805687sk_nat
@ bit_se2002935070580805687sk_nat ) ).
% mask_natural.rsp
thf(fact_10090_mask__integer_Orsp,axiom,
( bNF_re6650684261131312217nt_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ bit_se2000444600071755411sk_int
@ bit_se2000444600071755411sk_int ) ).
% mask_integer.rsp
thf(fact_10091_euclidean__size__integer_Orsp,axiom,
( bNF_re3715656647883201625at_nat
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ euclid4774559944035922753ze_int
@ euclid4774559944035922753ze_int ) ).
% euclidean_size_integer.rsp
thf(fact_10092_or__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ bit_se1409905431419307370or_int
@ bit_se1409905431419307370or_int ) ).
% or_integer.rsp
thf(fact_10093_or__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se1412395901928357646or_nat
@ bit_se1412395901928357646or_nat ) ).
% or_natural.rsp
thf(fact_10094_minus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ minus_minus_int
@ minus_minus_int ) ).
% minus_integer.rsp
thf(fact_10095_minus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ minus_minus_nat
@ minus_minus_nat ) ).
% minus_natural.rsp
thf(fact_10096_sub_Orsp,axiom,
( bNF_re8402795839162346335um_int
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ( bNF_re1822329894187522285nt_int
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ^ [M6: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) )
@ ^ [M6: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ).
% sub.rsp
thf(fact_10097_times__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ times_times_nat
@ times_times_nat ) ).
% times_natural.rsp
thf(fact_10098_times__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ times_times_int
@ times_times_int ) ).
% times_integer.rsp
thf(fact_10099_not__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ bit_ri7919022796975470100ot_int
@ bit_ri7919022796975470100ot_int ) ).
% not_integer.rsp
thf(fact_10100_Real_Opositive_Otransfer,axiom,
( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_10101_euclidean__size__natural_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ euclid4777050414544973029ze_nat
@ euclid4777050414544973029ze_nat ) ).
% euclidean_size_natural.rsp
thf(fact_10102_unset__bit__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ bit_se4205575877204974255it_nat
@ bit_se4205575877204974255it_nat ) ).
% unset_bit_natural.rsp
thf(fact_10103_Rat_Opositive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_10104_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit1 @ Y ) )
= ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_10105_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_10106_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X @ Y ) ) ) ).
% pow.simps(2)
thf(fact_10107_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_10108_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% zero_int.transfer
thf(fact_10109_int__transfer,axiom,
( bNF_re6830278522597306478at_int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ pcr_int
@ ^ [N3: nat] : ( product_Pair_nat_nat @ N3 @ zero_zero_nat )
@ semiri1314217659103216013at_int ) ).
% int_transfer
thf(fact_10110_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% one_int.transfer
thf(fact_10111_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_10112_Rat_Opositive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_10113_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_10114_vimage__Suc__insert__0,axiom,
! [A4: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A4 ) )
= ( vimage_nat_nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_10115_vimage__Suc__insert__Suc,axiom,
! [N: nat,A4: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A4 ) )
= ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A4 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_10116_finite__vimage__Suc__iff,axiom,
! [F4: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
= ( finite_finite_nat @ F4 ) ) ).
% finite_vimage_Suc_iff
thf(fact_10117_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_We3818239936649020644el_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_10118_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_10119_set__encode__vimage__Suc,axiom,
! [A4: set_nat] :
( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A4 ) )
= ( divide_divide_nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_10120_pairs__le__eq__Sigma,axiom,
! [M: nat] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ M ) ) )
= ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
@ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_10121_Bseq__monoseq__convergent_H__inc,axiom,
! [F: nat > real,M7: nat] :
( ( bfun_nat_real
@ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M7 ) )
@ at_top_nat )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M7 @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_real @ ( F @ M3 ) @ ( F @ N2 ) ) ) )
=> ( topolo7531315842566124627t_real @ F ) ) ) ).
% Bseq_monoseq_convergent'_inc
thf(fact_10122_Bseq__mono__convergent,axiom,
! [X7: nat > real] :
( ( bfun_nat_real @ X7 @ at_top_nat )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
=> ( topolo7531315842566124627t_real @ X7 ) ) ) ).
% Bseq_mono_convergent
thf(fact_10123_convergent__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% convergent_realpow
thf(fact_10124_Bseq__monoseq__convergent_H__dec,axiom,
! [F: nat > real,M7: nat] :
( ( bfun_nat_real
@ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M7 ) )
@ at_top_nat )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M7 @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M3 ) ) ) )
=> ( topolo7531315842566124627t_real @ F ) ) ) ).
% Bseq_monoseq_convergent'_dec
thf(fact_10125_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
@ ^ [Uu3: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_10126_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% natLeq_def
thf(fact_10127_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_10128_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_10129_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top_nat @ dvd_dvd_nat
@ ^ [M6: nat,N3: nat] :
( ( dvd_dvd_nat @ M6 @ N3 )
& ( M6 != N3 ) )
@ zero_zero_nat ) ).
% gcd_nat.ordering_top_axioms
thf(fact_10130_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_10131_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_10132_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_10133_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_10134_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).
% less_int.rsp
thf(fact_10135_less__eq__enat__def,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [M6: extended_enat] :
( extended_case_enat_o
@ ^ [N1: nat] :
( extended_case_enat_o
@ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
@ $false
@ M6 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_10136_less__enat__def,axiom,
( ord_le72135733267957522d_enat
= ( ^ [M6: extended_enat,N3: extended_enat] :
( extended_case_enat_o
@ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N3 )
@ $false
@ M6 ) ) ) ).
% less_enat_def
thf(fact_10137_division__segment__integer__def,axiom,
( euclid6289375185220004616nteger
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ euclid3395696857347342551nt_int ) ) ).
% division_segment_integer_def
thf(fact_10138_uminus__integer__def,axiom,
( uminus1351360451143612070nteger
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ uminus_uminus_int ) ) ).
% uminus_integer_def
thf(fact_10139_abs__integer__def,axiom,
( abs_abs_Code_integer
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ abs_abs_int ) ) ).
% abs_integer_def
thf(fact_10140_sgn__integer__def,axiom,
( sgn_sgn_Code_integer
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ sgn_sgn_int ) ) ).
% sgn_integer_def
thf(fact_10141_not__integer__def,axiom,
( bit_ri7632146776885996613nteger
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ bit_ri7919022796975470100ot_int ) ) ).
% not_integer_def
thf(fact_10142_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_10143_xor__integer__def,axiom,
( bit_se3222712562003087583nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se6526347334894502574or_int ) ) ).
% xor_integer_def
thf(fact_10144_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% Code_Numeral.dup_code(1)
thf(fact_10145_dup_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( code_dup @ X ) )
= ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% dup.rep_eq
thf(fact_10146_dup_Oabs__eq,axiom,
! [X: int] :
( ( code_dup @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).
% dup.abs_eq
thf(fact_10147_plus__integer__def,axiom,
( plus_p5714425477246183910nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).
% plus_integer_def
thf(fact_10148_times__integer__def,axiom,
( times_3573771949741848930nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).
% times_integer_def
thf(fact_10149_minus__integer__def,axiom,
( minus_8373710615458151222nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ minus_minus_int ) ) ).
% minus_integer_def
thf(fact_10150_divide__integer__def,axiom,
( divide6298287555418463151nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ divide_divide_int ) ) ).
% divide_integer_def
thf(fact_10151_modulo__integer__def,axiom,
( modulo364778990260209775nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ modulo_modulo_int ) ) ).
% modulo_integer_def
thf(fact_10152_and__integer__def,axiom,
( bit_se3949692690581998587nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se725231765392027082nd_int ) ) ).
% and_integer_def
thf(fact_10153_or__integer__def,axiom,
( bit_se1080825931792720795nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se1409905431419307370or_int ) ) ).
% or_integer_def
thf(fact_10154_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_10155_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_10156_Code__Numeral_Osub__code_I7_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(7)
thf(fact_10157_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(6)
thf(fact_10158_Code__Numeral_Osub__code_I1_J,axiom,
( ( code_sub @ one @ one )
= zero_z3403309356797280102nteger ) ).
% Code_Numeral.sub_code(1)
thf(fact_10159_sub_Orep__eq,axiom,
! [X: num,Xa2: num] :
( ( code_int_of_integer @ ( code_sub @ X @ Xa2 ) )
= ( minus_minus_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Xa2 ) ) ) ).
% sub.rep_eq
thf(fact_10160_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus_int @ ( numeral_numeral_int @ Xa4 ) @ ( numeral_numeral_int @ X3 ) ) ) ) ) ).
% sub.abs_eq
thf(fact_10161_less__RealD,axiom,
! [Y7: nat > rat,X: real] :
( ( cauchy @ Y7 )
=> ( ( ord_less_real @ X @ ( real2 @ Y7 ) )
=> ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) ) ) ) ).
% less_RealD
thf(fact_10162_of__rat__dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Q3: rat] :
( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q3 ) )
& ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y ) ) ) ).
% of_rat_dense
thf(fact_10163_le__RealI,axiom,
! [Y7: nat > rat,X: real] :
( ( cauchy @ Y7 )
=> ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq_real @ X @ ( real2 @ Y7 ) ) ) ) ).
% le_RealI
thf(fact_10164_Real__leI,axiom,
! [X7: nat > rat,Y: real] :
( ( cauchy @ X7 )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X7 @ N2 ) ) @ Y )
=> ( ord_less_eq_real @ ( real2 @ X7 ) @ Y ) ) ) ).
% Real_leI
thf(fact_10165_num__of__integer_Otransfer,axiom,
( bNF_re6718328864250387230um_num @ code_pcr_integer
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ( comp_nat_num_int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_10166_not__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ bit_ri7919022796975470100ot_int @ bit_ri7632146776885996613nteger ).
% not_integer.transfer
thf(fact_10167_times__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).
% times_integer.transfer
thf(fact_10168_minus__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ minus_minus_int @ minus_8373710615458151222nteger ).
% minus_integer.transfer
thf(fact_10169_nat__of__integer_Otransfer,axiom,
( bNF_re2807294637932363402at_nat @ code_pcr_integer
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ nat2
@ code_nat_of_integer ) ).
% nat_of_integer.transfer
thf(fact_10170_or__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se1409905431419307370or_int @ bit_se1080825931792720795nteger ).
% or_integer.transfer
thf(fact_10171_euclidean__size__integer_Otransfer,axiom,
( bNF_re2807294637932363402at_nat @ code_pcr_integer
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ euclid4774559944035922753ze_int
@ euclid6377331345833325938nteger ) ).
% euclidean_size_integer.transfer
thf(fact_10172_mask__integer_Otransfer,axiom,
( bNF_re4153400068438556298nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ code_pcr_integer
@ bit_se2000444600071755411sk_int
@ bit_se2119862282449309892nteger ) ).
% mask_integer.transfer
thf(fact_10173_modulo__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ modulo_modulo_int @ modulo364778990260209775nteger ).
% modulo_integer.transfer
thf(fact_10174_divide__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ divide_divide_int @ divide6298287555418463151nteger ).
% divide_integer.transfer
thf(fact_10175_unset__bit__integer_Otransfer,axiom,
( bNF_re4935368626670024657nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
@ bit_se4203085406695923979it_int
@ bit_se8260200283734997820nteger ) ).
% unset_bit_integer.transfer
thf(fact_10176_take__bit__integer_Otransfer,axiom,
( bNF_re4935368626670024657nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
@ bit_se2923211474154528505it_int
@ bit_se1745604003318907178nteger ) ).
% take_bit_integer.transfer
thf(fact_10177_division__segment__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ euclid3395696857347342551nt_int @ euclid6289375185220004616nteger ).
% division_segment_integer.transfer
thf(fact_10178_xor__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se6526347334894502574or_int @ bit_se3222712562003087583nteger ).
% xor_integer.transfer
thf(fact_10179_bit__integer_Otransfer,axiom,
( bNF_re4711666741709854504_nat_o @ code_pcr_integer
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ bit_se1146084159140164899it_int
@ bit_se9216721137139052372nteger ) ).
% bit_integer.transfer
thf(fact_10180_uminus__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ uminus_uminus_int @ uminus1351360451143612070nteger ).
% uminus_integer.transfer
thf(fact_10181_plus__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ plus_plus_int @ plus_p5714425477246183910nteger ).
% plus_integer.transfer
thf(fact_10182_set__bit__integer_Otransfer,axiom,
( bNF_re4935368626670024657nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
@ bit_se7879613467334960850it_int
@ bit_se2793503036327961859nteger ) ).
% set_bit_integer.transfer
thf(fact_10183_flip__bit__integer_Otransfer,axiom,
( bNF_re4935368626670024657nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
@ bit_se2159334234014336723it_int
@ bit_se1345352211410354436nteger ) ).
% flip_bit_integer.transfer
thf(fact_10184_abs__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ abs_abs_int @ abs_abs_Code_integer ).
% abs_integer.transfer
thf(fact_10185_push__bit__integer_Otransfer,axiom,
( bNF_re4935368626670024657nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
@ bit_se545348938243370406it_int
@ bit_se7788150548672797655nteger ) ).
% push_bit_integer.transfer
thf(fact_10186_and__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se725231765392027082nd_int @ bit_se3949692690581998587nteger ).
% and_integer.transfer
thf(fact_10187_integer_Oid__abs__transfer,axiom,
( bNF_re982302072995117890nteger
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ code_pcr_integer
@ ^ [X3: int] : X3
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_10188_drop__bit__integer_Otransfer,axiom,
( bNF_re4935368626670024657nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
@ bit_se8568078237143864401it_int
@ bit_se3928097537394005634nteger ) ).
% drop_bit_integer.transfer
thf(fact_10189_integer_Orep__transfer,axiom,
( bNF_re3804157879324367682nt_int @ code_pcr_integer
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [X3: int] : X3
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_10190_less__integer_Otransfer,axiom,
( bNF_re6321650412969554871eger_o @ code_pcr_integer
@ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ord_less_int
@ ord_le6747313008572928689nteger ) ).
% less_integer.transfer
thf(fact_10191_sgn__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ sgn_sgn_int @ sgn_sgn_Code_integer ).
% sgn_integer.transfer
thf(fact_10192_less__eq__integer_Otransfer,axiom,
( bNF_re6321650412969554871eger_o @ code_pcr_integer
@ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ord_less_eq_int
@ ord_le3102999989581377725nteger ) ).
% less_eq_integer.transfer
thf(fact_10193_integer_Obi__total,axiom,
bi_tot1331153423839324337nteger @ code_pcr_integer ).
% integer.bi_total
thf(fact_10194_one__integer_Otransfer,axiom,
code_pcr_integer @ one_one_int @ one_one_Code_integer ).
% one_integer.transfer
thf(fact_10195_zero__integer_Otransfer,axiom,
code_pcr_integer @ zero_zero_int @ zero_z3403309356797280102nteger ).
% zero_integer.transfer
thf(fact_10196_dup_Otransfer,axiom,
( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
@ code_dup ) ).
% dup.transfer
thf(fact_10197_sub_Otransfer,axiom,
( bNF_re7876454716742015248nteger
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ( bNF_re6501075790457514782nteger
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ code_pcr_integer )
@ ^ [M6: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_10198_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_10199_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_10200_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_10201_char_Osize_I2_J,axiom,
! [X15: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size_char @ ( char2 @ X15 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_10202_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_10203_char_Osize__gen,axiom,
! [X15: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X15 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_10204_integer__of__nat_Otransfer,axiom,
( bNF_re4153400068438556298nteger
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ code_pcr_integer
@ semiri1314217659103216013at_int
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_10205_nat__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
= N ) ).
% nat_of_integer_integer_of_nat
thf(fact_10206_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_10207_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiri1314217659103216013at_int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_10208_integer__of__nat__eq__of__nat,axiom,
code_integer_of_nat = semiri4939895301339042750nteger ).
% integer_of_nat_eq_of_nat
thf(fact_10209_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ zero_zero_nat )
= zero_z3403309356797280102nteger ) ).
% integer_of_nat_0
thf(fact_10210_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X3: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X3 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_10211_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% integer_of_nat_numeral
thf(fact_10212_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ one_one_nat )
= one_one_Code_integer ) ).
% integer_of_nat_1
thf(fact_10213_compute__powr__real,axiom,
( powr_real2
= ( ^ [B: real,I3: real] :
( if_real @ ( ord_less_eq_real @ B @ zero_zero_real )
@ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real2 @ B @ I3 ) )
@ ( if_real
@ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I3 ) )
= I3 )
@ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I3 ) @ ( power_power_real @ B @ ( nat2 @ ( archim6058952711729229775r_real @ I3 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I3 ) ) ) ) ) )
@ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real2 @ B @ I3 ) ) ) ) ) ) ).
% compute_powr_real
% Helper facts (42)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P: real > $o] :
( ( P @ ( fChoice_real @ P ) )
= ( ? [X8: real] : ( P @ X8 ) ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: int > int,Y: int > int] :
( ( if_int_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: int > int,Y: int > int] :
( ( if_int_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] :
( ( if_option_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] :
( ( if_option_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $true @ X @ Y )
= 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,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
= 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,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: code_integer > code_integer,Y: code_integer > code_integer] :
( ( if_Cod4779417660136461971nteger @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: code_integer > code_integer,Y: code_integer > code_integer] :
( ( if_Cod4779417660136461971nteger @ $true @ X @ Y )
= 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,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ ( vEBT_Leaf @ a @ b ) ) @ xa @ one_one_nat ).
%------------------------------------------------------------------------------