TPTP Problem File: SLH0881^1.p
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- 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 : Median_Method/0000_Median/prob_00418_015627__14863784_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1417 (1097 unt; 144 typ; 0 def)
% Number of atoms : 2225 (1620 equ; 0 cnn)
% Maximal formula atoms : 28 ( 1 avg)
% Number of connectives : 8550 ( 163 ~; 64 |; 71 &;7935 @)
% ( 0 <=>; 317 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 4 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 222 ( 222 >; 0 *; 0 +; 0 <<)
% Number of symbols : 129 ( 126 usr; 21 con; 0-3 aty)
% Number of variables : 2439 ( 86 ^;2329 !; 24 ?;2439 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:46:17.171
%------------------------------------------------------------------------------
% 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__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__Extended____Nonnegative____Real__Oennreal,type,
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thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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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,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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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 ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (126)
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Real__Oereal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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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,
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thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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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,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
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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,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
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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,
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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__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
times_2438108612031896577d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
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__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J,type,
times_3382592830563027286l_num1: set_Nu382503245525567899l_num1 > set_Nu382503245525567899l_num1 > set_Nu382503245525567899l_num1 ).
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__Extended____Real__Oereal,type,
uminus27091377158695749_ereal: extended_ereal > extended_ereal ).
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__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__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
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__Real__Oreal,type,
zero_zero_real: 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_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Omedian_001tf__a,type,
median_a: nat > ( nat > a ) > a ).
thf(sy_c_Num_OBitM,type,
bitM: 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__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__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,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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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,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Num_Onum_OOne,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Parity_Oadjust__mod,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Extended____Real__Oereal,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Int__Oint,type,
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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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_f,type,
f: nat > a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1267)
thf(fact_0_assms_I1_J,axiom,
interval_a @ i ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
( ord_less_nat @ n
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( ord_less_nat @ K @ n )
& ( member_a @ ( f @ K ) @ i ) ) ) ) ) ) ).
% assms(2)
thf(fact_2_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_3_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_4_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_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_14_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_15_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_16_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_17_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_18_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_19_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ S )
& ? [Y: nat] :
( ( member_nat @ Y @ S )
& ( X != Y )
& ! [Z2: nat] :
( ( member_nat @ Z2 @ S )
=> ( ( Z2 = X )
| ( Z2 = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_20_mult__numeral__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_21_mult__numeral__1,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_22_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_23_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_24_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_25_mult__numeral__1__right,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_26_mult__numeral__1__right,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_27_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_28_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_29_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_30_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_31_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_32_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_33_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_34_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_35_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_36_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_37_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_38_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_39_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_40_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_41_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_42_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_43_verit__comp__simplify1_I1_J,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_44_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_45_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_46_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_47_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_48_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_49_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_50_set__times__intro,axiom,
! [A: extended_enat,C: set_Extended_enat,B: extended_enat,D: set_Extended_enat] :
( ( member_Extended_enat @ A @ C )
=> ( ( member_Extended_enat @ B @ D )
=> ( member_Extended_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ ( times_2438108612031896577d_enat @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_51_set__times__intro,axiom,
! [A: real,C: set_real,B: real,D: set_real] :
( ( member_real @ A @ C )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_52_set__times__intro,axiom,
! [A: num,C: set_num,B: num,D: set_num] :
( ( member_num @ A @ C )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( times_times_num @ A @ B ) @ ( times_times_set_num @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_53_set__times__intro,axiom,
! [A: nat,C: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_54_set__times__intro,axiom,
! [A: int,C: set_int,B: int,D: set_int] :
( ( member_int @ A @ C )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_55_set__times__intro,axiom,
! [A: numera4273646738625120315l_num1,C: set_Nu382503245525567899l_num1,B: numera4273646738625120315l_num1,D: set_Nu382503245525567899l_num1] :
( ( member8231810200707485668l_num1 @ A @ C )
=> ( ( member8231810200707485668l_num1 @ B @ D )
=> ( member8231810200707485668l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_3382592830563027286l_num1 @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_56_dbl__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu5816564918971239084l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) )
= ( numera7754357348821619680l_num1 @ ( bit0 @ K2 ) ) ) ).
% dbl_simps(5)
thf(fact_57_dbl__simps_I5_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) )
= ( numeral_numeral_real @ ( bit0 @ K2 ) ) ) ).
% dbl_simps(5)
thf(fact_58_dbl__simps_I5_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) ).
% dbl_simps(5)
thf(fact_59_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_60_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_61_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_62_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_63_mult__commute__abs,axiom,
! [C2: extended_enat] :
( ( ^ [X: extended_enat] : ( times_7803423173614009249d_enat @ X @ C2 ) )
= ( times_7803423173614009249d_enat @ C2 ) ) ).
% mult_commute_abs
thf(fact_64_mult__commute__abs,axiom,
! [C2: real] :
( ( ^ [X: real] : ( times_times_real @ X @ C2 ) )
= ( times_times_real @ C2 ) ) ).
% mult_commute_abs
thf(fact_65_mult__commute__abs,axiom,
! [C2: nat] :
( ( ^ [X: nat] : ( times_times_nat @ X @ C2 ) )
= ( times_times_nat @ C2 ) ) ).
% mult_commute_abs
thf(fact_66_mult__commute__abs,axiom,
! [C2: int] :
( ( ^ [X: int] : ( times_times_int @ X @ C2 ) )
= ( times_times_int @ C2 ) ) ).
% mult_commute_abs
thf(fact_67_mult__commute__abs,axiom,
! [C2: numera4273646738625120315l_num1] :
( ( ^ [X: numera4273646738625120315l_num1] : ( times_2938166955517408246l_num1 @ X @ C2 ) )
= ( times_2938166955517408246l_num1 @ C2 ) ) ).
% mult_commute_abs
thf(fact_68_left__add__twice,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ A @ B ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_69_left__add__twice,axiom,
! [A: extended_enat,B: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_70_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_71_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_72_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_73_mult__2__right,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ Z @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2_right
thf(fact_74_mult__2__right,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_75_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_76_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_77_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_78_mult__2,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_79_mult__2,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2
thf(fact_80_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_81_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_82_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_83_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_84_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_set__plus__intro,axiom,
! [A: extended_enat,C: set_Extended_enat,B: extended_enat,D: set_Extended_enat] :
( ( member_Extended_enat @ A @ C )
=> ( ( member_Extended_enat @ B @ D )
=> ( member_Extended_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3482335003337316477d_enat @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_95_set__plus__intro,axiom,
! [A: real,C: set_real,B: real,D: set_real] :
( ( member_real @ A @ C )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_96_set__plus__intro,axiom,
! [A: num,C: set_num,B: num,D: set_num] :
( ( member_num @ A @ C )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( plus_plus_num @ A @ B ) @ ( plus_plus_set_num @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_97_set__plus__intro,axiom,
! [A: nat,C: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_98_set__plus__intro,axiom,
! [A: int,C: set_int,B: int,D: set_int] :
( ( member_int @ A @ C )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_set_int @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_99_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_100_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_101_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_115_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_119_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_120_distrib__left__numeral,axiom,
! [V: num,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ B @ C2 ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_121_distrib__left__numeral,axiom,
! [V: num,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C2 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_122_distrib__left__numeral,axiom,
! [V: num,B: real,C2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_123_distrib__left__numeral,axiom,
! [V: num,B: nat,C2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_124_distrib__left__numeral,axiom,
! [V: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_125_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_126_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_127_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_128_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_129_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_130_mult__minus1__right,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ Z @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ Z ) ) ).
% mult_minus1_right
thf(fact_131_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_132_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_133_mult__minus1,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ Z )
= ( uminus1336558196688952754l_num1 @ Z ) ) ).
% mult_minus1
thf(fact_134_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_135_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_136_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_137_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_138_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_139_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_140_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_141_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_142_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_143_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_144_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y3: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y3 ) )
= ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(167)
thf(fact_145_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(167)
thf(fact_146_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(167)
thf(fact_147_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_148_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_149_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_150_dbl__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu5816564918971239084l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu5816564918971239084l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) ) ) ).
% dbl_simps(1)
thf(fact_151_dbl__simps_I1_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).
% dbl_simps(1)
thf(fact_152_dbl__simps_I1_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
= ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).
% dbl_simps(1)
thf(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y3: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y3 ) )
= ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% semiring_norm(171)
thf(fact_168_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y3: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% semiring_norm(171)
thf(fact_169_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y3: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% semiring_norm(171)
thf(fact_170_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y3: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y3 ) )
= ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(170)
thf(fact_171_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y3: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(170)
thf(fact_172_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y3: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(170)
thf(fact_173_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y3: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Y3 ) )
= ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(169)
thf(fact_174_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y3: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(169)
thf(fact_175_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y3: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(169)
thf(fact_176_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_177_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_178_one__add__one,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_179_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_180_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_181_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_182_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_183_dbl__simps_I3_J,axiom,
( ( neg_nu5816564918971239084l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_184_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_185_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_186_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_187_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_188_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_189_dbl__simps_I4_J,axiom,
( ( neg_nu5816564918971239084l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_190_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_191_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_192_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_193_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_194_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_195_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% dbl_def
thf(fact_196_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% dbl_def
thf(fact_197_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_198_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_199_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_200_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_201_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_202_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_203_set__plus__elim,axiom,
! [X4: extended_enat,A2: set_Extended_enat,B2: set_Extended_enat] :
( ( member_Extended_enat @ X4 @ ( plus_p3482335003337316477d_enat @ A2 @ B2 ) )
=> ~ ! [A3: extended_enat,B3: extended_enat] :
( ( X4
= ( plus_p3455044024723400733d_enat @ A3 @ B3 ) )
=> ( ( member_Extended_enat @ A3 @ A2 )
=> ~ ( member_Extended_enat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_204_set__plus__elim,axiom,
! [X4: real,A2: set_real,B2: set_real] :
( ( member_real @ X4 @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A3: real,B3: real] :
( ( X4
= ( plus_plus_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_205_set__plus__elim,axiom,
! [X4: num,A2: set_num,B2: set_num] :
( ( member_num @ X4 @ ( plus_plus_set_num @ A2 @ B2 ) )
=> ~ ! [A3: num,B3: num] :
( ( X4
= ( plus_plus_num @ A3 @ B3 ) )
=> ( ( member_num @ A3 @ A2 )
=> ~ ( member_num @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_206_set__plus__elim,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X4 @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X4
= ( plus_plus_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_207_set__plus__elim,axiom,
! [X4: int,A2: set_int,B2: set_int] :
( ( member_int @ X4 @ ( plus_plus_set_int @ A2 @ B2 ) )
=> ~ ! [A3: int,B3: int] :
( ( X4
= ( plus_plus_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A2 )
=> ~ ( member_int @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_208_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_209_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_210_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_211_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_212_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_213_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_214_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_215_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_216_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) @ one_on7795324986448017462l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_217_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X4 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X4 ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_218_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X4 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_219_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_220_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_221_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_222_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_223_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_224_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_less_numeral
thf(fact_225_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_226_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_227_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_228_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_229_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_230_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_231_uminus__numeral__One,axiom,
( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) )
= ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% uminus_numeral_One
thf(fact_232_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_233_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_234_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_235_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_236_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_237_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_238_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_239_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_240_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_241_less__numeral__extra_I4_J,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).
% less_numeral_extra(4)
thf(fact_242_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_243_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_244_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_245_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_246_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_247_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_248_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_249_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_250_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_251_numeral__times__minus__swap,axiom,
! [W: num,X4: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ ( uminus1336558196688952754l_num1 @ X4 ) )
= ( times_2938166955517408246l_num1 @ X4 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_252_numeral__times__minus__swap,axiom,
! [W: num,X4: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X4 ) )
= ( times_times_int @ X4 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_253_numeral__times__minus__swap,axiom,
! [W: num,X4: real] :
( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X4 ) )
= ( times_times_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_254_numeral__Bit0,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ).
% numeral_Bit0
thf(fact_255_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_256_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_257_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_258_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_259_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% not_numeral_less_one
thf(fact_260_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_261_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_262_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_263_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_264_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_265_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_266_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_267_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_268_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_269_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ).
% numeral_code(2)
thf(fact_270_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_code(2)
thf(fact_271_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_272_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_273_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_274_mult__1s__ring__1_I2_J,axiom,
! [B: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ B @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) ) )
= ( uminus1336558196688952754l_num1 @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_275_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_276_mult__1s__ring__1_I2_J,axiom,
! [B: real] :
( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_277_mult__1s__ring__1_I1_J,axiom,
! [B: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) ) @ B )
= ( uminus1336558196688952754l_num1 @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_278_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_279_mult__1s__ring__1_I1_J,axiom,
! [B: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_280_set__times__elim,axiom,
! [X4: extended_enat,A2: set_Extended_enat,B2: set_Extended_enat] :
( ( member_Extended_enat @ X4 @ ( times_2438108612031896577d_enat @ A2 @ B2 ) )
=> ~ ! [A3: extended_enat,B3: extended_enat] :
( ( X4
= ( times_7803423173614009249d_enat @ A3 @ B3 ) )
=> ( ( member_Extended_enat @ A3 @ A2 )
=> ~ ( member_Extended_enat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_281_set__times__elim,axiom,
! [X4: real,A2: set_real,B2: set_real] :
( ( member_real @ X4 @ ( times_times_set_real @ A2 @ B2 ) )
=> ~ ! [A3: real,B3: real] :
( ( X4
= ( times_times_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_282_set__times__elim,axiom,
! [X4: num,A2: set_num,B2: set_num] :
( ( member_num @ X4 @ ( times_times_set_num @ A2 @ B2 ) )
=> ~ ! [A3: num,B3: num] :
( ( X4
= ( times_times_num @ A3 @ B3 ) )
=> ( ( member_num @ A3 @ A2 )
=> ~ ( member_num @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_283_set__times__elim,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X4 @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X4
= ( times_times_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_284_set__times__elim,axiom,
! [X4: int,A2: set_int,B2: set_int] :
( ( member_int @ X4 @ ( times_times_set_int @ A2 @ B2 ) )
=> ~ ! [A3: int,B3: int] :
( ( X4
= ( times_times_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A2 )
=> ~ ( member_int @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_285_set__times__elim,axiom,
! [X4: numera4273646738625120315l_num1,A2: set_Nu382503245525567899l_num1,B2: set_Nu382503245525567899l_num1] :
( ( member8231810200707485668l_num1 @ X4 @ ( times_3382592830563027286l_num1 @ A2 @ B2 ) )
=> ~ ! [A3: numera4273646738625120315l_num1,B3: numera4273646738625120315l_num1] :
( ( X4
= ( times_2938166955517408246l_num1 @ A3 @ B3 ) )
=> ( ( member8231810200707485668l_num1 @ A3 @ A2 )
=> ~ ( member8231810200707485668l_num1 @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_286_one__less__numeral,axiom,
! [N: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral
thf(fact_287_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_288_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_289_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_290_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_291_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_292_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_293_mult__minus__left,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ A ) @ B )
= ( uminus1336558196688952754l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_294_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_295_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_296_minus__mult__minus,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ A ) @ ( uminus1336558196688952754l_num1 @ B ) )
= ( times_2938166955517408246l_num1 @ A @ B ) ) ).
% minus_mult_minus
thf(fact_297_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_298_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_299_mult__minus__right,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( uminus1336558196688952754l_num1 @ B ) )
= ( uminus1336558196688952754l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_300_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_301_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_302_vector__space__over__itself_Oscale__minus__left,axiom,
! [A: real,X4: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ X4 )
= ( uminus_uminus_real @ ( times_times_real @ A @ X4 ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_303_vector__space__over__itself_Oscale__minus__right,axiom,
! [A: real,X4: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ X4 ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ X4 ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_304_sum__sqs__eq,axiom,
! [X4: real,Y3: real] :
( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
= ( times_times_real @ X4 @ ( times_times_real @ Y3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( Y3 = X4 ) ) ).
% sum_sqs_eq
thf(fact_305_sum__sqs__eq,axiom,
! [X4: int,Y3: int] :
( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
= ( times_times_int @ X4 @ ( times_times_int @ Y3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( Y3 = X4 ) ) ).
% sum_sqs_eq
thf(fact_306_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_307_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_308_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_309_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_310_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_311_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_312_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_313_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_314_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_315_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_316_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_317_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_318_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_319_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_320_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_321_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_322_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_323_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_324_mult__1,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat @ A )
= A ) ).
% mult_1
thf(fact_325_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_326_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_327_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_328_mult__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 @ A )
= A ) ).
% mult_1
thf(fact_329_vector__space__over__itself_Oscale__one,axiom,
! [X4: real] :
( ( times_times_real @ one_one_real @ X4 )
= X4 ) ).
% vector_space_over_itself.scale_one
thf(fact_330_mult_Oright__neutral,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ one_on7984719198319812577d_enat )
= A ) ).
% mult.right_neutral
thf(fact_331_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_332_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_333_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_334_mult_Oright__neutral,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% mult.right_neutral
thf(fact_335_linorder__neqE__linordered__idom,axiom,
! [X4: real,Y3: real] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ Y3 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_336_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y3: int] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ Y3 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_337_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X4 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X4 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_338_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X4 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X4 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_339_mult_Oleft__commute,axiom,
! [B: extended_enat,A: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ B @ ( times_7803423173614009249d_enat @ A @ C2 ) )
= ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_340_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_341_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_342_mult_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_343_mult_Oleft__commute,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ B @ ( times_2938166955517408246l_num1 @ A @ C2 ) )
= ( times_2938166955517408246l_num1 @ A @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_344_mult_Ocommute,axiom,
( times_7803423173614009249d_enat
= ( ^ [A4: extended_enat,B4: extended_enat] : ( times_7803423173614009249d_enat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_345_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_346_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_347_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_348_mult_Ocommute,axiom,
( times_2938166955517408246l_num1
= ( ^ [A4: numera4273646738625120315l_num1,B4: numera4273646738625120315l_num1] : ( times_2938166955517408246l_num1 @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_349_mult_Oassoc,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C2 )
= ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_350_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_351_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_352_mult_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_353_mult_Oassoc,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ C2 )
= ( times_2938166955517408246l_num1 @ A @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_354_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C2 )
= ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_355_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_356_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_357_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_358_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ C2 )
= ( times_2938166955517408246l_num1 @ A @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_359_one__reorient,axiom,
! [X4: numera4273646738625120315l_num1] :
( ( one_on7795324986448017462l_num1 = X4 )
= ( X4 = one_on7795324986448017462l_num1 ) ) ).
% one_reorient
thf(fact_360_one__reorient,axiom,
! [X4: extended_enat] :
( ( one_on7984719198319812577d_enat = X4 )
= ( X4 = one_on7984719198319812577d_enat ) ) ).
% one_reorient
thf(fact_361_one__reorient,axiom,
! [X4: real] :
( ( one_one_real = X4 )
= ( X4 = one_one_real ) ) ).
% one_reorient
thf(fact_362_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_363_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_364_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_365_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_366_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_367_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_368_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_369_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_370_add_Oleft__commute,axiom,
! [B: extended_enat,A: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C2 ) )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_371_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_372_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_373_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_374_add_Ocommute,axiom,
( plus_p3455044024723400733d_enat
= ( ^ [A4: extended_enat,B4: extended_enat] : ( plus_p3455044024723400733d_enat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_375_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_376_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_377_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_378_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_379_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_380_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_381_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_382_add_Oassoc,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C2 )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_383_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_384_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_385_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_386_group__cancel_Oadd2,axiom,
! [B2: extended_enat,K2: extended_enat,B: extended_enat,A: extended_enat] :
( ( B2
= ( plus_p3455044024723400733d_enat @ K2 @ B ) )
=> ( ( plus_p3455044024723400733d_enat @ A @ B2 )
= ( plus_p3455044024723400733d_enat @ K2 @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_387_group__cancel_Oadd2,axiom,
! [B2: real,K2: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K2 @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_388_group__cancel_Oadd2,axiom,
! [B2: nat,K2: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_389_group__cancel_Oadd2,axiom,
! [B2: int,K2: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_390_group__cancel_Oadd1,axiom,
! [A2: extended_enat,K2: extended_enat,A: extended_enat,B: extended_enat] :
( ( A2
= ( plus_p3455044024723400733d_enat @ K2 @ A ) )
=> ( ( plus_p3455044024723400733d_enat @ A2 @ B )
= ( plus_p3455044024723400733d_enat @ K2 @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_391_group__cancel_Oadd1,axiom,
! [A2: real,K2: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K2 @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_392_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_393_group__cancel_Oadd1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_394_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: extended_enat,J: extended_enat,K2: extended_enat,L: extended_enat] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_p3455044024723400733d_enat @ I2 @ K2 )
= ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_395_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_real @ I2 @ K2 )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_396_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I2 @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_397_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I2 @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_398_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C2 )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_399_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_400_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_401_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_402_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_403_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_404_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_405_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_406_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_407_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_408_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_409_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_410_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_411_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_412_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_413_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_414_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_415_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_416_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_417_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_418_add__strict__mono,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat,D2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ C2 @ D2 )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ C2 ) @ ( plus_p3455044024723400733d_enat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_419_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_420_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_421_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_422_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_423_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_424_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_425_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_426_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_427_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_428_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_429_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_430_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_431_mult_Ocomm__neutral,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ one_on7984719198319812577d_enat )
= A ) ).
% mult.comm_neutral
thf(fact_432_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_433_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_434_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_435_mult_Ocomm__neutral,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% mult.comm_neutral
thf(fact_436_comm__monoid__mult__class_Omult__1,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_437_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_438_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_439_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_440_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_441_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X4: real,Y3: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X4 @ Y3 ) )
= ( plus_plus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_442_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X4 )
= ( plus_plus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ B @ X4 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_443_combine__common__factor,axiom,
! [A: extended_enat,E: extended_enat,B: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ E ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ B @ E ) @ C2 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_444_combine__common__factor,axiom,
! [A: real,E: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_445_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_446_combine__common__factor,axiom,
! [A: int,E: int,B: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_447_combine__common__factor,axiom,
! [A: numera4273646738625120315l_num1,E: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ E ) @ ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ B @ E ) @ C2 ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_448_distrib__right,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C2 )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C2 ) @ ( times_7803423173614009249d_enat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_449_distrib__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_450_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_451_distrib__right,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_452_distrib__right,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C2 )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ C2 ) @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_453_distrib__left,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C2 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ ( times_7803423173614009249d_enat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_454_distrib__left,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_455_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_456_distrib__left,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_457_distrib__left,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ B @ C2 ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_2938166955517408246l_num1 @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_458_comm__semiring__class_Odistrib,axiom,
! [A: extended_enat,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C2 )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C2 ) @ ( times_7803423173614009249d_enat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_459_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_460_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_461_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_462_comm__semiring__class_Odistrib,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C2 )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ C2 ) @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_463_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_464_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_465_ring__class_Oring__distribs_I1_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ B @ C2 ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_2938166955517408246l_num1 @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_466_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_467_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_468_ring__class_Oring__distribs_I2_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C2 )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ C2 ) @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_469_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_470_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_471_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_472_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_473_minus__mult__commute,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( uminus1336558196688952754l_num1 @ A ) @ B )
= ( times_2938166955517408246l_num1 @ A @ ( uminus1336558196688952754l_num1 @ B ) ) ) ).
% minus_mult_commute
thf(fact_474_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_475_minus__mult__commute,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_476_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_477_square__eq__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_478_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_479_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_480_group__cancel_Oneg1,axiom,
! [A2: int,K2: int,A: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_481_group__cancel_Oneg1,axiom,
! [A2: real,K2: real,A: real] :
( ( A2
= ( plus_plus_real @ K2 @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_482_exhaust__2,axiom,
! [X4: numera2417102609627094330l_num1] :
( ( X4 = one_on3868389512446148991l_num1 )
| ( X4
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_483_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X5: numera2417102609627094330l_num1] : ( P2 @ X5 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_484_lambda__one,axiom,
( ( ^ [X: extended_enat] : X )
= ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat ) ) ).
% lambda_one
thf(fact_485_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_486_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_487_lambda__one,axiom,
( ( ^ [X: int] : X )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_488_lambda__one,axiom,
( ( ^ [X: numera4273646738625120315l_num1] : X )
= ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% lambda_one
thf(fact_489_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_490_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_491_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_492_add__mono1,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ B @ one_on7984719198319812577d_enat ) ) ) ).
% add_mono1
thf(fact_493_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_494_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_495_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_496_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_497_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_498_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_499_square__eq__1__iff,axiom,
! [X4: int] :
( ( ( times_times_int @ X4 @ X4 )
= one_one_int )
= ( ( X4 = one_one_int )
| ( X4
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_500_square__eq__1__iff,axiom,
! [X4: real] :
( ( ( times_times_real @ X4 @ X4 )
= one_one_real )
= ( ( X4 = one_one_real )
| ( X4
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_501_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_502_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_503_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_504_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_505_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_506_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_507_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_508_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_509_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_510_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_511_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_512_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_513_linorder__neqE__nat,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_514_add__lessD1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
=> ( ord_less_nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_515_add__less__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_516_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_517_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_518_add__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_519_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_520_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_521_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_522_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_523_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_524_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_eq__numeral__iff__iszero_I7_J,axiom,
! [X4: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) )
= one_on7795324986448017462l_num1 )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ X4 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_538_eq__numeral__iff__iszero_I7_J,axiom,
! [X4: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= one_one_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X4 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_539_eq__numeral__iff__iszero_I7_J,axiom,
! [X4: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) )
= one_one_real )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ X4 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_540_eq__numeral__iff__iszero_I8_J,axiom,
! [Y3: num] :
( ( one_on7795324986448017462l_num1
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y3 ) ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_541_eq__numeral__iff__iszero_I8_J,axiom,
! [Y3: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_542_eq__numeral__iff__iszero_I8_J,axiom,
! [Y3: num] :
( ( one_one_real
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y3 ) ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_543_crossproduct__eq,axiom,
! [W: real,Y3: real,X4: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X4 @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X4 @ Y3 ) ) )
= ( ( W = X4 )
| ( Y3 = Z ) ) ) ).
% crossproduct_eq
thf(fact_544_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X4: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X4 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y3 ) ) )
= ( ( W = X4 )
| ( Y3 = Z ) ) ) ).
% crossproduct_eq
thf(fact_545_crossproduct__eq,axiom,
! [W: int,Y3: int,X4: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X4 @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X4 @ Y3 ) ) )
= ( ( W = X4 )
| ( Y3 = Z ) ) ) ).
% crossproduct_eq
thf(fact_546_crossproduct__noteq,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ( A != B )
& ( C2 != D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D2 ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_547_crossproduct__noteq,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ( A != B )
& ( C2 != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_548_crossproduct__noteq,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ( A != B )
& ( C2 != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_549_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_550_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_551_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_552_sub__num__simps_I6_J,axiom,
! [K2: num,L: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K2 ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K2 @ L ) ) ) ).
% sub_num_simps(6)
thf(fact_553_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_554_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_555_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_556_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_557_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_558_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_559_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y3: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y3 ) )
= ( plus_p1441664204671982194l_num1 @ ( neg_nu3067386718351260922l_num1 @ V @ W ) @ Y3 ) ) ).
% semiring_norm(165)
thf(fact_560_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y3: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y3 ) ) ).
% semiring_norm(165)
thf(fact_561_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y3: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
= ( plus_plus_real @ ( neg_numeral_sub_real @ V @ W ) @ Y3 ) ) ).
% semiring_norm(165)
thf(fact_562_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y3: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Y3 ) )
= ( plus_p1441664204671982194l_num1 @ ( neg_nu3067386718351260922l_num1 @ W @ V ) @ Y3 ) ) ).
% semiring_norm(166)
thf(fact_563_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y3 ) ) ).
% semiring_norm(166)
thf(fact_564_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
= ( plus_plus_real @ ( neg_numeral_sub_real @ W @ V ) @ Y3 ) ) ).
% semiring_norm(166)
thf(fact_565_eq__numeral__iff__iszero_I1_J,axiom,
! [X4: num,Y3: num] :
( ( ( numera7754357348821619680l_num1 @ X4 )
= ( numera7754357348821619680l_num1 @ Y3 ) )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ X4 @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_566_eq__numeral__iff__iszero_I1_J,axiom,
! [X4: num,Y3: num] :
( ( ( numeral_numeral_real @ X4 )
= ( numeral_numeral_real @ Y3 ) )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ X4 @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_567_eq__numeral__iff__iszero_I1_J,axiom,
! [X4: num,Y3: num] :
( ( ( numeral_numeral_int @ X4 )
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X4 @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_568_eq__numeral__iff__iszero_I4_J,axiom,
! [X4: num,Y3: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y3 ) ) )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ Y3 @ X4 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_569_eq__numeral__iff__iszero_I4_J,axiom,
! [X4: num,Y3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y3 @ X4 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_570_eq__numeral__iff__iszero_I4_J,axiom,
! [X4: num,Y3: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y3 ) ) )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ Y3 @ X4 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_571_not__iszero__1,axiom,
~ ( ring_14756928180729810983l_num1 @ one_on7795324986448017462l_num1 ) ).
% not_iszero_1
thf(fact_572_not__iszero__1,axiom,
~ ( ring_1_iszero_real @ one_one_real ) ).
% not_iszero_1
thf(fact_573_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_574_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_real @ ( numeral_numeral_real @ W ) ) ).
% not_iszero_numeral
thf(fact_575_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ).
% not_iszero_numeral
thf(fact_576_eq__numeral__iff__iszero_I5_J,axiom,
! [X4: num] :
( ( ( numera7754357348821619680l_num1 @ X4 )
= one_on7795324986448017462l_num1 )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ X4 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_577_eq__numeral__iff__iszero_I5_J,axiom,
! [X4: num] :
( ( ( numeral_numeral_real @ X4 )
= one_one_real )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ X4 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_578_eq__numeral__iff__iszero_I5_J,axiom,
! [X4: num] :
( ( ( numeral_numeral_int @ X4 )
= one_one_int )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X4 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_579_eq__numeral__iff__iszero_I6_J,axiom,
! [Y3: num] :
( ( one_on7795324986448017462l_num1
= ( numera7754357348821619680l_num1 @ Y3 ) )
= ( ring_14756928180729810983l_num1 @ ( neg_nu3067386718351260922l_num1 @ one @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_580_eq__numeral__iff__iszero_I6_J,axiom,
! [Y3: num] :
( ( one_one_real
= ( numeral_numeral_real @ Y3 ) )
= ( ring_1_iszero_real @ ( neg_numeral_sub_real @ one @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_581_eq__numeral__iff__iszero_I6_J,axiom,
! [Y3: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_582_not__iszero__neg__1,axiom,
~ ( ring_14756928180729810983l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% not_iszero_neg_1
thf(fact_583_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_584_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_iszero_neg_1
thf(fact_585_not__iszero__Numeral1,axiom,
~ ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ one ) ) ).
% not_iszero_Numeral1
thf(fact_586_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_real @ ( numeral_numeral_real @ one ) ) ).
% not_iszero_Numeral1
thf(fact_587_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ one ) ) ).
% not_iszero_Numeral1
thf(fact_588_not__iszero__neg__Numeral1,axiom,
~ ( ring_14756928180729810983l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_589_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_590_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_591_eq__numeral__iff__iszero_I2_J,axiom,
! [X4: num,Y3: num] :
( ( ( numera7754357348821619680l_num1 @ X4 )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ Y3 ) ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ X4 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_592_eq__numeral__iff__iszero_I2_J,axiom,
! [X4: num,Y3: num] :
( ( ( numeral_numeral_int @ X4 )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X4 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_593_eq__numeral__iff__iszero_I2_J,axiom,
! [X4: num,Y3: num] :
( ( ( numeral_numeral_real @ X4 )
= ( uminus_uminus_real @ ( numeral_numeral_real @ Y3 ) ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ X4 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_594_eq__numeral__iff__iszero_I3_J,axiom,
! [X4: num,Y3: num] :
( ( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) )
= ( numera7754357348821619680l_num1 @ Y3 ) )
= ( ring_14756928180729810983l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ X4 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_595_eq__numeral__iff__iszero_I3_J,axiom,
! [X4: num,Y3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) )
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X4 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_596_eq__numeral__iff__iszero_I3_J,axiom,
! [X4: num,Y3: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) )
= ( numeral_numeral_real @ Y3 ) )
= ( ring_1_iszero_real @ ( numeral_numeral_real @ ( plus_plus_num @ X4 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_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_605_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_606_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_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_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_617_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_618_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_619_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_620_diff__diff__left,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_621_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_622_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_623_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_624_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_625_add__diff__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_626_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_627_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_628_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_629_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_630_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_631_add__diff__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_632_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_633_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_634_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_635_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_636_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_637_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_638_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_639_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_640_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_641_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_642_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_643_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_644_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_645_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_646_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_647_right__diff__distrib__numeral,axiom,
! [V: num,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( minus_5410813661909488930l_num1 @ B @ C2 ) )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C2 ) ) ) ).
% right_diff_distrib_numeral
thf(fact_648_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C2 ) ) ) ).
% right_diff_distrib_numeral
thf(fact_649_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).
% right_diff_distrib_numeral
thf(fact_650_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_651_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_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_662_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_663_sub__num__simps_I9_J,axiom,
! [K2: num,L: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K2 ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K2 @ L ) ) ) ).
% sub_num_simps(9)
thf(fact_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_675_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_676_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_677_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_sub__num__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu3067386718351260922l_num1 @ ( bit1 @ K2 ) @ one )
= ( numera7754357348821619680l_num1 @ ( bit0 @ K2 ) ) ) ).
% sub_num_simps(5)
thf(fact_686_sub__num__simps_I5_J,axiom,
! [K2: num] :
( ( neg_numeral_sub_real @ ( bit1 @ K2 ) @ one )
= ( numeral_numeral_real @ ( bit0 @ K2 ) ) ) ).
% sub_num_simps(5)
thf(fact_687_sub__num__simps_I5_J,axiom,
! [K2: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K2 ) @ one )
= ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) ).
% sub_num_simps(5)
thf(fact_688_sub__num__simps_I4_J,axiom,
! [K2: num] :
( ( neg_nu3067386718351260922l_num1 @ ( bit0 @ K2 ) @ one )
= ( numera7754357348821619680l_num1 @ ( bitM @ K2 ) ) ) ).
% sub_num_simps(4)
thf(fact_689_sub__num__simps_I4_J,axiom,
! [K2: num] :
( ( neg_numeral_sub_real @ ( bit0 @ K2 ) @ one )
= ( numeral_numeral_real @ ( bitM @ K2 ) ) ) ).
% sub_num_simps(4)
thf(fact_690_sub__num__simps_I4_J,axiom,
! [K2: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K2 ) @ one )
= ( numeral_numeral_int @ ( bitM @ K2 ) ) ) ).
% sub_num_simps(4)
thf(fact_691_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_692_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_693_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_694_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_695_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_696_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_697_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_698_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_699_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_700_diff__eq__diff__eq,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D2 ) )
=> ( ( A = B )
= ( C2 = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_701_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( A = B )
= ( C2 = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_702_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_703_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_704_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_705_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_706_diff__strict__mono,axiom,
! [A: real,B: real,D2: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D2 @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_707_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_708_diff__eq__diff__less,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D2 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_709_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_710_diff__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_711_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_712_diff__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_713_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_714_left__diff__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_715_left__diff__distrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_716_left__diff__distrib,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ A @ B ) @ C2 )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ A @ C2 ) @ ( times_2938166955517408246l_num1 @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_717_right__diff__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_718_right__diff__distrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_719_right__diff__distrib,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( minus_5410813661909488930l_num1 @ B @ C2 ) )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_2938166955517408246l_num1 @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_720_left__diff__distrib_H,axiom,
! [B: real,C2: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C2 ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_721_left__diff__distrib_H,axiom,
! [B: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_722_left__diff__distrib_H,axiom,
! [B: int,C2: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C2 ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_723_left__diff__distrib_H,axiom,
! [B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ B @ C2 ) @ A )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ B @ A ) @ ( times_2938166955517408246l_num1 @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_724_right__diff__distrib_H,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_725_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_726_right__diff__distrib_H,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_727_right__diff__distrib_H,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( minus_5410813661909488930l_num1 @ B @ C2 ) )
= ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_2938166955517408246l_num1 @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_728_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X4 )
= ( minus_minus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ B @ X4 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_729_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: real,X4: real,Y3: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X4 @ Y3 ) )
= ( minus_minus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_730_group__cancel_Osub1,axiom,
! [A2: real,K2: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K2 @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K2 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_731_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_732_diff__eq__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_733_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_734_eq__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( A
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_735_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_736_add__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_737_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_738_diff__diff__eq2,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_739_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_740_diff__add__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_741_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_742_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_743_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_744_add__implies__diff,axiom,
! [C2: real,B: real,A: real] :
( ( ( plus_plus_real @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_745_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_746_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_747_diff__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_748_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_749_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_750_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_751_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_752_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_753_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_754_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_755_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_756_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_757_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_758_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_759_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_760_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_761_eq__iff__iszero__diff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X: real,Y: real] : ( ring_1_iszero_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_762_eq__iff__iszero__diff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X: int,Y: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X @ Y ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_763_numeral__BitM,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bitM @ N ) )
= ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_BitM
thf(fact_764_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_765_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_766_num_Oexhaust,axiom,
! [Y3: num] :
( ( Y3 != one )
=> ( ! [X22: num] :
( Y3
!= ( bit0 @ X22 ) )
=> ~ ! [X33: num] :
( Y3
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_767_forall__4,axiom,
( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
! [X5: numera4273646738625120315l_num1] : ( P2 @ X5 ) )
= ( ^ [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_768_exhaust__4,axiom,
! [X4: numera4273646738625120315l_num1] :
( ( X4 = one_on7795324986448017462l_num1 )
| ( X4
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X4
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X4
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_769_forall__3,axiom,
( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
! [X5: numera6367994245245682809l_num1] : ( P2 @ X5 ) )
= ( ^ [P3: numera6367994245245682809l_num1 > $o] :
( ( P3 @ one_on7819281148064737470l_num1 )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_770_exhaust__3,axiom,
! [X4: numera6367994245245682809l_num1] :
( ( X4 = one_on7819281148064737470l_num1 )
| ( X4
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X4
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_771_diff__less__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_772_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_773_less__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_774_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_775_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_776_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_777_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_778_eq__add__iff1,axiom,
! [A: real,E: real,C2: real,B: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_779_eq__add__iff1,axiom,
! [A: int,E: int,C2: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_780_eq__add__iff1,axiom,
! [A: numera4273646738625120315l_num1,E: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,D2: numera4273646738625120315l_num1] :
( ( ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ E ) @ C2 )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ B @ E ) @ D2 ) )
= ( ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ A @ B ) @ E ) @ C2 )
= D2 ) ) ).
% eq_add_iff1
thf(fact_781_eq__add__iff2,axiom,
! [A: real,E: real,C2: real,B: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_782_eq__add__iff2,axiom,
! [A: int,E: int,C2: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( C2
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_783_eq__add__iff2,axiom,
! [A: numera4273646738625120315l_num1,E: numera4273646738625120315l_num1,C2: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,D2: numera4273646738625120315l_num1] :
( ( ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ E ) @ C2 )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ B @ E ) @ D2 ) )
= ( C2
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ B @ A ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_784_square__diff__square__factored,axiom,
! [X4: real,Y3: real] :
( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
= ( times_times_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ X4 @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_785_square__diff__square__factored,axiom,
! [X4: int,Y3: int] :
( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
= ( times_times_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( minus_minus_int @ X4 @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_786_square__diff__square__factored,axiom,
! [X4: numera4273646738625120315l_num1,Y3: numera4273646738625120315l_num1] :
( ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ X4 @ X4 ) @ ( times_2938166955517408246l_num1 @ Y3 @ Y3 ) )
= ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ X4 @ Y3 ) @ ( minus_5410813661909488930l_num1 @ X4 @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_787_group__cancel_Osub2,axiom,
! [B2: int,K2: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K2 @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_788_group__cancel_Osub2,axiom,
! [B2: real,K2: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K2 @ B ) )
=> ( ( minus_minus_real @ A @ B2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_789_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_790_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_791_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_792_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_793_pth__2,axiom,
( minus_minus_real
= ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% pth_2
thf(fact_794_less__diff__conv,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_795_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_796_neg__numeral__class_Osub__def,axiom,
( neg_nu3067386718351260922l_num1
= ( ^ [K: num,L2: num] : ( minus_5410813661909488930l_num1 @ ( numera7754357348821619680l_num1 @ K ) @ ( numera7754357348821619680l_num1 @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_797_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_real
= ( ^ [K: num,L2: num] : ( minus_minus_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_real @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_798_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_799_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_800_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_801_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_802_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_803_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_804_less__add__iff1,axiom,
! [A: real,E: real,C2: real,B: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_805_less__add__iff1,axiom,
! [A: int,E: int,C2: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_806_less__add__iff2,axiom,
! [A: real,E: real,C2: real,B: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_807_less__add__iff2,axiom,
! [A: int,E: int,C2: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_808_square__diff__one__factored,axiom,
! [X4: real] :
( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X4 @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_809_square__diff__one__factored,axiom,
! [X4: int] :
( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X4 @ one_one_int ) @ ( minus_minus_int @ X4 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_810_square__diff__one__factored,axiom,
! [X4: numera4273646738625120315l_num1] :
( ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ X4 @ X4 ) @ one_on7795324986448017462l_num1 )
= ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ X4 @ one_on7795324986448017462l_num1 ) @ ( minus_5410813661909488930l_num1 @ X4 @ one_on7795324986448017462l_num1 ) ) ) ).
% square_diff_one_factored
thf(fact_811_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit1 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_code(3)
thf(fact_812_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_code(3)
thf(fact_813_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_814_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_815_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_816_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_817_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_818_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_819_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_820_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_821_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5172728937851396970l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_822_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_823_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_824_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_825_mult__diff__mult,axiom,
! [X4: real,Y3: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X4 @ Y3 ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_826_mult__diff__mult,axiom,
! [X4: int,Y3: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X4 @ Y3 ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X4 @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X4 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_827_mult__diff__mult,axiom,
! [X4: numera4273646738625120315l_num1,Y3: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ X4 @ Y3 ) @ ( times_2938166955517408246l_num1 @ A @ B ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ X4 @ ( minus_5410813661909488930l_num1 @ Y3 @ B ) ) @ ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ X4 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_828_sub__num__simps_I8_J,axiom,
! [K2: num,L: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K2 ) @ ( bit0 @ L ) )
= ( neg_nu5851722552734809277nc_int @ ( neg_numeral_sub_int @ K2 @ L ) ) ) ).
% sub_num_simps(8)
thf(fact_829_sub__num__simps_I7_J,axiom,
! [K2: num,L: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K2 ) @ ( bit1 @ L ) )
= ( neg_nu3811975205180677377ec_int @ ( neg_numeral_sub_int @ K2 @ L ) ) ) ).
% sub_num_simps(7)
thf(fact_830_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7886226890278435366l_num1 @ one_on7795324986448017462l_num1 )
= one_on7795324986448017462l_num1 ) ).
% dbl_dec_simps(3)
thf(fact_831_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_832_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_833_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_834_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_835_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_836_dbl__inc__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu5172728937851396970l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) )
= ( numera7754357348821619680l_num1 @ ( bit1 @ K2 ) ) ) ).
% dbl_inc_simps(5)
thf(fact_837_dbl__inc__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K2 ) )
= ( numeral_numeral_real @ ( bit1 @ K2 ) ) ) ).
% dbl_inc_simps(5)
thf(fact_838_dbl__inc__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) ).
% dbl_inc_simps(5)
thf(fact_839_dbl__dec__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu7886226890278435366l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) )
= ( numera7754357348821619680l_num1 @ ( bitM @ K2 ) ) ) ).
% dbl_dec_simps(5)
thf(fact_840_dbl__dec__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K2 ) )
= ( numeral_numeral_real @ ( bitM @ K2 ) ) ) ).
% dbl_dec_simps(5)
thf(fact_841_dbl__dec__simps_I5_J,axiom,
! [K2: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ ( bitM @ K2 ) ) ) ).
% dbl_dec_simps(5)
thf(fact_842_dbl__dec__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu7886226890278435366l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu5172728937851396970l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_843_dbl__dec__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_844_dbl__dec__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
= ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_845_dbl__inc__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu5172728937851396970l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu7886226890278435366l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_846_dbl__inc__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_847_dbl__inc__simps_I1_J,axiom,
! [K2: num] :
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
= ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_848_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_849_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_850_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_851_dbl__inc__def,axiom,
( neg_nu5172728937851396970l_num1
= ( ^ [X: numera4273646738625120315l_num1] : ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ X @ X ) @ one_on7795324986448017462l_num1 ) ) ) ).
% dbl_inc_def
thf(fact_852_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_853_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_854_dbl__dec__def,axiom,
( neg_nu7886226890278435366l_num1
= ( ^ [X: numera4273646738625120315l_num1] : ( minus_5410813661909488930l_num1 @ ( plus_p1441664204671982194l_num1 @ X @ X ) @ one_on7795324986448017462l_num1 ) ) ) ).
% dbl_dec_def
thf(fact_855_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_856_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_857_add__diff__add,axiom,
! [A: real,C2: real,B: real,D2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D2 ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C2 @ D2 ) ) ) ).
% add_diff_add
thf(fact_858_add__diff__add,axiom,
! [A: int,C2: int,B: int,D2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C2 @ D2 ) ) ) ).
% add_diff_add
thf(fact_859_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_860_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_861_xor__numerals_I4_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_862_xor__numerals_I4_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_863_xor__numerals_I6_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_864_xor__numerals_I6_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_865_or__numerals_I4_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% or_numerals(4)
thf(fact_866_or__numerals_I4_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% or_numerals(4)
thf(fact_867_or__numerals_I6_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% or_numerals(6)
thf(fact_868_or__numerals_I6_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% or_numerals(6)
thf(fact_869_or__numerals_I7_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% or_numerals(7)
thf(fact_870_or__numerals_I7_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% or_numerals(7)
thf(fact_871_and__numerals_I7_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% and_numerals(7)
thf(fact_872_and__numerals_I7_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% and_numerals(7)
thf(fact_873_and_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
= A ) ).
% and.left_neutral
thf(fact_874_and_Oright__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= A ) ).
% and.right_neutral
thf(fact_875_bit_Oconj__one__right,axiom,
! [X4: int] :
( ( bit_se725231765392027082nd_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
= X4 ) ).
% bit.conj_one_right
thf(fact_876_bit_Odisj__one__left,axiom,
! [X4: int] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.disj_one_left
thf(fact_877_bit_Odisj__one__right,axiom,
! [X4: int] :
( ( bit_se1409905431419307370or_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.disj_one_right
thf(fact_878_and__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= one_one_int ) ).
% and_numerals(2)
thf(fact_879_and__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= one_one_nat ) ).
% and_numerals(2)
thf(fact_880_and__numerals_I8_J,axiom,
! [X4: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
= one_one_int ) ).
% and_numerals(8)
thf(fact_881_and__numerals_I8_J,axiom,
! [X4: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
= one_one_nat ) ).
% and_numerals(8)
thf(fact_882_or__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(2)
thf(fact_883_or__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(2)
thf(fact_884_or__numerals_I8_J,axiom,
! [X4: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% or_numerals(8)
thf(fact_885_or__numerals_I8_J,axiom,
! [X4: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).
% or_numerals(8)
thf(fact_886_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_887_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_888_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_889_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_890_and__numerals_I3_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% and_numerals(3)
thf(fact_891_and__numerals_I3_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% and_numerals(3)
thf(fact_892_or__numerals_I3_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% or_numerals(3)
thf(fact_893_or__numerals_I3_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% or_numerals(3)
thf(fact_894_xor__numerals_I3_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% xor_numerals(3)
thf(fact_895_xor__numerals_I3_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% xor_numerals(3)
thf(fact_896_or__numerals_I5_J,axiom,
! [X4: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% or_numerals(5)
thf(fact_897_or__numerals_I5_J,axiom,
! [X4: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).
% or_numerals(5)
thf(fact_898_or__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(1)
thf(fact_899_or__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% or_numerals(1)
thf(fact_900_xor__numerals_I8_J,axiom,
! [X4: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ X4 ) ) ) ).
% xor_numerals(8)
thf(fact_901_xor__numerals_I8_J,axiom,
! [X4: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).
% xor_numerals(8)
thf(fact_902_xor__numerals_I5_J,axiom,
! [X4: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).
% xor_numerals(5)
thf(fact_903_xor__numerals_I5_J,axiom,
! [X4: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% xor_numerals(5)
thf(fact_904_xor__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit0 @ Y3 ) ) ) ).
% xor_numerals(2)
thf(fact_905_xor__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% xor_numerals(2)
thf(fact_906_xor__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% xor_numerals(1)
thf(fact_907_xor__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% xor_numerals(1)
thf(fact_908_and__numerals_I6_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% and_numerals(6)
thf(fact_909_and__numerals_I6_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% and_numerals(6)
thf(fact_910_and__numerals_I4_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% and_numerals(4)
thf(fact_911_and__numerals_I4_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% and_numerals(4)
thf(fact_912_xor__numerals_I7_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% xor_numerals(7)
thf(fact_913_xor__numerals_I7_J,axiom,
! [X4: num,Y3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% xor_numerals(7)
thf(fact_914_int__less__induct,axiom,
! [I2: int,K2: int,P: int > $o] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_915_int__gr__induct,axiom,
! [K2: int,I2: int,P: int > $o] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_916_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_917_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( ( bit_se725231765392027082nd_int @ A @ B )
= ( uminus_uminus_int @ one_one_int ) )
= ( ( A
= ( uminus_uminus_int @ one_one_int ) )
& ( B
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_918_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_919_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_920_int__bit__induct,axiom,
! [P: int > $o,K2: 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 @ K2 ) ) ) ) ) ).
% int_bit_induct
thf(fact_921_inf__period_I1_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K3: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D ) ) ) )
=> ( ! [X3: real,K3: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D ) ) ) )
=> ! [X6: real,K4: real] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D ) ) )
& ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_922_inf__period_I1_J,axiom,
! [P: int > $o,D: int,Q: int > $o] :
( ! [X3: int,K3: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ! [X6: int,K4: int] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D ) ) )
& ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_923_inf__period_I1_J,axiom,
! [P: numera4273646738625120315l_num1 > $o,D: numera4273646738625120315l_num1,Q: numera4273646738625120315l_num1 > $o] :
( ! [X3: numera4273646738625120315l_num1,K3: numera4273646738625120315l_num1] :
( ( P @ X3 )
= ( P @ ( minus_5410813661909488930l_num1 @ X3 @ ( times_2938166955517408246l_num1 @ K3 @ D ) ) ) )
=> ( ! [X3: numera4273646738625120315l_num1,K3: numera4273646738625120315l_num1] :
( ( Q @ X3 )
= ( Q @ ( minus_5410813661909488930l_num1 @ X3 @ ( times_2938166955517408246l_num1 @ K3 @ D ) ) ) )
=> ! [X6: numera4273646738625120315l_num1,K4: numera4273646738625120315l_num1] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_5410813661909488930l_num1 @ X6 @ ( times_2938166955517408246l_num1 @ K4 @ D ) ) )
& ( Q @ ( minus_5410813661909488930l_num1 @ X6 @ ( times_2938166955517408246l_num1 @ K4 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_924_inf__period_I2_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K3: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D ) ) ) )
=> ( ! [X3: real,K3: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D ) ) ) )
=> ! [X6: real,K4: real] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D ) ) )
| ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_925_inf__period_I2_J,axiom,
! [P: int > $o,D: int,Q: int > $o] :
( ! [X3: int,K3: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ! [X6: int,K4: int] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D ) ) )
| ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_926_inf__period_I2_J,axiom,
! [P: numera4273646738625120315l_num1 > $o,D: numera4273646738625120315l_num1,Q: numera4273646738625120315l_num1 > $o] :
( ! [X3: numera4273646738625120315l_num1,K3: numera4273646738625120315l_num1] :
( ( P @ X3 )
= ( P @ ( minus_5410813661909488930l_num1 @ X3 @ ( times_2938166955517408246l_num1 @ K3 @ D ) ) ) )
=> ( ! [X3: numera4273646738625120315l_num1,K3: numera4273646738625120315l_num1] :
( ( Q @ X3 )
= ( Q @ ( minus_5410813661909488930l_num1 @ X3 @ ( times_2938166955517408246l_num1 @ K3 @ D ) ) ) )
=> ! [X6: numera4273646738625120315l_num1,K4: numera4273646738625120315l_num1] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_5410813661909488930l_num1 @ X6 @ ( times_2938166955517408246l_num1 @ K4 @ D ) ) )
| ( Q @ ( minus_5410813661909488930l_num1 @ X6 @ ( times_2938166955517408246l_num1 @ K4 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_927_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_928_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_929_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_930_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_931_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_932_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_933_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_934_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_935_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X4: real,B: real] :
( ( ( times_times_real @ A @ X4 )
= ( times_times_real @ B @ X4 ) )
= ( ( A = B )
| ( X4 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_936_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X4: real,Y3: real] :
( ( ( times_times_real @ A @ X4 )
= ( times_times_real @ A @ Y3 ) )
= ( ( X4 = Y3 )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_937_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_938_vector__space__over__itself_Oscale__zero__left,axiom,
! [X4: real] :
( ( times_times_real @ zero_zero_real @ X4 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_939_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X4: real] :
( ( ( times_times_real @ A @ X4 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X4 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_940_mult__eq__0__iff,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ( times_7803423173614009249d_enat @ A @ B )
= zero_z5237406670263579293d_enat )
= ( ( A = zero_z5237406670263579293d_enat )
| ( B = zero_z5237406670263579293d_enat ) ) ) ).
% mult_eq_0_iff
thf(fact_941_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_942_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_943_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_944_mult__zero__right,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% mult_zero_right
thf(fact_945_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_946_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_947_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_948_mult__zero__right,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ zero_z2241845390563828978l_num1 )
= zero_z2241845390563828978l_num1 ) ).
% mult_zero_right
thf(fact_949_mult__zero__left,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat @ A )
= zero_z5237406670263579293d_enat ) ).
% mult_zero_left
thf(fact_950_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_951_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_952_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_953_mult__zero__left,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ zero_z2241845390563828978l_num1 @ A )
= zero_z2241845390563828978l_num1 ) ).
% mult_zero_left
thf(fact_954_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_955_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_956_add__0,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal @ A )
= A ) ).
% add_0
thf(fact_957_add__0,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
= A ) ).
% add_0
thf(fact_958_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_959_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_960_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_961_zero__eq__add__iff__both__eq__0,axiom,
! [X4: extended_enat,Y3: extended_enat] :
( ( zero_z5237406670263579293d_enat
= ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) )
= ( ( X4 = zero_z5237406670263579293d_enat )
& ( Y3 = zero_z5237406670263579293d_enat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_962_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y3 ) )
= ( ( X4 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_963_add__eq__0__iff__both__eq__0,axiom,
! [X4: extended_enat,Y3: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ X4 @ Y3 )
= zero_z5237406670263579293d_enat )
= ( ( X4 = zero_z5237406670263579293d_enat )
& ( Y3 = zero_z5237406670263579293d_enat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_964_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y3: nat] :
( ( ( plus_plus_nat @ X4 @ Y3 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_978_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_979_add_Oright__neutral,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ zero_z2744965634713055877_ereal )
= A ) ).
% add.right_neutral
thf(fact_980_add_Oright__neutral,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
= A ) ).
% add.right_neutral
thf(fact_981_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_982_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_983_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_984_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_985_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_986_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_987_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_988_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_989_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_990_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_991_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_992_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_993_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_994_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_995_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_996_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_997_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_998_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_999_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_1000_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_1001_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_1002_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_1003_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1004_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1005_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_1006_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1007_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1008_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1009_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1010_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1011_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_1012_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1013_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1014_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_1015_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1016_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1017_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1018_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1019_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1020_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1021_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1022_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1023_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1024_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1025_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1026_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1027_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1028_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1029_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1030_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1031_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1032_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1033_diff__numeral__special_I9_J,axiom,
( ( minus_5410813661909488930l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= zero_z2241845390563828978l_num1 ) ).
% diff_numeral_special(9)
thf(fact_1034_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1035_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1036_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_1037_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1038_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1039_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1040_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1041_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_1042_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_1043_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1044_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1045_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1046_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_1047_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1048_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1049_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_1050_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_1051_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1052_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_1053_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_real @ one @ one )
= zero_zero_real ) ).
% sub_num_simps(1)
thf(fact_1054_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_1055_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5172728937851396970l_num1 @ zero_z2241845390563828978l_num1 )
= one_on7795324986448017462l_num1 ) ).
% dbl_inc_simps(2)
thf(fact_1056_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1057_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7886226890278435366l_num1 @ zero_z2241845390563828978l_num1 )
= ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% dbl_dec_simps(2)
thf(fact_1068_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1069_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1070_and__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
= zero_zero_int ) ).
% and_numerals(1)
thf(fact_1071_and__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
= zero_zero_nat ) ).
% and_numerals(1)
thf(fact_1072_and__numerals_I5_J,axiom,
! [X4: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
= zero_zero_int ) ).
% and_numerals(5)
thf(fact_1073_and__numerals_I5_J,axiom,
! [X4: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
= zero_zero_nat ) ).
% and_numerals(5)
thf(fact_1074_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_1075_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_1076_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_1077_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_1078_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_1079_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_1080_zero__reorient,axiom,
! [X4: extended_ereal] :
( ( zero_z2744965634713055877_ereal = X4 )
= ( X4 = zero_z2744965634713055877_ereal ) ) ).
% zero_reorient
thf(fact_1081_zero__reorient,axiom,
! [X4: extended_enat] :
( ( zero_z5237406670263579293d_enat = X4 )
= ( X4 = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_1082_zero__reorient,axiom,
! [X4: real] :
( ( zero_zero_real = X4 )
= ( X4 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1083_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1084_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1085_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1086_zero__less__iff__neq__zero,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1087_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1088_gr__implies__not__zero,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ M @ N )
=> ( N != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_1089_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1090_not__less__zero,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_less_zero
thf(fact_1091_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1092_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1093_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1094_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1095_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one @ one )
= one ) ).
% or_not_num_neg.simps(1)
thf(fact_1096_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1097_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_1098_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
= ( bit0 @ one ) ) ).
% or_not_num_neg.simps(4)
thf(fact_1099_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1100_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_1101_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
= one ) ).
% or_not_num_neg.simps(7)
thf(fact_1102_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_1103_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_1104_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_1105_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_1106_or__not__num__neg_Oelims,axiom,
! [X4: num,Xa: num,Y3: num] :
( ( ( bit_or_not_num_neg @ X4 @ Xa )
= Y3 )
=> ( ( ( X4 = one )
=> ( ( Xa = one )
=> ( Y3 != one ) ) )
=> ( ( ( X4 = one )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( Y3
!= ( bit1 @ M3 ) ) ) )
=> ( ( ( X4 = one )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( Y3
!= ( bit1 @ M3 ) ) ) )
=> ( ( ? [N2: num] :
( X4
= ( bit0 @ N2 ) )
=> ( ( Xa = one )
=> ( Y3
!= ( bit0 @ one ) ) ) )
=> ( ! [N2: num] :
( ( X4
= ( bit0 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
=> ( ! [N2: num] :
( ( X4
= ( bit0 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( Y3
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
=> ( ( ? [N2: num] :
( X4
= ( bit1 @ N2 ) )
=> ( ( Xa = one )
=> ( Y3 != one ) ) )
=> ( ! [N2: num] :
( ( X4
= ( bit1 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
=> ~ ! [N2: num] :
( ( X4
= ( bit1 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_1107_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1108_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1109_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1110_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1111_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_1112_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1113_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1114_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1115_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1116_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1117_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1118_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1119_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1120_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1121_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1122_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1123_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1124_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1125_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1126_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1127_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1128_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1129_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1130_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1131_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1132_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_1133_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1134_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1135_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_1136_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1137_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1138_diff__gr0__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ).
% diff_gr0_ennreal
thf(fact_1139_ennreal__zero__less__mult__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( times_1893300245718287421nnreal @ A @ B ) )
= ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A )
& ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B ) ) ) ).
% ennreal_zero_less_mult_iff
thf(fact_1140_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1141_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_1142_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1143_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1144_mult__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1145_mult__less__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1146_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1147_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1148_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_1149_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_1150_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_1151_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1152_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1153_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1154_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_1155_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1156_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_1157_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_1158_ennreal__minus__zero,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% ennreal_minus_zero
thf(fact_1159_zero__minus__ennreal,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% zero_minus_ennreal
thf(fact_1160_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_1161_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_1162_ennreal__mult__left__cong,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ( A != zero_z7100319975126383169nnreal )
=> ( B = C2 ) )
=> ( ( times_1893300245718287421nnreal @ A @ B )
= ( times_1893300245718287421nnreal @ A @ C2 ) ) ) ).
% ennreal_mult_left_cong
thf(fact_1163_ennreal__mult__right__cong,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ( A != zero_z7100319975126383169nnreal )
=> ( B = C2 ) )
=> ( ( times_1893300245718287421nnreal @ B @ A )
= ( times_1893300245718287421nnreal @ C2 @ A ) ) ) ).
% ennreal_mult_right_cong
thf(fact_1164_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_1165_diff__add__eq__diff__diff__swap__ennreal,axiom,
! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ X4 @ ( plus_p1859984266308609217nnreal @ Y3 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ X4 @ Y3 ) @ Z ) ) ).
% diff_add_eq_diff_diff_swap_ennreal
thf(fact_1166_diff__diff__commute__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C2 )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ C2 ) @ B ) ) ).
% diff_diff_commute_ennreal
thf(fact_1167_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_1168_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_1169_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_1170_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_1171_xor__minus__numerals_I1_J,axiom,
! [N: num,K2: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K2 )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K2 ) ) ) ).
% xor_minus_numerals(1)
thf(fact_1172_xor__minus__numerals_I2_J,axiom,
! [K2: int,N: num] :
( ( bit_se6526347334894502574or_int @ K2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K2 @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_1173_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_1174_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_1175_not__int__code_I1_J,axiom,
( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% not_int_code(1)
thf(fact_1176_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K: int] : ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_1177_and__not__numerals_I1_J,axiom,
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= zero_zero_int ) ).
% and_not_numerals(1)
thf(fact_1178_or__not__numerals_I1_J,axiom,
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(1)
thf(fact_1179_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= one_one_int ) ).
% and_not_numerals(2)
thf(fact_1180_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_1181_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_1182_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% or_not_numerals(4)
thf(fact_1183_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_1184_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_1185_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_1186_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_1187_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_1188_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_1189_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_1190_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(7)
thf(fact_1191_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_1192_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_1193_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_1194_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_1195_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1196_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1197_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_1198_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1199_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1200_dvd__diff__nat,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1201_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1202_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1203_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1204_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1205_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1206_four__x__squared,axiom,
! [X4: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_1207_dvd__mult__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1208_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1209_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_1210_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1211_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1212_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1213_ereal__power__uminus,axiom,
! [N: nat,X4: extended_ereal] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_1054015426188190660_ereal @ ( uminus27091377158695749_ereal @ X4 ) @ N )
= ( power_1054015426188190660_ereal @ X4 @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_1054015426188190660_ereal @ ( uminus27091377158695749_ereal @ X4 ) @ N )
= ( uminus27091377158695749_ereal @ ( power_1054015426188190660_ereal @ X4 @ N ) ) ) ) ) ).
% ereal_power_uminus
thf(fact_1214_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X3: nat,Y6: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y6 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1215_real__add__minus__iff,axiom,
! [X4: real,A: real] :
( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X4 = A ) ) ).
% real_add_minus_iff
thf(fact_1216_ereal__minus_I8_J,axiom,
! [X4: extended_ereal] :
( ( minus_2816186181549245109_ereal @ zero_z2744965634713055877_ereal @ X4 )
= ( uminus27091377158695749_ereal @ X4 ) ) ).
% ereal_minus(8)
thf(fact_1217_ereal__minus_I6_J,axiom,
! [X4: extended_ereal,Y3: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X4 @ ( uminus27091377158695749_ereal @ Y3 ) )
= ( plus_p7876563987511257093_ereal @ X4 @ Y3 ) ) ).
% ereal_minus(6)
thf(fact_1218_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= ( uminus27091377158695749_ereal @ B ) )
= ( A = B ) ) ).
% ereal_uminus_eq_iff
thf(fact_1219_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus27091377158695749_ereal @ ( uminus27091377158695749_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_1220_ereal__minus__less__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ A ) ) ).
% ereal_minus_less_minus
thf(fact_1221_ereal__mult__minus__left,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_left
thf(fact_1222_ereal__mult__minus__right,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ ( uminus27091377158695749_ereal @ B ) )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_right
thf(fact_1223_neg__0__less__iff__less__erea,axiom,
! [A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% neg_0_less_iff_less_erea
thf(fact_1224_ereal__uminus__zero__iff,axiom,
! [A: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= zero_z2744965634713055877_ereal )
= ( A = zero_z2744965634713055877_ereal ) ) ).
% ereal_uminus_zero_iff
thf(fact_1225_ereal__uminus__zero,axiom,
( ( uminus27091377158695749_ereal @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_uminus_zero
thf(fact_1226_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int,T2: int] : ( dvd_dvd_int @ D3 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1227_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D3 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_1228_real__arch__pow,axiom,
! [X4: real,Y3: real] :
( ( ord_less_real @ one_one_real @ X4 )
=> ? [N2: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X4 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1229_real__arch__pow__inv,axiom,
! [Y3: real,X4: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ X4 @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N2 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_1230_real__0__less__add__iff,axiom,
! [X4: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).
% real_0_less_add_iff
thf(fact_1231_real__add__less__0__iff,axiom,
! [X4: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
= ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).
% real_add_less_0_iff
thf(fact_1232_ereal__m1__less__0,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ one_on4623092294121504201_ereal ) @ zero_z2744965634713055877_ereal ).
% ereal_m1_less_0
thf(fact_1233_minus__ereal__def,axiom,
( minus_2816186181549245109_ereal
= ( ^ [X: extended_ereal,Y: extended_ereal] : ( plus_p7876563987511257093_ereal @ X @ ( uminus27091377158695749_ereal @ Y ) ) ) ) ).
% minus_ereal_def
thf(fact_1234_ereal__add__uminus__conv__diff,axiom,
! [X4: extended_ereal,Y3: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ X4 ) @ Y3 )
= ( minus_2816186181549245109_ereal @ Y3 @ X4 ) ) ).
% ereal_add_uminus_conv_diff
thf(fact_1235_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= B )
= ( A
= ( uminus27091377158695749_ereal @ B ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_1236_ereal__less__uminus__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ ( uminus27091377158695749_ereal @ A ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_1237_ereal__uminus__less__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ B ) @ A ) ) ).
% ereal_uminus_less_reorder
thf(fact_1238_even__not__iff__int,axiom,
! [K2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K2 ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).
% even_not_iff_int
thf(fact_1239_even__and__iff__int,axiom,
! [K2: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K2 @ L ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_1240_even__diff__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K2 @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).
% even_diff_iff
thf(fact_1241_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_1242_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1243_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X3: nat,Y6: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y6 ) @ D4 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y6 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_1244_bezout__lemma__nat,axiom,
! [D2: nat,A: nat,B: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A )
=> ( ( dvd_dvd_nat @ D2 @ B )
=> ( ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) )
| ( ( times_times_nat @ B @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D2 ) ) )
=> ? [X3: nat,Y6: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y6 ) @ D2 ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y6 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1245_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X3: nat,Y6: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y6 ) )
= D4 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y6 ) )
= D4 ) ) ) ).
% bezout1_nat
thf(fact_1246_ereal__uminus__divide,axiom,
! [X4: extended_ereal,Y3: extended_ereal] :
( ( divide8893690120176169980_ereal @ ( uminus27091377158695749_ereal @ X4 ) @ Y3 )
= ( uminus27091377158695749_ereal @ ( divide8893690120176169980_ereal @ X4 @ Y3 ) ) ) ).
% ereal_uminus_divide
thf(fact_1247_ennreal__zero__divide,axiom,
! [X4: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ zero_z7100319975126383169nnreal @ X4 )
= zero_z7100319975126383169nnreal ) ).
% ennreal_zero_divide
thf(fact_1248_one__divide__one__divide__ennreal,axiom,
! [C2: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ one_on2969667320475766781nnreal @ ( divide4826598186094686858nnreal @ one_on2969667320475766781nnreal @ C2 ) )
= C2 ) ).
% one_divide_one_divide_ennreal
thf(fact_1249_ereal__divide__one,axiom,
! [X4: extended_ereal] :
( ( divide8893690120176169980_ereal @ X4 @ one_on4623092294121504201_ereal )
= X4 ) ).
% ereal_divide_one
thf(fact_1250_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_1251_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_1252_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_1253_ereal__0__less__1,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).
% ereal_0_less_1
thf(fact_1254_ereal__one__not__less__zero__ereal,axiom,
~ ( ord_le1188267648640031866_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).
% ereal_one_not_less_zero_ereal
thf(fact_1255_div__mod__decomp__int,axiom,
! [A2: int,N: int] :
( A2
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1256_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1257_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_1258_ennreal__times__divide,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ ( divide4826598186094686858nnreal @ B @ C2 ) )
= ( divide4826598186094686858nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C2 ) ) ).
% ennreal_times_divide
thf(fact_1259_ennreal__divide__times,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( divide4826598186094686858nnreal @ A @ B ) @ C2 )
= ( times_1893300245718287421nnreal @ A @ ( divide4826598186094686858nnreal @ C2 @ B ) ) ) ).
% ennreal_divide_times
thf(fact_1260_div__less__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A2 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1261_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo_nat @ M3 @ N2 ) )
=> ( P @ M3 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1262_zdiv__mono__strict,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A2 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B2 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1263_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1264_add__divide__distrib__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C2 )
= ( plus_p1859984266308609217nnreal @ ( divide4826598186094686858nnreal @ A @ C2 ) @ ( divide4826598186094686858nnreal @ B @ C2 ) ) ) ).
% add_divide_distrib_ennreal
thf(fact_1265_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M4: nat,N3: nat] :
( if_nat
@ ( ( M4 = zero_zero_nat )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_1266_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M4 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y3: int] :
( ( if_int @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y3: int] :
( ( if_int @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y3: nat] :
( ( if_nat @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y3: nat] :
( ( if_nat @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ ( median_a @ n @ f ) @ i ).
%------------------------------------------------------------------------------