TPTP Problem File: SLH0764^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Cotangent_PFD_Formula/0007_Cotangent_PFD_Formula/prob_00531_021226__14112520_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1458 (1015 unt; 176 typ; 0 def)
% Number of atoms : 2274 (1667 equ; 0 cnn)
% Maximal formula atoms : 11 ( 1 avg)
% Number of connectives : 8052 ( 230 ~; 36 |; 82 &;7286 @)
% ( 0 <=>; 418 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 255 ( 255 >; 0 *; 0 +; 0 <<)
% Number of symbols : 161 ( 158 usr; 29 con; 0-3 aty)
% Number of variables : 2369 ( 98 ^;2245 !; 26 ?;2369 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:59:58.905
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__Set__Oset_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J,type,
set_Nu382503245525567899l_num1: $tType ).
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
numera4273646738625120315l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
set_Nu795013586925006960l_num1: $tType ).
thf(ty_n_t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
numera6367994245245682809l_num1: $tType ).
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2417102609627094330l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Num__Onum_J_J,type,
set_set_num: $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__Real__Oreal_J,type,
set_real: $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__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $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 (158)
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Real__Oreal,type,
cotang1502006655779026648d_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
minus_5410813661909488930l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
minus_838314146864362899l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
minus_minus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
one_on7795324986448017462l_num1: numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
one_on7819281148064737470l_num1: numera6367994245245682809l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
plus_p1441664204671982194l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
plus_p2313304076027620419l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Num__Onum_J,type,
plus_plus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
plus_p2327523748178379809et_int: set_set_int > set_set_int > set_set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
plus_p4817606893110106565et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Num__Onum_J_J,type,
plus_p532826482549453327et_num: set_set_num > set_set_num > set_set_num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
plus_p7620395444238123297t_real: set_set_real > set_set_real > set_set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
times_2938166955517408246l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
times_8498157372700349887l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Num__Onum_J,type,
times_times_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
uminus1336558196688952754l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
uminus7224005126491068675l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
zero_z2241845390563828978l_num1: numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
zero_z5982384998485459395l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Int__Oint_J,type,
zero_zero_set_int: set_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
ring_19056730708217498732l_num1: set_Nu382503245525567899l_num1 ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
ring_14745913572136535497l_num1: set_Nu795013586925006960l_num1 ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
neg_nu5816564918971239084l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
neg_nu5590746349488142217l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
neg_numeral_dbl_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
neg_nu7886226890278435366l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
neg_nu228592723992507279l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
neg_nu5172728937851396970l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
neg_nu4048618728508742987l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
neg_numeral_sub_int: num > num > int ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
neg_nu3067386718351260922l_num1: num > num > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
neg_nu3733408198258700219l_num1: num > num > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Real__Oreal,type,
neg_numeral_sub_real: num > num > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
numera7754357348821619680l_num1: num > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2161328050825114965l_num1: num > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
numera6112219686443703444l_num1: num > numera6367994245245682809l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
ring_1_iszero_int: int > $o ).
thf(sy_c_Num_Oring__1__class_Oiszero_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
ring_14756928180729810983l_num1: numera4273646738625120315l_num1 > $o ).
thf(sy_c_Num_Oring__1__class_Oiszero_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
ring_16753260291671729742l_num1: numera2417102609627094330l_num1 > $o ).
thf(sy_c_Num_Oring__1__class_Oiszero_001t__Real__Oreal,type,
ring_1_iszero_real: real > $o ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Num__Onum_M_Eo_J,type,
bot_bot_num_o: num > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
bot_bot_set_num: set_num ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Parity_Oadjust__mod,type,
adjust_mod: num > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
insert_num: num > set_num > set_num ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Int__Oint,type,
set_elt_set_plus_int: int > set_int > set_int ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Nat__Onat,type,
set_elt_set_plus_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Num__Onum,type,
set_elt_set_plus_num: num > set_num > set_num ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Real__Oreal,type,
set_el4531060646461604733s_real: real > set_real > set_real ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Int__Oint_J,type,
set_el7516368150128015667et_int: set_int > set_set_int > set_set_int ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Nat__Onat_J,type,
set_el2470847132782436567et_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Num__Onum_J,type,
set_el3179091069731274273et_num: set_num > set_set_num > set_set_num ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Real__Oreal_J,type,
set_el4273275173557528499t_real: set_real > set_set_real > set_set_real ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Int__Oint,type,
set_el2930815339941905629es_int: int > set_int > set_int ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Nat__Onat,type,
set_el2933305810450955905es_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Num__Onum,type,
set_el8714009633461510347es_num: num > set_num > set_num ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
set_el7870430730610807626l_num1: numera4273646738625120315l_num1 > set_Nu382503245525567899l_num1 > set_Nu382503245525567899l_num1 ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
set_el2854875225590388075l_num1: numera2417102609627094330l_num1 > set_Nu795013586925006960l_num1 > set_Nu795013586925006960l_num1 ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Real__Oreal,type,
set_el1043507519367895261s_real: real > set_real > set_real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
member8231810200707485668l_num1: numera4273646738625120315l_num1 > set_Nu382503245525567899l_num1 > $o ).
thf(sy_c_member_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
member2815666790699981905l_num1: numera2417102609627094330l_num1 > set_Nu795013586925006960l_num1 > $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__Set__Oset_It__Num__Onum_J,type,
member_set_num: set_num > set_set_num > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_v_f____,type,
f: real > real ).
thf(sy_v_g____,type,
g: real > real ).
thf(sy_v_h____,type,
h: real > real ).
thf(sy_v_x,type,
x: real ).
thf(sy_v_xa____,type,
xa: real ).
% Relevant facts (1274)
thf(fact_0_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_2_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_3_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_4_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_5_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_6_set__plus__intro,axiom,
! [A: set_int,C2: set_set_int,B: set_int,D: set_set_int] :
( ( member_set_int @ A @ C2 )
=> ( ( member_set_int @ B @ D )
=> ( member_set_int @ ( plus_plus_set_int @ A @ B ) @ ( plus_p2327523748178379809et_int @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_7_set__plus__intro,axiom,
! [A: set_nat,C2: set_set_nat,B: set_nat,D: set_set_nat] :
( ( member_set_nat @ A @ C2 )
=> ( ( member_set_nat @ B @ D )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_8_set__plus__intro,axiom,
! [A: set_num,C2: set_set_num,B: set_num,D: set_set_num] :
( ( member_set_num @ A @ C2 )
=> ( ( member_set_num @ B @ D )
=> ( member_set_num @ ( plus_plus_set_num @ A @ B ) @ ( plus_p532826482549453327et_num @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_9_set__plus__intro,axiom,
! [A: set_real,C2: set_set_real,B: set_real,D: set_set_real] :
( ( member_set_real @ A @ C2 )
=> ( ( member_set_real @ B @ D )
=> ( member_set_real @ ( plus_plus_set_real @ A @ B ) @ ( plus_p7620395444238123297t_real @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_10_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_11_set__plus__intro,axiom,
! [A: num,C2: set_num,B: num,D: set_num] :
( ( member_num @ A @ C2 )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( plus_plus_num @ A @ B ) @ ( plus_plus_set_num @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_12_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_13_set__plus__intro,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_set_int @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_14_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( plus_plus_set_int @ ( plus_plus_set_int @ A @ B ) @ C )
= ( plus_plus_set_int @ A @ ( plus_plus_set_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_15_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_16_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( plus_plus_set_real @ ( plus_plus_set_real @ A @ B ) @ C )
= ( plus_plus_set_real @ A @ ( plus_plus_set_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_17_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_18_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_19_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_20_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_21_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_22_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_23_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_24_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_25_group__cancel_Oadd1,axiom,
! [A2: set_int,K: set_int,A: set_int,B: set_int] :
( ( A2
= ( plus_plus_set_int @ K @ A ) )
=> ( ( plus_plus_set_int @ A2 @ B )
= ( plus_plus_set_int @ K @ ( plus_plus_set_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_26_group__cancel_Oadd1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( plus_plus_set_nat @ K @ A ) )
=> ( ( plus_plus_set_nat @ A2 @ B )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_27_group__cancel_Oadd1,axiom,
! [A2: set_real,K: set_real,A: set_real,B: set_real] :
( ( A2
= ( plus_plus_set_real @ K @ A ) )
=> ( ( plus_plus_set_real @ A2 @ B )
= ( plus_plus_set_real @ K @ ( plus_plus_set_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_28_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_29_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_30_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_31_group__cancel_Oadd2,axiom,
! [B2: set_int,K: set_int,B: set_int,A: set_int] :
( ( B2
= ( plus_plus_set_int @ K @ B ) )
=> ( ( plus_plus_set_int @ A @ B2 )
= ( plus_plus_set_int @ K @ ( plus_plus_set_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_32_group__cancel_Oadd2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( plus_plus_set_nat @ K @ B ) )
=> ( ( plus_plus_set_nat @ A @ B2 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_33_group__cancel_Oadd2,axiom,
! [B2: set_real,K: set_real,B: set_real,A: set_real] :
( ( B2
= ( plus_plus_set_real @ K @ B ) )
=> ( ( plus_plus_set_real @ A @ B2 )
= ( plus_plus_set_real @ K @ ( plus_plus_set_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_34_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_35_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_36_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_37_add_Oassoc,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( plus_plus_set_int @ ( plus_plus_set_int @ A @ B ) @ C )
= ( plus_plus_set_int @ A @ ( plus_plus_set_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_38_add_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_39_add_Oassoc,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( plus_plus_set_real @ ( plus_plus_set_real @ A @ B ) @ C )
= ( plus_plus_set_real @ A @ ( plus_plus_set_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_40_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_41_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_42_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_43_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_44_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_45_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_46_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_47_one__reorient,axiom,
! [X: numera2417102609627094330l_num1] :
( ( one_on3868389512446148991l_num1 = X )
= ( X = one_on3868389512446148991l_num1 ) ) ).
% one_reorient
thf(fact_48_one__reorient,axiom,
! [X: numera4273646738625120315l_num1] :
( ( one_on7795324986448017462l_num1 = X )
= ( X = one_on7795324986448017462l_num1 ) ) ).
% one_reorient
thf(fact_49_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_50_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_51_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_52_set__plus__elim,axiom,
! [X: set_int,A2: set_set_int,B2: set_set_int] :
( ( member_set_int @ X @ ( plus_p2327523748178379809et_int @ A2 @ B2 ) )
=> ~ ! [A3: set_int,B3: set_int] :
( ( X
= ( plus_plus_set_int @ A3 @ B3 ) )
=> ( ( member_set_int @ A3 @ A2 )
=> ~ ( member_set_int @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_53_set__plus__elim,axiom,
! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ X @ ( plus_p4817606893110106565et_nat @ A2 @ B2 ) )
=> ~ ! [A3: set_nat,B3: set_nat] :
( ( X
= ( plus_plus_set_nat @ A3 @ B3 ) )
=> ( ( member_set_nat @ A3 @ A2 )
=> ~ ( member_set_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_54_set__plus__elim,axiom,
! [X: set_num,A2: set_set_num,B2: set_set_num] :
( ( member_set_num @ X @ ( plus_p532826482549453327et_num @ A2 @ B2 ) )
=> ~ ! [A3: set_num,B3: set_num] :
( ( X
= ( plus_plus_set_num @ A3 @ B3 ) )
=> ( ( member_set_num @ A3 @ A2 )
=> ~ ( member_set_num @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_55_set__plus__elim,axiom,
! [X: set_real,A2: set_set_real,B2: set_set_real] :
( ( member_set_real @ X @ ( plus_p7620395444238123297t_real @ A2 @ B2 ) )
=> ~ ! [A3: set_real,B3: set_real] :
( ( X
= ( plus_plus_set_real @ A3 @ B3 ) )
=> ( ( member_set_real @ A3 @ A2 )
=> ~ ( member_set_real @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_56_set__plus__elim,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( member_int @ X @ ( plus_plus_set_int @ A2 @ B2 ) )
=> ~ ! [A3: int,B3: int] :
( ( X
= ( plus_plus_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A2 )
=> ~ ( member_int @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_57_set__plus__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( plus_plus_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_58_set__plus__elim,axiom,
! [X: num,A2: set_num,B2: set_num] :
( ( member_num @ X @ ( plus_plus_set_num @ A2 @ B2 ) )
=> ~ ! [A3: num,B3: num] :
( ( X
= ( plus_plus_num @ A3 @ B3 ) )
=> ( ( member_num @ A3 @ A2 )
=> ~ ( member_num @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_59_set__plus__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A3: real,B3: real] :
( ( X
= ( plus_plus_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_60_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_61_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_62_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_63_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_64_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_65_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_66_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_67_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_68_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_69_add_Oleft__commute,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( plus_plus_set_int @ B @ ( plus_plus_set_int @ A @ C ) )
= ( plus_plus_set_int @ A @ ( plus_plus_set_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_70_add_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ B @ ( plus_plus_set_nat @ A @ C ) )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_71_add_Oleft__commute,axiom,
! [B: set_real,A: set_real,C: set_real] :
( ( plus_plus_set_real @ B @ ( plus_plus_set_real @ A @ C ) )
= ( plus_plus_set_real @ A @ ( plus_plus_set_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_72_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_73_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_74_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_75_add_Ocommute,axiom,
( plus_plus_set_int
= ( ^ [A4: set_int,B4: set_int] : ( plus_plus_set_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_76_add_Ocommute,axiom,
( plus_plus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( plus_plus_set_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_77_add_Ocommute,axiom,
( plus_plus_set_real
= ( ^ [A4: set_real,B4: set_real] : ( plus_plus_set_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_78_dbl__inc__def,axiom,
( neg_nu4048618728508742987l_num1
= ( ^ [X2: numera2417102609627094330l_num1] : ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ X2 @ X2 ) @ one_on3868389512446148991l_num1 ) ) ) ).
% dbl_inc_def
thf(fact_79_dbl__inc__def,axiom,
( neg_nu5172728937851396970l_num1
= ( ^ [X2: numera4273646738625120315l_num1] : ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ X2 @ X2 ) @ one_on7795324986448017462l_num1 ) ) ) ).
% dbl_inc_def
thf(fact_80_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_81_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_82_dbl__dec__simps_I3_J,axiom,
( ( neg_nu228592723992507279l_num1 @ one_on3868389512446148991l_num1 )
= one_on3868389512446148991l_num1 ) ).
% dbl_dec_simps(3)
thf(fact_83_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7886226890278435366l_num1 @ one_on7795324986448017462l_num1 )
= one_on7795324986448017462l_num1 ) ).
% dbl_dec_simps(3)
thf(fact_84_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_85_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_86_set__plus__rearrange,axiom,
! [A: set_int,C2: set_set_int,B: set_int,D: set_set_int] :
( ( plus_p2327523748178379809et_int @ ( set_el7516368150128015667et_int @ A @ C2 ) @ ( set_el7516368150128015667et_int @ B @ D ) )
= ( set_el7516368150128015667et_int @ ( plus_plus_set_int @ A @ B ) @ ( plus_p2327523748178379809et_int @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_87_set__plus__rearrange,axiom,
! [A: set_nat,C2: set_set_nat,B: set_nat,D: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ ( set_el2470847132782436567et_nat @ A @ C2 ) @ ( set_el2470847132782436567et_nat @ B @ D ) )
= ( set_el2470847132782436567et_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_88_set__plus__rearrange,axiom,
! [A: set_real,C2: set_set_real,B: set_real,D: set_set_real] :
( ( plus_p7620395444238123297t_real @ ( set_el4273275173557528499t_real @ A @ C2 ) @ ( set_el4273275173557528499t_real @ B @ D ) )
= ( set_el4273275173557528499t_real @ ( plus_plus_set_real @ A @ B ) @ ( plus_p7620395444238123297t_real @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_89_set__plus__rearrange,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( plus_plus_set_int @ ( set_elt_set_plus_int @ A @ C2 ) @ ( set_elt_set_plus_int @ B @ D ) )
= ( set_elt_set_plus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_set_int @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_90_set__plus__rearrange,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( plus_plus_set_nat @ ( set_elt_set_plus_nat @ A @ C2 ) @ ( set_elt_set_plus_nat @ B @ D ) )
= ( set_elt_set_plus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_91_set__plus__rearrange,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( plus_plus_set_real @ ( set_el4531060646461604733s_real @ A @ C2 ) @ ( set_el4531060646461604733s_real @ B @ D ) )
= ( set_el4531060646461604733s_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_92_assms,axiom,
~ ( member_real @ x @ ring_1_Ints_real ) ).
% assms
thf(fact_93_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_94_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_95_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ X ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ X ) @ one_on3868389512446148991l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_96_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ X ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X ) @ one_on7795324986448017462l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_97_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_98_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_99_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_100_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_101_set__plus__intro2,axiom,
! [B: real,C2: set_real,A: real] :
( ( member_real @ B @ C2 )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( set_el4531060646461604733s_real @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_102_set__plus__intro2,axiom,
! [B: num,C2: set_num,A: num] :
( ( member_num @ B @ C2 )
=> ( member_num @ ( plus_plus_num @ A @ B ) @ ( set_elt_set_plus_num @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_103_set__plus__intro2,axiom,
! [B: nat,C2: set_nat,A: nat] :
( ( member_nat @ B @ C2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( set_elt_set_plus_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_104_set__plus__intro2,axiom,
! [B: int,C2: set_int,A: int] :
( ( member_int @ B @ C2 )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ( set_elt_set_plus_int @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_105_set__plus__intro2,axiom,
! [B: set_int,C2: set_set_int,A: set_int] :
( ( member_set_int @ B @ C2 )
=> ( member_set_int @ ( plus_plus_set_int @ A @ B ) @ ( set_el7516368150128015667et_int @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_106_set__plus__intro2,axiom,
! [B: set_nat,C2: set_set_nat,A: set_nat] :
( ( member_set_nat @ B @ C2 )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( set_el2470847132782436567et_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_107_set__plus__intro2,axiom,
! [B: set_num,C2: set_set_num,A: set_num] :
( ( member_set_num @ B @ C2 )
=> ( member_set_num @ ( plus_plus_set_num @ A @ B ) @ ( set_el3179091069731274273et_num @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_108_set__plus__intro2,axiom,
! [B: set_real,C2: set_set_real,A: set_real] :
( ( member_set_real @ B @ C2 )
=> ( member_set_real @ ( plus_plus_set_real @ A @ B ) @ ( set_el4273275173557528499t_real @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_109_set__one__times,axiom,
! [C2: set_Nu795013586925006960l_num1] :
( ( set_el2854875225590388075l_num1 @ one_on3868389512446148991l_num1 @ C2 )
= C2 ) ).
% set_one_times
thf(fact_110_set__one__times,axiom,
! [C2: set_Nu382503245525567899l_num1] :
( ( set_el7870430730610807626l_num1 @ one_on7795324986448017462l_num1 @ C2 )
= C2 ) ).
% set_one_times
thf(fact_111_set__one__times,axiom,
! [C2: set_real] :
( ( set_el1043507519367895261s_real @ one_one_real @ C2 )
= C2 ) ).
% set_one_times
thf(fact_112_set__one__times,axiom,
! [C2: set_nat] :
( ( set_el2933305810450955905es_nat @ one_one_nat @ C2 )
= C2 ) ).
% set_one_times
thf(fact_113_set__one__times,axiom,
! [C2: set_int] :
( ( set_el2930815339941905629es_int @ one_one_int @ C2 )
= C2 ) ).
% set_one_times
thf(fact_114_sumset__empty_I2_J,axiom,
! [A2: set_int] :
( ( plus_plus_set_int @ bot_bot_set_int @ A2 )
= bot_bot_set_int ) ).
% sumset_empty(2)
thf(fact_115_sumset__empty_I2_J,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% sumset_empty(2)
thf(fact_116_sumset__empty_I2_J,axiom,
! [A2: set_num] :
( ( plus_plus_set_num @ bot_bot_set_num @ A2 )
= bot_bot_set_num ) ).
% sumset_empty(2)
thf(fact_117_sumset__empty_I2_J,axiom,
! [A2: set_real] :
( ( plus_plus_set_real @ bot_bot_set_real @ A2 )
= bot_bot_set_real ) ).
% sumset_empty(2)
thf(fact_118_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_119_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_120_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_121_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_122_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_123_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_124_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_125_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_126_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_127_add__numeral__left,axiom,
! [V: num,W: num,Z: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_128_add__numeral__left,axiom,
! [V: num,W: num,Z: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_129_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_130_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_131_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_132_Ints__add__iff1,axiom,
! [X: real,Y: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y ) @ ring_1_Ints_real )
= ( member_real @ Y @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff1
thf(fact_133_Ints__add__iff1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( member_int @ ( plus_plus_int @ X @ Y ) @ ring_1_Ints_int )
= ( member_int @ Y @ ring_1_Ints_int ) ) ) ).
% Ints_add_iff1
thf(fact_134_Ints__add__iff2,axiom,
! [Y: real,X: real] :
( ( member_real @ Y @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff2
thf(fact_135_Ints__add__iff2,axiom,
! [Y: int,X: int] :
( ( member_int @ Y @ ring_1_Ints_int )
=> ( ( member_int @ ( plus_plus_int @ X @ Y ) @ ring_1_Ints_int )
= ( member_int @ X @ ring_1_Ints_int ) ) ) ).
% Ints_add_iff2
thf(fact_136_sumset__empty_I1_J,axiom,
! [A2: set_int] :
( ( plus_plus_set_int @ A2 @ bot_bot_set_int )
= bot_bot_set_int ) ).
% sumset_empty(1)
thf(fact_137_sumset__empty_I1_J,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% sumset_empty(1)
thf(fact_138_sumset__empty_I1_J,axiom,
! [A2: set_num] :
( ( plus_plus_set_num @ A2 @ bot_bot_set_num )
= bot_bot_set_num ) ).
% sumset_empty(1)
thf(fact_139_sumset__empty_I1_J,axiom,
! [A2: set_real] :
( ( plus_plus_set_real @ A2 @ bot_bot_set_real )
= bot_bot_set_real ) ).
% sumset_empty(1)
thf(fact_140_set__plus__rearrange2,axiom,
! [A: real,B: real,C2: set_real] :
( ( set_el4531060646461604733s_real @ A @ ( set_el4531060646461604733s_real @ B @ C2 ) )
= ( set_el4531060646461604733s_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_141_set__plus__rearrange2,axiom,
! [A: nat,B: nat,C2: set_nat] :
( ( set_elt_set_plus_nat @ A @ ( set_elt_set_plus_nat @ B @ C2 ) )
= ( set_elt_set_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_142_set__plus__rearrange2,axiom,
! [A: int,B: int,C2: set_int] :
( ( set_elt_set_plus_int @ A @ ( set_elt_set_plus_int @ B @ C2 ) )
= ( set_elt_set_plus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_143_set__plus__rearrange2,axiom,
! [A: set_int,B: set_int,C2: set_set_int] :
( ( set_el7516368150128015667et_int @ A @ ( set_el7516368150128015667et_int @ B @ C2 ) )
= ( set_el7516368150128015667et_int @ ( plus_plus_set_int @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_144_set__plus__rearrange2,axiom,
! [A: set_nat,B: set_nat,C2: set_set_nat] :
( ( set_el2470847132782436567et_nat @ A @ ( set_el2470847132782436567et_nat @ B @ C2 ) )
= ( set_el2470847132782436567et_nat @ ( plus_plus_set_nat @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_145_set__plus__rearrange2,axiom,
! [A: set_real,B: set_real,C2: set_set_real] :
( ( set_el4273275173557528499t_real @ A @ ( set_el4273275173557528499t_real @ B @ C2 ) )
= ( set_el4273275173557528499t_real @ ( plus_plus_set_real @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_146_set__times__plus__distrib2,axiom,
! [A: int,B2: set_int,C2: set_int] :
( ( set_el2930815339941905629es_int @ A @ ( plus_plus_set_int @ B2 @ C2 ) )
= ( plus_plus_set_int @ ( set_el2930815339941905629es_int @ A @ B2 ) @ ( set_el2930815339941905629es_int @ A @ C2 ) ) ) ).
% set_times_plus_distrib2
thf(fact_147_set__times__plus__distrib2,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( set_el2933305810450955905es_nat @ A @ ( plus_plus_set_nat @ B2 @ C2 ) )
= ( plus_plus_set_nat @ ( set_el2933305810450955905es_nat @ A @ B2 ) @ ( set_el2933305810450955905es_nat @ A @ C2 ) ) ) ).
% set_times_plus_distrib2
thf(fact_148_set__times__plus__distrib2,axiom,
! [A: real,B2: set_real,C2: set_real] :
( ( set_el1043507519367895261s_real @ A @ ( plus_plus_set_real @ B2 @ C2 ) )
= ( plus_plus_set_real @ ( set_el1043507519367895261s_real @ A @ B2 ) @ ( set_el1043507519367895261s_real @ A @ C2 ) ) ) ).
% set_times_plus_distrib2
thf(fact_149_set__plus__rearrange4,axiom,
! [C2: set_int,A: int,D: set_int] :
( ( plus_plus_set_int @ C2 @ ( set_elt_set_plus_int @ A @ D ) )
= ( set_elt_set_plus_int @ A @ ( plus_plus_set_int @ C2 @ D ) ) ) ).
% set_plus_rearrange4
thf(fact_150_set__plus__rearrange4,axiom,
! [C2: set_nat,A: nat,D: set_nat] :
( ( plus_plus_set_nat @ C2 @ ( set_elt_set_plus_nat @ A @ D ) )
= ( set_elt_set_plus_nat @ A @ ( plus_plus_set_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange4
thf(fact_151_set__plus__rearrange4,axiom,
! [C2: set_real,A: real,D: set_real] :
( ( plus_plus_set_real @ C2 @ ( set_el4531060646461604733s_real @ A @ D ) )
= ( set_el4531060646461604733s_real @ A @ ( plus_plus_set_real @ C2 @ D ) ) ) ).
% set_plus_rearrange4
thf(fact_152_set__plus__rearrange3,axiom,
! [A: int,B2: set_int,C2: set_int] :
( ( plus_plus_set_int @ ( set_elt_set_plus_int @ A @ B2 ) @ C2 )
= ( set_elt_set_plus_int @ A @ ( plus_plus_set_int @ B2 @ C2 ) ) ) ).
% set_plus_rearrange3
thf(fact_153_set__plus__rearrange3,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( plus_plus_set_nat @ ( set_elt_set_plus_nat @ A @ B2 ) @ C2 )
= ( set_elt_set_plus_nat @ A @ ( plus_plus_set_nat @ B2 @ C2 ) ) ) ).
% set_plus_rearrange3
thf(fact_154_set__plus__rearrange3,axiom,
! [A: real,B2: set_real,C2: set_real] :
( ( plus_plus_set_real @ ( set_el4531060646461604733s_real @ A @ B2 ) @ C2 )
= ( set_el4531060646461604733s_real @ A @ ( plus_plus_set_real @ B2 @ C2 ) ) ) ).
% set_plus_rearrange3
thf(fact_155_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X2: real] :
~ ( member_real @ X2 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_156_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_157_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X2: int] :
~ ( member_int @ X2 @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_158_all__not__in__conv,axiom,
! [A2: set_num] :
( ( ! [X2: num] :
~ ( member_num @ X2 @ A2 ) )
= ( A2 = bot_bot_set_num ) ) ).
% all_not_in_conv
thf(fact_159_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_160_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_161_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_162_empty__iff,axiom,
! [C: num] :
~ ( member_num @ C @ bot_bot_set_num ) ).
% empty_iff
thf(fact_163_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_164_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
! [A: num,P: num > $o] :
( ( member_num @ A @ ( collect_num @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A2: set_num] :
( ( collect_num
@ ^ [X2: num] : ( member_num @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_171_Ints__numeral,axiom,
! [N: num] : ( member8231810200707485668l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ring_19056730708217498732l_num1 ) ).
% Ints_numeral
thf(fact_172_Ints__numeral,axiom,
! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% Ints_numeral
thf(fact_173_Ints__numeral,axiom,
! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% Ints_numeral
thf(fact_174_Ints__add,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_add
thf(fact_175_Ints__add,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_add
thf(fact_176_Ints__1,axiom,
member2815666790699981905l_num1 @ one_on3868389512446148991l_num1 @ ring_14745913572136535497l_num1 ).
% Ints_1
thf(fact_177_Ints__1,axiom,
member8231810200707485668l_num1 @ one_on7795324986448017462l_num1 @ ring_19056730708217498732l_num1 ).
% Ints_1
thf(fact_178_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_179_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_180_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7886226890278435366l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu5172728937851396970l_num1 @ ( numera7754357348821619680l_num1 @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_181_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_182_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_183_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5172728937851396970l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu7886226890278435366l_num1 @ ( numera7754357348821619680l_num1 @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_184_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_185_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_186_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_187_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_188_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_189_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_190_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_191_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_192_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_193_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_194_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_195_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_196_odd__h,axiom,
! [X: real] :
( ( h @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( h @ X ) ) ) ).
% odd_h
thf(fact_197_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_198_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_199_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_200_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_201_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_202_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_203_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_204_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_205_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_206_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_207_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_208_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_209_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_210_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_211_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_212_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_213_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( uminus1336558196688952754l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_214_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_215_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_216_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5816564918971239084l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu5816564918971239084l_num1 @ ( numera7754357348821619680l_num1 @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_217_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_218_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_219_dbl__inc__simps_I4_J,axiom,
( ( neg_nu4048618728508742987l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% dbl_inc_simps(4)
thf(fact_220_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5172728937851396970l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% dbl_inc_simps(4)
thf(fact_221_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_222_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_223_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_224_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_225_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_226_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_227_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_228_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_229_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_230_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_231_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_232_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_233_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_234_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_235_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_236_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_237_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_238_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_239_Ints__minus,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_240_Ints__minus,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_minus
thf(fact_241_minus__in__Ints__iff,axiom,
! [X: real] :
( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_242_minus__in__Ints__iff,axiom,
! [X: int] :
( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
= ( member_int @ X @ ring_1_Ints_int ) ) ).
% minus_in_Ints_iff
thf(fact_243_uminus__numeral__One,axiom,
( ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ one ) )
= ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% uminus_numeral_One
thf(fact_244_uminus__numeral__One,axiom,
( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) )
= ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% uminus_numeral_One
thf(fact_245_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_246_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_247_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_248_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_249_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_250_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_251_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_252_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_253_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_254_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_255_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_256_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_257_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_258_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_259_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_260_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_261_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_262_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_263_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_264_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_265_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_266_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_267_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_268_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_269_set__neg__intro,axiom,
! [A: numera2417102609627094330l_num1,C2: set_Nu795013586925006960l_num1] :
( ( member2815666790699981905l_num1 @ A @ ( set_el2854875225590388075l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ C2 ) )
=> ( member2815666790699981905l_num1 @ ( uminus7224005126491068675l_num1 @ A ) @ C2 ) ) ).
% set_neg_intro
thf(fact_270_set__neg__intro,axiom,
! [A: numera4273646738625120315l_num1,C2: set_Nu382503245525567899l_num1] :
( ( member8231810200707485668l_num1 @ A @ ( set_el7870430730610807626l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ C2 ) )
=> ( member8231810200707485668l_num1 @ ( uminus1336558196688952754l_num1 @ A ) @ C2 ) ) ).
% set_neg_intro
thf(fact_271_set__neg__intro,axiom,
! [A: real,C2: set_real] :
( ( member_real @ A @ ( set_el1043507519367895261s_real @ ( uminus_uminus_real @ one_one_real ) @ C2 ) )
=> ( member_real @ ( uminus_uminus_real @ A ) @ C2 ) ) ).
% set_neg_intro
thf(fact_272_set__neg__intro,axiom,
! [A: int,C2: set_int] :
( ( member_int @ A @ ( set_el2930815339941905629es_int @ ( uminus_uminus_int @ one_one_int ) @ C2 ) )
=> ( member_int @ ( uminus_uminus_int @ A ) @ C2 ) ) ).
% set_neg_intro
thf(fact_273_set__neg__intro2,axiom,
! [A: numera2417102609627094330l_num1,C2: set_Nu795013586925006960l_num1] :
( ( member2815666790699981905l_num1 @ A @ C2 )
=> ( member2815666790699981905l_num1 @ ( uminus7224005126491068675l_num1 @ A ) @ ( set_el2854875225590388075l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ C2 ) ) ) ).
% set_neg_intro2
thf(fact_274_set__neg__intro2,axiom,
! [A: numera4273646738625120315l_num1,C2: set_Nu382503245525567899l_num1] :
( ( member8231810200707485668l_num1 @ A @ C2 )
=> ( member8231810200707485668l_num1 @ ( uminus1336558196688952754l_num1 @ A ) @ ( set_el7870430730610807626l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ C2 ) ) ) ).
% set_neg_intro2
thf(fact_275_set__neg__intro2,axiom,
! [A: real,C2: set_real] :
( ( member_real @ A @ C2 )
=> ( member_real @ ( uminus_uminus_real @ A ) @ ( set_el1043507519367895261s_real @ ( uminus_uminus_real @ one_one_real ) @ C2 ) ) ) ).
% set_neg_intro2
thf(fact_276_set__neg__intro2,axiom,
! [A: int,C2: set_int] :
( ( member_int @ A @ C2 )
=> ( member_int @ ( uminus_uminus_int @ A ) @ ( set_el2930815339941905629es_int @ ( uminus_uminus_int @ one_one_int ) @ C2 ) ) ) ).
% set_neg_intro2
thf(fact_277_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_278_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_279_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_280_emptyE,axiom,
! [A: num] :
~ ( member_num @ A @ bot_bot_set_num ) ).
% emptyE
thf(fact_281_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_282_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_283_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_284_equals0D,axiom,
! [A2: set_num,A: num] :
( ( A2 = bot_bot_set_num )
=> ~ ( member_num @ A @ A2 ) ) ).
% equals0D
thf(fact_285_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_286_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_287_equals0I,axiom,
! [A2: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_288_equals0I,axiom,
! [A2: set_num] :
( ! [Y2: num] :
~ ( member_num @ Y2 @ A2 )
=> ( A2 = bot_bot_set_num ) ) ).
% equals0I
thf(fact_289_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_290_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_291_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X2: int] : ( member_int @ X2 @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_292_ex__in__conv,axiom,
! [A2: set_num] :
( ( ? [X2: num] : ( member_num @ X2 @ A2 ) )
= ( A2 != bot_bot_set_num ) ) ).
% ex_in_conv
thf(fact_293_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y ) )
= ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_294_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_295_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_296_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_297_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_298_dbl__simps_I4_J,axiom,
( ( neg_nu5590746349488142217l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_299_dbl__simps_I4_J,axiom,
( ( neg_nu5816564918971239084l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_300_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_301_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_302_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_p2313304076027620419l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( neg_nu3733408198258700219l_num1 @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_303_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( neg_nu3067386718351260922l_num1 @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_304_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
= ( neg_numeral_sub_real @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_305_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_306_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( neg_nu3733408198258700219l_num1 @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_307_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( neg_nu3067386718351260922l_num1 @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_308_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
= ( neg_numeral_sub_real @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_309_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_310_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_p2313304076027620419l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ M ) ) @ one_on3868389512446148991l_num1 )
= ( neg_nu3733408198258700219l_num1 @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_311_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ one_on7795324986448017462l_num1 )
= ( neg_nu3067386718351260922l_num1 @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_312_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
= ( neg_numeral_sub_real @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_313_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_314_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ M ) ) )
= ( neg_nu3733408198258700219l_num1 @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_315_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) )
= ( neg_nu3067386718351260922l_num1 @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_316_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) )
= ( neg_numeral_sub_real @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_317_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_318_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ X ) )
= one_on3868389512446148991l_num1 )
= ( ring_16753260291671729742l_num1 @ ( numera2161328050825114965l_num1 @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_319_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X ) )
= one_on7795324986448017462l_num1 )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_320_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X ) )
= one_one_real )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_321_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= one_one_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_322_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_323_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_324_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_325_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_326_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_327_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_328_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_14756928180729810983l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_329_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_330_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_331_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5816564918971239084l_num1 @ ( numera7754357348821619680l_num1 @ K ) )
= ( numera7754357348821619680l_num1 @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_332_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_333_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_334_sub__num__simps_I6_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(6)
thf(fact_335_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Y ) )
= ( plus_p1441664204671982194l_num1 @ ( neg_nu3067386718351260922l_num1 @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_336_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Y ) )
= ( plus_plus_real @ ( neg_numeral_sub_real @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_337_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_338_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y ) )
= ( plus_p1441664204671982194l_num1 @ ( neg_nu3067386718351260922l_num1 @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_339_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( plus_plus_real @ ( neg_numeral_sub_real @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_340_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_341_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( neg_nu3067386718351260922l_num1 @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_342_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( neg_numeral_sub_real @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_343_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_344_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( neg_nu3067386718351260922l_num1 @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_345_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( neg_numeral_sub_real @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_346_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_347_one__add__one,axiom,
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_348_one__add__one,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_349_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_350_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_351_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_352_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_353_dbl__simps_I3_J,axiom,
( ( neg_nu5590746349488142217l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_354_dbl__simps_I3_J,axiom,
( ( neg_nu5816564918971239084l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_355_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_356_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_357_add__neg__numeral__special_I9_J,axiom,
( ( plus_p2313304076027620419l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_358_add__neg__numeral__special_I9_J,axiom,
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_359_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_360_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_361_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_362_eq__numeral__iff__iszero_I1_J,axiom,
! [X: num,Y: num] :
( ( ( numera7754357348821619680l_num1 @ X )
= ( numera7754357348821619680l_num1 @ Y ) )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ X @ Y ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_363_eq__numeral__iff__iszero_I1_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_real @ X )
= ( numeral_numeral_real @ Y ) )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ X @ Y ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_364_eq__numeral__iff__iszero_I1_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_int @ X )
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ Y ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_365_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y ) ) )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_366_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y ) ) )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_367_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_368_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_369_not__iszero__1,axiom,
~ ( ring_16753260291671729742l_num1 @ one_on3868389512446148991l_num1 ) ).
% not_iszero_1
thf(fact_370_not__iszero__1,axiom,
~ ( ring_14756928180729810983l_num1 @ one_on7795324986448017462l_num1 ) ).
% not_iszero_1
thf(fact_371_not__iszero__1,axiom,
~ ( ring_1_iszero_real @ one_one_real ) ).
% not_iszero_1
thf(fact_372_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_373_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_real @ ( numeral_numeral_real @ W ) ) ).
% not_iszero_numeral
thf(fact_374_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ).
% not_iszero_numeral
thf(fact_375_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_on3868389512446148991l_num1
= ( numera2161328050825114965l_num1 @ Y ) )
= ( ring_16753260291671729742l_num1 @ ( neg_nu3733408198258700219l_num1 @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_376_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_on7795324986448017462l_num1
= ( numera7754357348821619680l_num1 @ Y ) )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_377_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_real
= ( numeral_numeral_real @ Y ) )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_378_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_379_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numera2161328050825114965l_num1 @ X )
= one_on3868389512446148991l_num1 )
= ( ring_16753260291671729742l_num1 @ ( neg_nu3733408198258700219l_num1 @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_380_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numera7754357348821619680l_num1 @ X )
= one_on7795324986448017462l_num1 )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_381_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numeral_numeral_real @ X )
= one_one_real )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_382_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numeral_numeral_int @ X )
= one_one_int )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_383_numeral__Bit0,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ).
% numeral_Bit0
thf(fact_384_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_385_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_386_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_387_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_388_not__iszero__neg__1,axiom,
~ ( ring_16753260291671729742l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% not_iszero_neg_1
thf(fact_389_not__iszero__neg__1,axiom,
~ ( ring_14756928180729810983l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% not_iszero_neg_1
thf(fact_390_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_iszero_neg_1
thf(fact_391_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_392_not__iszero__Numeral1,axiom,
~ ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ one ) ) ).
% not_iszero_Numeral1
thf(fact_393_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_real @ ( numeral_numeral_real @ one ) ) ).
% not_iszero_Numeral1
thf(fact_394_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ one ) ) ).
% not_iszero_Numeral1
thf(fact_395_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_396_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_397_not__iszero__neg__Numeral1,axiom,
~ ( ring_14756928180729810983l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_398_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_399_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_400_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X ) )
= ( numera7754357348821619680l_num1 @ Y ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_401_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ Y ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_402_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_403_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numera7754357348821619680l_num1 @ X )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y ) ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_404_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_real @ X )
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y ) ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_405_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_int @ X )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_406_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_on3868389512446148991l_num1
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ Y ) ) )
= ( ring_16753260291671729742l_num1 @ ( numera2161328050825114965l_num1 @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_407_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_on7795324986448017462l_num1
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y ) ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_408_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_real
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y ) ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_409_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_410_nat__add__1__add__1,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_411_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X2: real] : ( member_real @ X2 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_412_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_413_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_414_bot__empty__eq,axiom,
( bot_bot_num_o
= ( ^ [X2: num] : ( member_num @ X2 @ bot_bot_set_num ) ) ) ).
% bot_empty_eq
thf(fact_415_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_nu3067386718351260922l_num1 @ one @ ( bit0 @ L ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_416_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_real @ one @ ( bit0 @ L ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_417_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_418_exhaust__2,axiom,
! [X: numera2417102609627094330l_num1] :
( ( X = one_on3868389512446148991l_num1 )
| ( X
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_419_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X3: numera2417102609627094330l_num1] : ( P2 @ X3 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_420_dbl__dec__simps_I4_J,axiom,
( ( neg_nu228592723992507279l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_421_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7886226890278435366l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_422_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_423_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_424_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ M ) ) @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( neg_nu3733408198258700219l_num1 @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_425_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( neg_nu3067386718351260922l_num1 @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_426_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( neg_numeral_sub_real @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_427_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_428_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ N ) ) )
= ( neg_nu3733408198258700219l_num1 @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_429_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( neg_nu3067386718351260922l_num1 @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_430_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( neg_numeral_sub_real @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_431_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_432_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ ( neg_nu3733408198258700219l_num1 @ M @ one ) ) @ one_on3868389512446148991l_num1 )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_433_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ ( neg_nu3067386718351260922l_num1 @ M @ one ) ) @ one_on7795324986448017462l_num1 )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_434_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( neg_numeral_sub_real @ M @ one ) ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_435_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_436_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_437_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y3: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y3 ) )
= ( X32 = Y3 ) ) ).
% verit_eq_simplify(9)
thf(fact_438_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_439_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_440_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_441_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_442_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_443_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_444_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_445_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_446_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_447_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_448_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_449_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_450_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_451_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_452_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_453_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_454_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_455_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_456_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_457_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_458_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_459_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_460_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_461_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_462_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_463_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_464_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( neg_nu3067386718351260922l_num1 @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_465_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( neg_numeral_sub_real @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_466_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_467_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_468_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_469_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5172728937851396970l_num1 @ ( numera7754357348821619680l_num1 @ K ) )
= ( numera7754357348821619680l_num1 @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_470_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_471_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_472_sub__num__simps_I9_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(9)
thf(fact_473_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu7886226890278435366l_num1 @ ( numera7754357348821619680l_num1 @ K ) )
= ( numera7754357348821619680l_num1 @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_474_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_475_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_476_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_477_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_478_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_479_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_480_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_481_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_482_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( neg_nu3067386718351260922l_num1 @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_483_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( neg_numeral_sub_real @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_484_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_485_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_486_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_487_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_488_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_489_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_490_sub__num__simps_I7_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_nu3811975205180677377ec_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(7)
thf(fact_491_sub__num__simps_I8_J,axiom,
! [K: num,L: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_nu5851722552734809277nc_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% sub_num_simps(8)
thf(fact_492_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_838314146864362899l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ N ) )
= ( neg_nu3733408198258700219l_num1 @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_493_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_5410813661909488930l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ N ) )
= ( neg_nu3067386718351260922l_num1 @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_494_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( neg_numeral_sub_real @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_495_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_496_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_838314146864362899l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ one_on3868389512446148991l_num1 )
= ( neg_nu3733408198258700219l_num1 @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_497_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ one_on7795324986448017462l_num1 )
= ( neg_nu3067386718351260922l_num1 @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_498_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ one_one_real )
= ( neg_numeral_sub_real @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_499_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_500_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3067386718351260922l_num1 @ ( bit1 @ K ) @ one )
= ( numera7754357348821619680l_num1 @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_501_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_real @ ( bit1 @ K ) @ one )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_502_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_503_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_nu3067386718351260922l_num1 @ ( bit0 @ K ) @ one )
= ( numera7754357348821619680l_num1 @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_504_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_numeral_sub_real @ ( bit0 @ K ) @ one )
= ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_505_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_506_dbl__inc__simps_I3_J,axiom,
( ( neg_nu4048618728508742987l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_507_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5172728937851396970l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_508_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_509_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_510_diff__numeral__special_I10_J,axiom,
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ one_on3868389512446148991l_num1 )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_511_diff__numeral__special_I10_J,axiom,
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ one_on7795324986448017462l_num1 )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_512_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_513_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_514_diff__numeral__special_I11_J,axiom,
( ( minus_838314146864362899l_num1 @ one_on3868389512446148991l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_515_diff__numeral__special_I11_J,axiom,
( ( minus_5410813661909488930l_num1 @ one_on7795324986448017462l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_516_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_517_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_518_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_838314146864362899l_num1 @ one_on3868389512446148991l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ N ) ) )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_519_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_5410813661909488930l_num1 @ one_on7795324986448017462l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_520_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_521_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_522_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ M ) ) @ one_on3868389512446148991l_num1 )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_523_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ one_on7795324986448017462l_num1 )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_524_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_525_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_526_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_nu3067386718351260922l_num1 @ one @ ( bit1 @ L ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_527_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_real @ one @ ( bit1 @ L ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_528_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_529_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_530_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_531_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_532_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_533_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_534_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_535_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_536_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_537_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_538_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_539_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_540_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_541_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_542_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_543_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_544_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_545_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_546_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_547_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_548_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_549_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_550_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_551_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_552_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_553_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_554_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_555_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_556_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_557_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_558_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_559_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_560_Ints__diff,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_561_Ints__diff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_diff
thf(fact_562_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_563_exhaust__4,axiom,
! [X: numera4273646738625120315l_num1] :
( ( X = one_on7795324986448017462l_num1 )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_564_exhaust__3,axiom,
! [X: numera6367994245245682809l_num1] :
( ( X = one_on7819281148064737470l_num1 )
| ( X
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_565_forall__4,axiom,
( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
! [X3: numera4273646738625120315l_num1] : ( P2 @ X3 ) )
= ( ^ [P3: numera4273646738625120315l_num1 > $o] :
( ( P3 @ one_on7795324986448017462l_num1 )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).
% forall_4
thf(fact_566_forall__3,axiom,
( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
! [X3: numera6367994245245682809l_num1] : ( P2 @ X3 ) )
= ( ^ [P3: numera6367994245245682809l_num1 > $o] :
( ( P3 @ one_on7819281148064737470l_num1 )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_567_set__plus__imp__minus,axiom,
! [A: real,B: real,C2: set_real] :
( ( member_real @ A @ ( set_el4531060646461604733s_real @ B @ C2 ) )
=> ( member_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% set_plus_imp_minus
thf(fact_568_set__plus__imp__minus,axiom,
! [A: int,B: int,C2: set_int] :
( ( member_int @ A @ ( set_elt_set_plus_int @ B @ C2 ) )
=> ( member_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% set_plus_imp_minus
thf(fact_569_set__minus__imp__plus,axiom,
! [A: real,B: real,C2: set_real] :
( ( member_real @ ( minus_minus_real @ A @ B ) @ C2 )
=> ( member_real @ A @ ( set_el4531060646461604733s_real @ B @ C2 ) ) ) ).
% set_minus_imp_plus
thf(fact_570_set__minus__imp__plus,axiom,
! [A: int,B: int,C2: set_int] :
( ( member_int @ ( minus_minus_int @ A @ B ) @ C2 )
=> ( member_int @ A @ ( set_elt_set_plus_int @ B @ C2 ) ) ) ).
% set_minus_imp_plus
thf(fact_571_set__minus__plus,axiom,
! [A: real,B: real,C2: set_real] :
( ( member_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( member_real @ A @ ( set_el4531060646461604733s_real @ B @ C2 ) ) ) ).
% set_minus_plus
thf(fact_572_set__minus__plus,axiom,
! [A: int,B: int,C2: set_int] :
( ( member_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( member_int @ A @ ( set_elt_set_plus_int @ B @ C2 ) ) ) ).
% set_minus_plus
thf(fact_573_eq__iff__iszero__diff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [X2: real,Y5: real] : ( ring_1_iszero_real @ ( minus_minus_real @ X2 @ Y5 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_574_eq__iff__iszero__diff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [X2: int,Y5: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X2 @ Y5 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_575_numeral__BitM,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bitM @ N ) )
= ( minus_838314146864362899l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ N ) ) @ one_on3868389512446148991l_num1 ) ) ).
% numeral_BitM
thf(fact_576_numeral__BitM,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bitM @ N ) )
= ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_BitM
thf(fact_577_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bitM @ N ) )
= ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% numeral_BitM
thf(fact_578_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_579_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_580_group__cancel_Osub2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_581_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_582_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_583_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_584_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_585_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_586_neg__numeral__class_Osub__def,axiom,
( neg_nu3067386718351260922l_num1
= ( ^ [K2: num,L2: num] : ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) @ ( numera7754357348821619680l_num1 @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_587_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_real
= ( ^ [K2: num,L2: num] : ( minus_minus_real @ ( numeral_numeral_real @ K2 ) @ ( numeral_numeral_real @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_588_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_589_numeral__Bit1,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit1 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) @ one_on3868389512446148991l_num1 ) ) ).
% numeral_Bit1
thf(fact_590_numeral__Bit1,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit1 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_Bit1
thf(fact_591_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_592_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_593_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_594_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_595_dbl__dec__def,axiom,
( neg_nu228592723992507279l_num1
= ( ^ [X2: numera2417102609627094330l_num1] : ( minus_838314146864362899l_num1 @ ( plus_p2313304076027620419l_num1 @ X2 @ X2 ) @ one_on3868389512446148991l_num1 ) ) ) ).
% dbl_dec_def
thf(fact_596_dbl__dec__def,axiom,
( neg_nu7886226890278435366l_num1
= ( ^ [X2: numera4273646738625120315l_num1] : ( minus_5410813661909488930l_num1 @ ( plus_p1441664204671982194l_num1 @ X2 @ X2 ) @ one_on7795324986448017462l_num1 ) ) ) ).
% dbl_dec_def
thf(fact_597_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_598_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_599_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_600_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_601_h__def,axiom,
( h
= ( ^ [X2: real] : ( if_real @ ( member_real @ X2 @ ring_1_Ints_real ) @ zero_zero_real @ ( minus_minus_real @ ( f @ X2 ) @ ( g @ X2 ) ) ) ) ) ).
% h_def
thf(fact_602_pth__2,axiom,
( minus_minus_real
= ( ^ [X2: real,Y5: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% pth_2
thf(fact_603_minus__eq__plus__uminus,axiom,
( minus_minus_real
= ( ^ [F: real,G: real] : ( plus_plus_real @ F @ ( uminus_uminus_real @ G ) ) ) ) ).
% minus_eq_plus_uminus
thf(fact_604_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_605_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y5: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_606_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_607_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_608_add__diff__add,axiom,
! [A: real,C: real,B: real,D2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_609_add__diff__add,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_610_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_611_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_612_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_613_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_614_Diff__iff,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_615_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_616_Diff__iff,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
= ( ( member_int @ C @ A2 )
& ~ ( member_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_617_Diff__iff,axiom,
! [C: num,A2: set_num,B2: set_num] :
( ( member_num @ C @ ( minus_minus_set_num @ A2 @ B2 ) )
= ( ( member_num @ C @ A2 )
& ~ ( member_num @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_618_DiffI,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B2 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_619_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_620_DiffI,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ A2 )
=> ( ~ ( member_int @ C @ B2 )
=> ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_621_DiffI,axiom,
! [C: num,A2: set_num,B2: set_num] :
( ( member_num @ C @ A2 )
=> ( ~ ( member_num @ C @ B2 )
=> ( member_num @ C @ ( minus_minus_set_num @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_622_add_Oright__neutral,axiom,
! [A: set_int] :
( ( plus_plus_set_int @ A @ zero_zero_set_int )
= A ) ).
% add.right_neutral
thf(fact_623_add_Oright__neutral,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ zero_zero_set_nat )
= A ) ).
% add.right_neutral
thf(fact_624_add_Oright__neutral,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ A @ zero_zero_set_real )
= A ) ).
% add.right_neutral
thf(fact_625_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_626_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_627_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_628_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_629_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_630_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_631_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_632_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_633_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_634_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_635_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_636_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_637_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_638_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_639_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_640_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_641_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_642_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_643_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_644_add__0,axiom,
! [A: set_int] :
( ( plus_plus_set_int @ zero_zero_set_int @ A )
= A ) ).
% add_0
thf(fact_645_add__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A )
= A ) ).
% add_0
thf(fact_646_add__0,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ zero_zero_set_real @ A )
= A ) ).
% add_0
thf(fact_647_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_648_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_649_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_650_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_651_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_652_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_653_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_654_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_655_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_656_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_657_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_658_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_659_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_660_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_661_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_662_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_663_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_664_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_665_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_666_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_667_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_668_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_669_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_670_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_671_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_672_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_673_set__zero__plus,axiom,
! [C2: set_real] :
( ( set_el4531060646461604733s_real @ zero_zero_real @ C2 )
= C2 ) ).
% set_zero_plus
thf(fact_674_set__zero__plus,axiom,
! [C2: set_nat] :
( ( set_elt_set_plus_nat @ zero_zero_nat @ C2 )
= C2 ) ).
% set_zero_plus
thf(fact_675_set__zero__plus,axiom,
! [C2: set_int] :
( ( set_elt_set_plus_int @ zero_zero_int @ C2 )
= C2 ) ).
% set_zero_plus
thf(fact_676_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_677_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_678_diff__numeral__special_I9_J,axiom,
( ( minus_838314146864362899l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
= zero_z5982384998485459395l_num1 ) ).
% diff_numeral_special(9)
thf(fact_679_diff__numeral__special_I9_J,axiom,
( ( minus_5410813661909488930l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= zero_z2241845390563828978l_num1 ) ).
% diff_numeral_special(9)
thf(fact_680_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_681_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_682_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_683_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_684_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_685_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_686_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_687_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_688_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_689_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_690_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_691_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_692_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_real @ one @ one )
= zero_zero_real ) ).
% sub_num_simps(1)
thf(fact_693_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_694_dbl__inc__simps_I2_J,axiom,
( ( neg_nu4048618728508742987l_num1 @ zero_z5982384998485459395l_num1 )
= one_on3868389512446148991l_num1 ) ).
% dbl_inc_simps(2)
thf(fact_695_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5172728937851396970l_num1 @ zero_z2241845390563828978l_num1 )
= one_on7795324986448017462l_num1 ) ).
% dbl_inc_simps(2)
thf(fact_696_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_697_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_698_add__neg__numeral__special_I8_J,axiom,
( ( plus_p2313304076027620419l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ one_on3868389512446148991l_num1 )
= zero_z5982384998485459395l_num1 ) ).
% add_neg_numeral_special(8)
thf(fact_699_add__neg__numeral__special_I8_J,axiom,
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ one_on7795324986448017462l_num1 )
= zero_z2241845390563828978l_num1 ) ).
% add_neg_numeral_special(8)
thf(fact_700_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_701_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_702_add__neg__numeral__special_I7_J,axiom,
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= zero_z5982384998485459395l_num1 ) ).
% add_neg_numeral_special(7)
thf(fact_703_add__neg__numeral__special_I7_J,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= zero_z2241845390563828978l_num1 ) ).
% add_neg_numeral_special(7)
thf(fact_704_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_705_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_706_diff__numeral__special_I12_J,axiom,
( ( minus_838314146864362899l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= zero_z5982384998485459395l_num1 ) ).
% diff_numeral_special(12)
thf(fact_707_diff__numeral__special_I12_J,axiom,
( ( minus_5410813661909488930l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= zero_z2241845390563828978l_num1 ) ).
% diff_numeral_special(12)
thf(fact_708_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_709_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_710_dbl__dec__simps_I2_J,axiom,
( ( neg_nu228592723992507279l_num1 @ zero_z5982384998485459395l_num1 )
= ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% dbl_dec_simps(2)
thf(fact_711_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7886226890278435366l_num1 @ zero_z2241845390563828978l_num1 )
= ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% dbl_dec_simps(2)
thf(fact_712_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_713_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_714__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_A_092_060int_062_A_092_060Longrightarrow_062_Ah_Ax_A_061_A0_092_060close_062,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( h @ X )
= zero_zero_real ) ) ).
% \<open>\<And>x. x \<in> \<int> \<Longrightarrow> h x = 0\<close>
thf(fact_715_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_p2313304076027620419l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ M ) ) @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_716_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_717_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_718_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_719_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_p2313304076027620419l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ N ) ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_720_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_721_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_722_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_723_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_838314146864362899l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( numera2161328050825114965l_num1 @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_724_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( numera7754357348821619680l_num1 @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_725_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_726_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_727_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_728_Multiseries__Expansion_Oreal__eqI,axiom,
! [A: real,B: real] :
( ( ( minus_minus_real @ A @ B )
= zero_zero_real )
=> ( A = B ) ) ).
% Multiseries_Expansion.real_eqI
thf(fact_729_DiffD2,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( member_real @ C @ B2 ) ) ).
% DiffD2
thf(fact_730_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_731_DiffD2,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ~ ( member_int @ C @ B2 ) ) ).
% DiffD2
thf(fact_732_DiffD2,axiom,
! [C: num,A2: set_num,B2: set_num] :
( ( member_num @ C @ ( minus_minus_set_num @ A2 @ B2 ) )
=> ~ ( member_num @ C @ B2 ) ) ).
% DiffD2
thf(fact_733_DiffD1,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_734_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_735_DiffD1,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ( member_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_736_DiffD1,axiom,
! [C: num,A2: set_num,B2: set_num] :
( ( member_num @ C @ ( minus_minus_set_num @ A2 @ B2 ) )
=> ( member_num @ C @ A2 ) ) ).
% DiffD1
thf(fact_737_DiffE,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% DiffE
thf(fact_738_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_739_DiffE,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ~ ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% DiffE
thf(fact_740_DiffE,axiom,
! [C: num,A2: set_num,B2: set_num] :
( ( member_num @ C @ ( minus_minus_set_num @ A2 @ B2 ) )
=> ~ ( ( member_num @ C @ A2 )
=> ( member_num @ C @ B2 ) ) ) ).
% DiffE
thf(fact_741_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_742_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_743_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_744_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_745_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_746_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_747_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_748_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_749_comm__monoid__add__class_Oadd__0,axiom,
! [A: set_int] :
( ( plus_plus_set_int @ zero_zero_set_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_750_comm__monoid__add__class_Oadd__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_751_comm__monoid__add__class_Oadd__0,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ zero_zero_set_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_752_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_753_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_754_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_755_add_Ocomm__neutral,axiom,
! [A: set_int] :
( ( plus_plus_set_int @ A @ zero_zero_set_int )
= A ) ).
% add.comm_neutral
thf(fact_756_add_Ocomm__neutral,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ zero_zero_set_nat )
= A ) ).
% add.comm_neutral
thf(fact_757_add_Ocomm__neutral,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ A @ zero_zero_set_real )
= A ) ).
% add.comm_neutral
thf(fact_758_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_759_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_760_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_761_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_762_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_763_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_764_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_765_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_766_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_767_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_768_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_769_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_770_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_771_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_772_iszero__0,axiom,
ring_1_iszero_real @ zero_zero_real ).
% iszero_0
thf(fact_773_iszero__0,axiom,
ring_1_iszero_int @ zero_zero_int ).
% iszero_0
thf(fact_774_iszero__def,axiom,
( ring_1_iszero_real
= ( ^ [Z3: real] : ( Z3 = zero_zero_real ) ) ) ).
% iszero_def
thf(fact_775_iszero__def,axiom,
( ring_1_iszero_int
= ( ^ [Z3: int] : ( Z3 = zero_zero_int ) ) ) ).
% iszero_def
thf(fact_776_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_777_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_778_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_779_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_780_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_781_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_782_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_783_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_784_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_785_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_786_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_787_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_788_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_789_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_790_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_791_Ints__double__eq__0__iff,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_792_Ints__double__eq__0__iff,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_793_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_794_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_795_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_796_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_z2241845390563828978l_num1
= ( numera7754357348821619680l_num1 @ Y ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_797_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_zero_real
= ( numeral_numeral_real @ Y ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_798_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_799_eq__numeral__iff__iszero_I9_J,axiom,
! [X: num] :
( ( ( numera7754357348821619680l_num1 @ X )
= zero_z2241845390563828978l_num1 )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ X ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_800_eq__numeral__iff__iszero_I9_J,axiom,
! [X: num] :
( ( ( numeral_numeral_real @ X )
= zero_zero_real )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ X ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_801_eq__numeral__iff__iszero_I9_J,axiom,
! [X: num] :
( ( ( numeral_numeral_int @ X )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_802_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_803_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_804_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_805_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_806_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_807_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_808_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_809_Ints__odd__nonzero,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_810_Ints__odd__nonzero,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
!= zero_zero_int ) ) ).
% Ints_odd_nonzero
thf(fact_811_numeral__inc,axiom,
! [X: num] :
( ( numera2161328050825114965l_num1 @ ( inc @ X ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ X ) @ one_on3868389512446148991l_num1 ) ) ).
% numeral_inc
thf(fact_812_numeral__inc,axiom,
! [X: num] :
( ( numera7754357348821619680l_num1 @ ( inc @ X ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_inc
thf(fact_813_numeral__inc,axiom,
! [X: num] :
( ( numera1916890842035813515d_enat @ ( inc @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_inc
thf(fact_814_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_real @ ( inc @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% numeral_inc
thf(fact_815_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_816_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_817_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X ) )
= zero_z2241845390563828978l_num1 )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_818_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X ) )
= zero_zero_real )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_819_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_820_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_z2241845390563828978l_num1
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y ) ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_821_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_real
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y ) ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_822_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_823_sub__inc__One__eq,axiom,
! [N: num] :
( ( neg_nu3067386718351260922l_num1 @ ( inc @ N ) @ one )
= ( numera7754357348821619680l_num1 @ N ) ) ).
% sub_inc_One_eq
thf(fact_824_sub__inc__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_real @ ( inc @ N ) @ one )
= ( numeral_numeral_real @ N ) ) ).
% sub_inc_One_eq
thf(fact_825_sub__inc__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( inc @ N ) @ one )
= ( numeral_numeral_int @ N ) ) ).
% sub_inc_One_eq
thf(fact_826_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus_real @ Y @ Z ) )
= ( Y
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_827_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_828_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_829_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_830_eq__add__iff,axiom,
! [X: real,Y: real] :
( ( X
= ( plus_plus_real @ X @ Y ) )
= ( Y = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_831_eq__add__iff,axiom,
! [X: int,Y: int] :
( ( X
= ( plus_plus_int @ X @ Y ) )
= ( Y = zero_zero_int ) ) ).
% eq_add_iff
thf(fact_832_zero__neq__one,axiom,
zero_z5982384998485459395l_num1 != one_on3868389512446148991l_num1 ).
% zero_neq_one
thf(fact_833_zero__neq__one,axiom,
zero_z2241845390563828978l_num1 != one_on7795324986448017462l_num1 ).
% zero_neq_one
thf(fact_834_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_835_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_836_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_837_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_838_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_839_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_840_not__one__eq,axiom,
( ( bit_ri7919022796975470100ot_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_841_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_842_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_843_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_844_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_845_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_846_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_847_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_848_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_849_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_850_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_851_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_852_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_853_bit_Ocompl__zero,axiom,
( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.compl_zero
thf(fact_854_bit_Ocompl__one,axiom,
( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% bit.compl_one
thf(fact_855_minus__not__numeral__eq,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% minus_not_numeral_eq
thf(fact_856_not__minus__numeral__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% not_minus_numeral_eq
thf(fact_857_not__int__div__2,axiom,
! [K: int] :
( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% not_int_div_2
thf(fact_858_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_859_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_860_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_861_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_862_not__add__distrib,axiom,
! [A: int,B: int] :
( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
= ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% not_add_distrib
thf(fact_863_not__diff__distrib,axiom,
! [A: int,B: int] :
( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
= ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% not_diff_distrib
thf(fact_864_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_865_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_866_sqr_Osimps_I1_J,axiom,
( ( sqr @ one )
= one ) ).
% sqr.simps(1)
thf(fact_867_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_868_minus__eq__not__plus__1,axiom,
( uminus_uminus_int
= ( ^ [A4: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A4 ) @ one_one_int ) ) ) ).
% minus_eq_not_plus_1
thf(fact_869_minus__eq__not__minus__1,axiom,
( uminus_uminus_int
= ( ^ [A4: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A4 @ one_one_int ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_870_not__eq__complement,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [A4: int] : ( minus_minus_int @ ( uminus_uminus_int @ A4 ) @ one_one_int ) ) ) ).
% not_eq_complement
thf(fact_871_minus__numeral__inc__eq,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% minus_numeral_inc_eq
thf(fact_872_not__numeral__Bit0__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_873_field__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_874_minus__numeral__eq__not__sub__one,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( bit_ri7919022796975470100ot_int @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_875_not__numeral__BitM__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_876_g__def,axiom,
( g
= ( ^ [X2: real] : ( plus_plus_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( cotang1502006655779026648d_real @ X2 ) ) ) ) ).
% g_def
thf(fact_877_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_878_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_879_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_880_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_881_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_882_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_883_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_884_cot__pfd__real__minus,axiom,
! [X: real] :
( ( cotang1502006655779026648d_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( cotang1502006655779026648d_real @ X ) ) ) ).
% cot_pfd_real_minus
thf(fact_885_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_886_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_887_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_888_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_889_not__int__code_I1_J,axiom,
( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% not_int_code(1)
thf(fact_890_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_891_cot__pfd__plus__1__real,axiom,
! [X: real] :
( ~ ( member_real @ X @ ring_1_Ints_real )
=> ( ( cotang1502006655779026648d_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_real @ ( minus_minus_real @ ( cotang1502006655779026648d_real @ X ) @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ X @ one_one_real ) ) ) @ ( divide_divide_real @ one_one_real @ X ) ) ) ) ).
% cot_pfd_plus_1_real
thf(fact_892_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_893_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_894_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_895_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_896_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_897_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_898_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_899_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_900_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_901_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_902_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_903_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_904_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_905_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_906_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_907_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_908_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_909_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_910_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_911_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_912_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_913_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_914_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_915_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_916_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_917_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_918_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_919_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_920_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_921_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_922_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_923_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_924_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_925_floor__minus__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_926_cot__pfd__funeq__real,axiom,
! [X: real] :
( ~ ( member_real @ X @ ring_1_Ints_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cotang1502006655779026648d_real @ X ) )
= ( plus_plus_real @ ( plus_plus_real @ ( cotang1502006655779026648d_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( cotang1502006655779026648d_real @ ( divide_divide_real @ ( plus_plus_real @ X @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ one_one_real ) ) ) ) ) ).
% cot_pfd_funeq_real
thf(fact_927_Parity_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L2: num,R: int] : ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ ( numeral_numeral_int @ L2 ) @ R ) ) ) ) ).
% Parity.adjust_mod_def
thf(fact_928_drop__bit__rec,axiom,
( bit_se8570568707652914677it_nat
= ( ^ [N3: nat,A4: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ A4 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_929_drop__bit__rec,axiom,
( bit_se8568078237143864401it_int
= ( ^ [N3: nat,A4: int] : ( if_int @ ( N3 = zero_zero_nat ) @ A4 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_930_set__times__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_931_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_932_set__times__intro,axiom,
! [A: num,C2: set_num,B: num,D: set_num] :
( ( member_num @ A @ C2 )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( times_times_num @ A @ B ) @ ( times_times_set_num @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_933_set__times__intro,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_934_insertCI,axiom,
! [A: real,B2: set_real,B: real] :
( ( ~ ( member_real @ A @ B2 )
=> ( A = B ) )
=> ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_935_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_936_insertCI,axiom,
! [A: int,B2: set_int,B: int] :
( ( ~ ( member_int @ A @ B2 )
=> ( A = B ) )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_937_insertCI,axiom,
! [A: num,B2: set_num,B: num] :
( ( ~ ( member_num @ A @ B2 )
=> ( A = B ) )
=> ( member_num @ A @ ( insert_num @ B @ B2 ) ) ) ).
% insertCI
thf(fact_938_insert__iff,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B @ A2 ) )
= ( ( A = B )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_939_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_940_insert__iff,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
= ( ( A = B )
| ( member_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_941_insert__iff,axiom,
! [A: num,B: num,A2: set_num] :
( ( member_num @ A @ ( insert_num @ B @ A2 ) )
= ( ( A = B )
| ( member_num @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_942_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_943_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_944_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_945_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_946_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_947_mult_Oright__neutral,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% mult.right_neutral
thf(fact_948_mult_Oright__neutral,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% mult.right_neutral
thf(fact_949_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_950_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_951_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_952_mult__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% mult_1
thf(fact_953_mult__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 @ A )
= A ) ).
% mult_1
thf(fact_954_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_955_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_956_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_957_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_958_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z ) )
= ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_959_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_960_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_961_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_962_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_963_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_964_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_965_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_966_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_967_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_968_vector__space__over__itself_Oscale__minus__right,axiom,
! [A: real,X: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_969_vector__space__over__itself_Oscale__minus__left,axiom,
! [A: real,X: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ X )
= ( uminus_uminus_real @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_970_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_971_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_972_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_973_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_974_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_975_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_976_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_977_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_978_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_979_singletonI,axiom,
! [A: num] : ( member_num @ A @ ( insert_num @ A @ bot_bot_set_num ) ) ).
% singletonI
thf(fact_980_insert__Diff1,axiom,
! [X: real,B2: set_real,A2: set_real] :
( ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_981_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_982_insert__Diff1,axiom,
! [X: int,B2: set_int,A2: set_int] :
( ( member_int @ X @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
= ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_983_insert__Diff1,axiom,
! [X: num,B2: set_num,A2: set_num] :
( ( member_num @ X @ B2 )
=> ( ( minus_minus_set_num @ ( insert_num @ X @ A2 ) @ B2 )
= ( minus_minus_set_num @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_984_Diff__insert0,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_985_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_986_Diff__insert0,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
= ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_987_Diff__insert0,axiom,
! [X: num,A2: set_num,B2: set_num] :
( ~ ( member_num @ X @ A2 )
=> ( ( minus_minus_set_num @ A2 @ ( insert_num @ X @ B2 ) )
= ( minus_minus_set_num @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_988_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% drop_bit_minus_one
thf(fact_989_drop__bit__drop__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se8568078237143864401it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
= ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% drop_bit_drop_bit
thf(fact_990_set__times__intro2,axiom,
! [B: real,C2: set_real,A: real] :
( ( member_real @ B @ C2 )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( set_el1043507519367895261s_real @ A @ C2 ) ) ) ).
% set_times_intro2
thf(fact_991_set__times__intro2,axiom,
! [B: nat,C2: set_nat,A: nat] :
( ( member_nat @ B @ C2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( set_el2933305810450955905es_nat @ A @ C2 ) ) ) ).
% set_times_intro2
thf(fact_992_set__times__intro2,axiom,
! [B: num,C2: set_num,A: num] :
( ( member_num @ B @ C2 )
=> ( member_num @ ( times_times_num @ A @ B ) @ ( set_el8714009633461510347es_num @ A @ C2 ) ) ) ).
% set_times_intro2
thf(fact_993_set__times__intro2,axiom,
! [B: int,C2: set_int,A: int] :
( ( member_int @ B @ C2 )
=> ( member_int @ ( times_times_int @ A @ B ) @ ( set_el2930815339941905629es_int @ A @ C2 ) ) ) ).
% set_times_intro2
thf(fact_994_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_995_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_996_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_997_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_998_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_999_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1000_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1001_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1002_distrib__right__numeral,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,V: num] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ B @ ( numera7754357348821619680l_num1 @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1003_distrib__right__numeral,axiom,
! [A: extended_enat,B: extended_enat,V: num] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1004_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1005_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1006_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1007_distrib__left__numeral,axiom,
! [V: num,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ B @ C ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1008_distrib__left__numeral,axiom,
! [V: num,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1009_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1010_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1011_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1012_left__diff__distrib__numeral,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,V: num] :
( ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ A @ B ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ B @ ( numera7754357348821619680l_num1 @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1013_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1014_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1015_right__diff__distrib__numeral,axiom,
! [V: num,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( minus_5410813661909488930l_num1 @ B @ C ) )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1016_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1017_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1018_mult__minus1__right,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ Z @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) )
= ( uminus7224005126491068675l_num1 @ Z ) ) ).
% mult_minus1_right
thf(fact_1019_mult__minus1__right,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ Z @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ Z ) ) ).
% mult_minus1_right
thf(fact_1020_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_1021_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_1022_mult__minus1,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) @ Z )
= ( uminus7224005126491068675l_num1 @ Z ) ) ).
% mult_minus1
thf(fact_1023_mult__minus1,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ Z )
= ( uminus1336558196688952754l_num1 @ Z ) ) ).
% mult_minus1
thf(fact_1024_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_1025_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_1026_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y ) )
= ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1027_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1028_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1029_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y ) )
= ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1030_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1031_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1032_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Y ) )
= ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(169)
thf(fact_1033_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(169)
thf(fact_1034_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(169)
thf(fact_1035_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1036_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_1037_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_1038_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1039_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_1040_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_1041_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1042_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_1043_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_1044_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1045_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1046_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1047_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1048_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1049_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1050_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1051_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1052_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1053_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1054_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1055_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1056_set__add__0,axiom,
! [A2: set_real] :
( ( plus_plus_set_real @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_1057_set__add__0,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_1058_set__add__0,axiom,
! [A2: set_int] :
( ( plus_plus_set_int @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_1059_set__add__0__right,axiom,
! [A2: set_real] :
( ( plus_plus_set_real @ A2 @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
= A2 ) ).
% set_add_0_right
thf(fact_1060_set__add__0__right,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= A2 ) ).
% set_add_0_right
thf(fact_1061_set__add__0__right,axiom,
! [A2: set_int] :
( ( plus_plus_set_int @ A2 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= A2 ) ).
% set_add_0_right
thf(fact_1062_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1063_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1064_floor__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_1065_floor__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_1066_floor__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_1067_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_1068_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_1069_insertE,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B @ A2 ) )
=> ( ( A != B )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_1070_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_1071_insertE,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_1072_insertE,axiom,
! [A: num,B: num,A2: set_num] :
( ( member_num @ A @ ( insert_num @ B @ A2 ) )
=> ( ( A != B )
=> ( member_num @ A @ A2 ) ) ) ).
% insertE
thf(fact_1073_insertI1,axiom,
! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% insertI1
thf(fact_1074_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_1075_insertI1,axiom,
! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% insertI1
thf(fact_1076_insertI1,axiom,
! [A: num,B2: set_num] : ( member_num @ A @ ( insert_num @ A @ B2 ) ) ).
% insertI1
thf(fact_1077_insertI2,axiom,
! [A: real,B2: set_real,B: real] :
( ( member_real @ A @ B2 )
=> ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1078_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1079_insertI2,axiom,
! [A: int,B2: set_int,B: int] :
( ( member_int @ A @ B2 )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1080_insertI2,axiom,
! [A: num,B2: set_num,B: num] :
( ( member_num @ A @ B2 )
=> ( member_num @ A @ ( insert_num @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1081_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B5: set_real] :
( ( A2
= ( insert_real @ X @ B5 ) )
=> ( member_real @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1082_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1083_Set_Oset__insert,axiom,
! [X: int,A2: set_int] :
( ( member_int @ X @ A2 )
=> ~ ! [B5: set_int] :
( ( A2
= ( insert_int @ X @ B5 ) )
=> ( member_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1084_Set_Oset__insert,axiom,
! [X: num,A2: set_num] :
( ( member_num @ X @ A2 )
=> ~ ! [B5: set_num] :
( ( A2
= ( insert_num @ X @ B5 ) )
=> ( member_num @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1085_insert__ident,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B2 )
=> ( ( ( insert_real @ X @ A2 )
= ( insert_real @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1086_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1087_insert__ident,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ A2 )
= ( insert_int @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1088_insert__ident,axiom,
! [X: num,A2: set_num,B2: set_num] :
( ~ ( member_num @ X @ A2 )
=> ( ~ ( member_num @ X @ B2 )
=> ( ( ( insert_num @ X @ A2 )
= ( insert_num @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1089_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1090_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1091_insert__absorb,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1092_insert__absorb,axiom,
! [A: num,A2: set_num] :
( ( member_num @ A @ A2 )
=> ( ( insert_num @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1093_insert__eq__iff,axiom,
! [A: real,A2: set_real,B: real,B2: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B @ B2 )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_real] :
( ( A2
= ( insert_real @ B @ C3 ) )
& ~ ( member_real @ B @ C3 )
& ( B2
= ( insert_real @ A @ C3 ) )
& ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1094_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1095_insert__eq__iff,axiom,
! [A: int,A2: set_int,B: int,B2: set_int] :
( ~ ( member_int @ A @ A2 )
=> ( ~ ( member_int @ B @ B2 )
=> ( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_int] :
( ( A2
= ( insert_int @ B @ C3 ) )
& ~ ( member_int @ B @ C3 )
& ( B2
= ( insert_int @ A @ C3 ) )
& ~ ( member_int @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1096_insert__eq__iff,axiom,
! [A: num,A2: set_num,B: num,B2: set_num] :
( ~ ( member_num @ A @ A2 )
=> ( ~ ( member_num @ B @ B2 )
=> ( ( ( insert_num @ A @ A2 )
= ( insert_num @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_num] :
( ( A2
= ( insert_num @ B @ C3 ) )
& ~ ( member_num @ B @ C3 )
& ( B2
= ( insert_num @ A @ C3 ) )
& ~ ( member_num @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1097_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B5: set_real] :
( ( A2
= ( insert_real @ A @ B5 ) )
& ~ ( member_real @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1098_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1099_mk__disjoint__insert,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ? [B5: set_int] :
( ( A2
= ( insert_int @ A @ B5 ) )
& ~ ( member_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1100_mk__disjoint__insert,axiom,
! [A: num,A2: set_num] :
( ( member_num @ A @ A2 )
=> ? [B5: set_num] :
( ( A2
= ( insert_num @ A @ B5 ) )
& ~ ( member_num @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1101_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1102_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1103_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1104_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1105_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1106_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1107_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_1108_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_1109_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_1110_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1111_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1112_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1113_set__times__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A2 @ B2 ) )
=> ~ ! [A3: real,B3: real] :
( ( X
= ( times_times_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1114_set__times__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( times_times_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1115_set__times__elim,axiom,
! [X: num,A2: set_num,B2: set_num] :
( ( member_num @ X @ ( times_times_set_num @ A2 @ B2 ) )
=> ~ ! [A3: num,B3: num] :
( ( X
= ( times_times_num @ A3 @ B3 ) )
=> ( ( member_num @ A3 @ A2 )
=> ~ ( member_num @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1116_set__times__elim,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A2 @ B2 ) )
=> ~ ! [A3: int,B3: int] :
( ( X
= ( times_times_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A2 )
=> ~ ( member_int @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1117_set__times__rearrange,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( times_times_set_real @ ( set_el1043507519367895261s_real @ A @ C2 ) @ ( set_el1043507519367895261s_real @ B @ D ) )
= ( set_el1043507519367895261s_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ).
% set_times_rearrange
thf(fact_1118_set__times__rearrange,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( times_times_set_nat @ ( set_el2933305810450955905es_nat @ A @ C2 ) @ ( set_el2933305810450955905es_nat @ B @ D ) )
= ( set_el2933305810450955905es_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ).
% set_times_rearrange
thf(fact_1119_set__times__rearrange,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( times_times_set_int @ ( set_el2930815339941905629es_int @ A @ C2 ) @ ( set_el2930815339941905629es_int @ B @ D ) )
= ( set_el2930815339941905629es_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C2 @ D ) ) ) ).
% set_times_rearrange
thf(fact_1120_singleton__iff,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1121_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1122_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1123_singleton__iff,axiom,
! [B: num,A: num] :
( ( member_num @ B @ ( insert_num @ A @ bot_bot_set_num ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1124_singletonD,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1125_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1126_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1127_singletonD,axiom,
! [B: num,A: num] :
( ( member_num @ B @ ( insert_num @ A @ bot_bot_set_num ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1128_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_1129_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_1130_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_1131_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_1132_mult__delta__left,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_1133_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_1134_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_1135_mult__delta__right,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_1136_comm__monoid__mult__class_Omult__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1137_comm__monoid__mult__class_Omult__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1138_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1139_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1140_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1141_mult_Ocomm__neutral,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% mult.comm_neutral
thf(fact_1142_mult_Ocomm__neutral,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% mult.comm_neutral
thf(fact_1143_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1144_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1145_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1146_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_1147_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_1148_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D2 ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1149_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1150_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1151_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1152_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1153_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1154_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1155_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1156_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1157_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1158_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_1159_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_1160_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_1161_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1162_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1163_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1164_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1165_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_1166_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_1167_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_1168_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_1169_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1170_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_1171_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_1172_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1173_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1174_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_1175_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_1176_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1177_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1178_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1179_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1180_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1181_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1182_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_1183_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_1184_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_1185_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1186_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1187_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_1188_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1189_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_1190_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3 != zero_zero_int )
=> ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1191_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_1192_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_1193_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_1194_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_1195_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_1196_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X4: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1197_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1198_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1199_zdvd__period,axiom,
! [A: int,D2: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D2 )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_1200_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1201_even__not__iff__int,axiom,
! [K: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% even_not_iff_int
thf(fact_1202_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1203_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1204_bezout__lemma__nat,axiom,
! [D2: nat,A: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D2 @ A )
=> ( ( dvd_dvd_nat @ D2 @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D2 ) ) )
=> ? [X4: nat,Y2: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y2 ) @ D2 ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1205_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X4: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D3 ) )
| ( ( times_times_nat @ B @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1206_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X4: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y2 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y2 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_1207_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_1208_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1209_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_1210_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1211_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1212_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_1213_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_1214_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_1215_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_1216_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_1217_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1218_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_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1227_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_1228_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1229_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_1230_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1231_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_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_1241_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_1242_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1243_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1244_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_1245_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_1246_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1247_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1248_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_1249_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_1250_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_1251_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_1252_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_1253_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_1254_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_1255_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1256_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1257_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_segment__bound__lemma,axiom,
! [B2: real,X: real,Y: real,U: real] :
( ( ord_less_eq_real @ B2 @ X )
=> ( ( ord_less_eq_real @ B2 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X ) @ ( times_times_real @ U @ Y ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1270_dvd__minus__add,axiom,
! [Q: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq_nat @ Q @ N )
=> ( ( ord_less_eq_nat @ Q @ ( times_times_nat @ R2 @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1271_half__bounded__equal,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( times_times_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real )
= ( X
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% half_bounded_equal
thf(fact_1272_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F2: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F2 @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X5 @ I3 ) @ one_one_nat )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X5 @ I3 )
= zero_zero_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X5 @ I3 )
= one_one_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L3 @ X5 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I3 ) @ ( F2 @ X5 @ I3 ) ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L3 @ X5 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F2 @ X5 @ I3 ) @ ( X5 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1273_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)
% Helper facts (7)
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_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( f @ ( plus_plus_real @ xa @ one_one_real ) )
= ( f @ xa ) ) ).
%------------------------------------------------------------------------------