TPTP Problem File: SLH0546^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Cotangent_PFD_Formula/0007_Cotangent_PFD_Formula/prob_00157_006085__13978446_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1391 ( 836 unt; 111 typ; 0 def)
% Number of atoms : 2920 (1521 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 7821 ( 253 ~; 50 |; 142 &;6622 @)
% ( 0 <=>; 754 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 252 ( 252 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 101 usr; 16 con; 0-3 aty)
% Number of variables : 2332 ( 106 ^;2194 !; 32 ?;2332 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:56:50.617
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (101)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
invers8013647133539491842omplex: complex > complex ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Extended____Nonnegative____Real__Oennreal,type,
invers7556275967461373580nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
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__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Extended____Nat__Oenat,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(sy_c_If_001t__Extended____Nonnegative____Real__Oennreal,type,
if_Ext9135588136721118450nnreal: $o > extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nonnegative____Real__Oennreal,type,
semiri6283507881447550617nnreal: nat > extend8495563244428889912nnreal ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
neg_numeral_dbl_real: real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
numera6690914467698888265omplex: num > complex ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nonnegative____Real__Oennreal,type,
numera4658534427948366547nnreal: num > extend8495563244428889912nnreal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nonnegative____Real__Oennreal,type,
power_6007165696250533058nnreal: extend8495563244428889912nnreal > nat > extend8495563244428889912nnreal ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Nonnegative____Real__Oennreal,type,
divide4826598186094686858nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Extended____Nat__Oenat,type,
dvd_dv3785147216227455552d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Extended____Nonnegative____Real__Oennreal,type,
dvd_dv1013850698770059486nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nat__Oenat,type,
zero_n1046097342994218471d_enat: $o > extended_enat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_n4168557817388953207nnreal: $o > extend8495563244428889912nnreal ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
zero_n3304061248610475627l_real: $o > real ).
thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
summable_complex: ( nat > complex ) > $o ).
thf(sy_c_Series_Osummable_001t__Extended____Nat__Oenat,type,
summab1538256873603986438d_enat: ( nat > extended_enat ) > $o ).
thf(sy_c_Series_Osummable_001t__Int__Oint,type,
summable_int: ( nat > int ) > $o ).
thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
summable_nat: ( nat > nat ) > $o ).
thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_R,type,
r: real ).
% Relevant facts (1266)
thf(fact_0__092_060open_0622_A_092_060le_062_A2_A_092_060Longrightarrow_062_Asummable_A_I_092_060lambda_062n_O_Ainverse_A_I_Iof__nat_An_J_092_060_094sup_0622_J_J_092_060close_062,axiom,
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( summable_complex
@ ^ [N: nat] : ( invers8013647133539491842omplex @ ( power_power_complex @ ( semiri8010041392384452111omplex @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% \<open>2 \<le> 2 \<Longrightarrow> summable (\<lambda>n. inverse ((of_nat n)\<^sup>2))\<close>
thf(fact_1__092_060open_0622_A_092_060le_062_A2_A_092_060Longrightarrow_062_Asummable_A_I_092_060lambda_062n_O_Ainverse_A_I_Iof__nat_An_J_092_060_094sup_0622_J_J_092_060close_062,axiom,
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( summable_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% \<open>2 \<le> 2 \<Longrightarrow> summable (\<lambda>n. inverse ((of_nat n)\<^sup>2))\<close>
thf(fact_2_norm__divide__numeral,axiom,
! [A: real,W: num] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_divide_numeral
thf(fact_3_norm__divide__numeral,axiom,
! [A: complex,W: num] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_divide_numeral
thf(fact_4_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
= ( semiri8010041392384452111omplex @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N2 )
= ( semiri4216267220026989637d_enat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_7_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_8_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_9_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X ) @ N2 )
= ( semiri6283507881447550617nnreal @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_10_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri8010041392384452111omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_11_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_12_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri4216267220026989637d_enat @ Y )
= ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_13_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_14_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_15_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ Y )
= ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_16_norm__of__nat,axiom,
! [N2: nat] :
( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% norm_of_nat
thf(fact_17_norm__of__nat,axiom,
! [N2: nat] :
( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% norm_of_nat
thf(fact_18_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
= ( numera6690914467698888265omplex @ N2 ) ) ).
% of_nat_numeral
thf(fact_19_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% of_nat_numeral
thf(fact_20_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N2 ) )
= ( numera1916890842035813515d_enat @ N2 ) ) ).
% of_nat_numeral
thf(fact_21_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_real @ N2 ) ) ).
% of_nat_numeral
thf(fact_22_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% of_nat_numeral
thf(fact_23_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ N2 ) )
= ( numera4658534427948366547nnreal @ N2 ) ) ).
% of_nat_numeral
thf(fact_24_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_25_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_26_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera6690914467698888265omplex @ N2 )
= one_one_complex )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_27_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_nat @ N2 )
= one_one_nat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_28_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_real @ N2 )
= one_one_real )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_29_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera1916890842035813515d_enat @ N2 )
= one_on7984719198319812577d_enat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_30_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_int @ N2 )
= one_one_int )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_31_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera4658534427948366547nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_32_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_33_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_34_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_real
= ( numeral_numeral_real @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_35_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_36_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_int
= ( numeral_numeral_int @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_37_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_on2969667320475766781nnreal
= ( numera4658534427948366547nnreal @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_38_square__norm__one,axiom,
! [X: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
=> ( ( real_V7735802525324610683m_real @ X )
= one_one_real ) ) ).
% square_norm_one
thf(fact_39_square__norm__one,axiom,
! [X: complex] :
( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex )
=> ( ( real_V1022390504157884413omplex @ X )
= one_one_real ) ) ).
% square_norm_one
thf(fact_40_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_41_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_42_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( power_power_nat @ M @ N2 ) )
= ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_43_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_44_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri4216267220026989637d_enat @ ( power_power_nat @ M @ N2 ) )
= ( power_8040749407984259932d_enat @ ( semiri4216267220026989637d_enat @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_45_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_46_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_47_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_48_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W )
= ( semiri6283507881447550617nnreal @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_49_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_50_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_8040749407984259932d_enat @ ( semiri4216267220026989637d_enat @ B ) @ W )
= ( semiri4216267220026989637d_enat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_51_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_52_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_53_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_54_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri6283507881447550617nnreal @ X )
= ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_55_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_56_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri4216267220026989637d_enat @ X )
= ( power_8040749407984259932d_enat @ ( semiri4216267220026989637d_enat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_57_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_58_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_59_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_60_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_61_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_62_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numera4658534427948366547nnreal @ M )
= ( numera4658534427948366547nnreal @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_63_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numera6690914467698888265omplex @ M )
= ( numera6690914467698888265omplex @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_64_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_65_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_66_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_67_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_68_power__one__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_69_power__one__right,axiom,
! [A: extended_enat] :
( ( power_8040749407984259932d_enat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_70_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_71_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_72_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_73_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_74_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_75_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_76_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_77_power__one,axiom,
! [N2: nat] :
( ( power_power_complex @ one_one_complex @ N2 )
= one_one_complex ) ).
% power_one
thf(fact_78_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_79_power__one,axiom,
! [N2: nat] :
( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ N2 )
= one_on2969667320475766781nnreal ) ).
% power_one
thf(fact_80_power__one,axiom,
! [N2: nat] :
( ( power_8040749407984259932d_enat @ one_on7984719198319812577d_enat @ N2 )
= one_on7984719198319812577d_enat ) ).
% power_one
thf(fact_81_norm__numeral,axiom,
! [W: num] :
( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_numeral
thf(fact_82_norm__numeral,axiom,
! [W: num] :
( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_numeral
thf(fact_83_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_84_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_85_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_86_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_87_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_88_inverse__eq__divide__numeral,axiom,
! [W: num] :
( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_89_inverse__eq__divide__numeral,axiom,
! [W: num] :
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_90_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_91_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_92_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_93_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_94_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_95_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_96_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_97_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_98_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_99_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_100_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_101_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_102_norm__inverse,axiom,
! [A: real] :
( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_inverse
thf(fact_103_norm__inverse,axiom,
! [A: complex] :
( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% norm_inverse
thf(fact_104_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_105_power__inverse,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( inverse_inverse_real @ A ) @ N2 )
= ( inverse_inverse_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_inverse
thf(fact_106_power__inverse,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N2 )
= ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_inverse
thf(fact_107_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_108_le__numeral__extra_I4_J,axiom,
ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ).
% le_numeral_extra(4)
thf(fact_109_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_110_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_111_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_112_power2__nat__le__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_113_power2__nat__le__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_114_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_115_power__increasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_116_power__increasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_117_power__increasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_118_inverse__numeral__1,axiom,
( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
= ( numeral_numeral_real @ one ) ) ).
% inverse_numeral_1
thf(fact_119_inverse__numeral__1,axiom,
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
= ( numera6690914467698888265omplex @ one ) ) ).
% inverse_numeral_1
thf(fact_120_one__le__numeral,axiom,
! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% one_le_numeral
thf(fact_121_one__le__numeral,axiom,
! [N2: num] : ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% one_le_numeral
thf(fact_122_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% one_le_numeral
thf(fact_123_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% one_le_numeral
thf(fact_124_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% one_le_numeral
thf(fact_125_one__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_126_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_127_one__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_128_power__divide,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
= ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_divide
thf(fact_129_power__divide,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_divide
thf(fact_130_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_131_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_132_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_133_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_134_numeral__One,axiom,
( ( numera4658534427948366547nnreal @ one )
= one_on2969667320475766781nnreal ) ).
% numeral_One
thf(fact_135_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_136_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_137_divide__numeral__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_138_power__one__over,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_139_power__one__over,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_140_norm__divide,axiom,
! [A: real,B: real] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_divide
thf(fact_141_norm__divide,axiom,
! [A: complex,B: complex] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_divide
thf(fact_142_norm__power,axiom,
! [X: real,N2: nat] :
( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) )
= ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% norm_power
thf(fact_143_norm__power,axiom,
! [X: complex,N2: nat] :
( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% norm_power
thf(fact_144_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_145_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_146_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex ) ).
% one_power2
thf(fact_147_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_148_one__power2,axiom,
( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on2969667320475766781nnreal ) ).
% one_power2
thf(fact_149_one__power2,axiom,
( ( power_8040749407984259932d_enat @ one_on7984719198319812577d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on7984719198319812577d_enat ) ).
% one_power2
thf(fact_150_inverse__power__summable,axiom,
! [S: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ S )
=> ( summable_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ N ) @ S ) ) ) ) ).
% inverse_power_summable
thf(fact_151_inverse__power__summable,axiom,
! [S: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ S )
=> ( summable_complex
@ ^ [N: nat] : ( invers8013647133539491842omplex @ ( power_power_complex @ ( semiri8010041392384452111omplex @ N ) @ S ) ) ) ) ).
% inverse_power_summable
thf(fact_152_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_153_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_154_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_155_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_156_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_157_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_161_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_162_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_163_inverse__divide,axiom,
! [A: complex,B: complex] :
( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ B @ A ) ) ).
% inverse_divide
thf(fact_164_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_165_inverse__1,axiom,
( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex ) ).
% inverse_1
thf(fact_166_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_167_inverse__eq__1__iff,axiom,
! [X: complex] :
( ( ( invers8013647133539491842omplex @ X )
= one_one_complex )
= ( X = one_one_complex ) ) ).
% inverse_eq_1_iff
thf(fact_168_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_169_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_170_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri6283507881447550617nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_171_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri8010041392384452111omplex @ N2 )
= one_one_complex )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_172_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri4216267220026989637d_enat @ N2 )
= one_on7984719198319812577d_enat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_173_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_174_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_175_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_176_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_on2969667320475766781nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_177_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_178_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_on7984719198319812577d_enat
= ( semiri4216267220026989637d_enat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_179_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_180_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_181_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_182_of__nat__1,axiom,
( ( semiri6283507881447550617nnreal @ one_one_nat )
= one_on2969667320475766781nnreal ) ).
% of_nat_1
thf(fact_183_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_184_of__nat__1,axiom,
( ( semiri4216267220026989637d_enat @ one_one_nat )
= one_on7984719198319812577d_enat ) ).
% of_nat_1
thf(fact_185_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_186_semiring__norm_I83_J,axiom,
! [N2: num] :
( one
!= ( bit0 @ N2 ) ) ).
% semiring_norm(83)
thf(fact_187_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_188_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_189_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_190_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_191_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_192_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_193_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_194_semiring__norm_I87_J,axiom,
! [M: num,N2: num] :
( ( ( bit0 @ M )
= ( bit0 @ N2 ) )
= ( M = N2 ) ) ).
% semiring_norm(87)
thf(fact_195_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_196_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_197_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_198_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= ( semiri8010041392384452111omplex @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_199_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri4216267220026989637d_enat @ M )
= ( semiri4216267220026989637d_enat @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_200_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_201_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_202_inverse__inverse__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_203_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_204_inverse__eq__iff__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_205_semiring__norm_I71_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(71)
thf(fact_206_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_207_assms,axiom,
ord_less_eq_real @ zero_zero_real @ r ).
% assms
thf(fact_208_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_209_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_210_norm__power__ineq,axiom,
! [X: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% norm_power_ineq
thf(fact_211_norm__power__ineq,axiom,
! [X: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% norm_power_ineq
thf(fact_212_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_213_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_214_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_215_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_216_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_217_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_218_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_219_inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_220_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ I ) @ ( semiri6283507881447550617nnreal @ J ) ) ) ).
% of_nat_mono
thf(fact_221_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ I ) @ ( semiri4216267220026989637d_enat @ J ) ) ) ).
% of_nat_mono
thf(fact_222_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_223_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_224_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_225_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_226_inverse__eq__divide,axiom,
( invers8013647133539491842omplex
= ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% inverse_eq_divide
thf(fact_227_not__summable__harmonic,axiom,
~ ( summable_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% not_summable_harmonic
thf(fact_228_not__summable__harmonic,axiom,
~ ( summable_complex
@ ^ [N: nat] : ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% not_summable_harmonic
thf(fact_229_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_230_enat__ord__number_I1_J,axiom,
! [M: num,N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% enat_ord_number(1)
thf(fact_231_two__realpow__ge__one,axiom,
! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% two_realpow_ge_one
thf(fact_232_summable__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N4: nat] :
! [N5: nat] :
( ( ord_less_eq_nat @ N4 @ N5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N5 ) ) @ ( G @ N5 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test
thf(fact_233_summable__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N4: nat] :
! [N5: nat] :
( ( ord_less_eq_nat @ N4 @ N5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N5 ) ) @ ( G @ N5 ) ) )
=> ( ( summable_real @ G )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test
thf(fact_234_summable__comparison__test_H,axiom,
! [G: nat > real,N3: nat,F: nat > real] :
( ( summable_real @ G )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N5 ) ) @ ( G @ N5 ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test'
thf(fact_235_summable__comparison__test_H,axiom,
! [G: nat > real,N3: nat,F: nat > complex] :
( ( summable_real @ G )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N5 ) ) @ ( G @ N5 ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test'
thf(fact_236_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_237_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_238_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_239_summable__norm__cancel,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
=> ( summable_real @ F ) ) ).
% summable_norm_cancel
thf(fact_240_summable__norm__cancel,axiom,
! [F: nat > complex] :
( ( summable_real
@ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
=> ( summable_complex @ F ) ) ).
% summable_norm_cancel
thf(fact_241_summable__norm__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N4: nat] :
! [N5: nat] :
( ( ord_less_eq_nat @ N4 @ N5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N5 ) ) @ ( G @ N5 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_242_summable__norm__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N4: nat] :
! [N5: nat] :
( ( ord_less_eq_nat @ N4 @ N5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N5 ) ) @ ( G @ N5 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_243_Multiseries__Expansion_Ointyness__simps_I3_J,axiom,
! [A: nat,N2: nat] :
( ( power_power_real @ ( semiri5074537144036343181t_real @ A ) @ N2 )
= ( semiri5074537144036343181t_real @ ( power_power_nat @ A @ N2 ) ) ) ).
% Multiseries_Expansion.intyness_simps(3)
thf(fact_244_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_245_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_246_dbl__simps_I3_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_247_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_248_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_249_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_250_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_251_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_252_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_253_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_254_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_255_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_256_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_257_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_258_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_259_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_260_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_261_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_262_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_263_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_264_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_265_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_266_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_267_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_268_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_269_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_270_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_271_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri4216267220026989637d_enat @ M )
= zero_z5237406670263579293d_enat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_272_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_273_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_274_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_275_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_z7100319975126383169nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_276_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_277_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_z5237406670263579293d_enat
= ( semiri4216267220026989637d_enat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_278_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_279_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_280_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_281_of__nat__0,axiom,
( ( semiri6283507881447550617nnreal @ zero_zero_nat )
= zero_z7100319975126383169nnreal ) ).
% of_nat_0
thf(fact_282_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_283_of__nat__0,axiom,
( ( semiri4216267220026989637d_enat @ zero_zero_nat )
= zero_z5237406670263579293d_enat ) ).
% of_nat_0
thf(fact_284_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_285_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_286_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_287_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_288_inverse__nonzero__iff__nonzero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_289_summable__single,axiom,
! [I: nat,F: nat > extended_enat] :
( summab1538256873603986438d_enat
@ ^ [R: nat] : ( if_Extended_enat @ ( R = I ) @ ( F @ R ) @ zero_z5237406670263579293d_enat ) ) ).
% summable_single
thf(fact_290_summable__single,axiom,
! [I: nat,F: nat > complex] :
( summable_complex
@ ^ [R: nat] : ( if_complex @ ( R = I ) @ ( F @ R ) @ zero_zero_complex ) ) ).
% summable_single
thf(fact_291_summable__single,axiom,
! [I: nat,F: nat > real] :
( summable_real
@ ^ [R: nat] : ( if_real @ ( R = I ) @ ( F @ R ) @ zero_zero_real ) ) ).
% summable_single
thf(fact_292_summable__single,axiom,
! [I: nat,F: nat > nat] :
( summable_nat
@ ^ [R: nat] : ( if_nat @ ( R = I ) @ ( F @ R ) @ zero_zero_nat ) ) ).
% summable_single
thf(fact_293_summable__single,axiom,
! [I: nat,F: nat > int] :
( summable_int
@ ^ [R: nat] : ( if_int @ ( R = I ) @ ( F @ R ) @ zero_zero_int ) ) ).
% summable_single
thf(fact_294_summable__zero,axiom,
( summab1538256873603986438d_enat
@ ^ [N: nat] : zero_z5237406670263579293d_enat ) ).
% summable_zero
thf(fact_295_summable__zero,axiom,
( summable_complex
@ ^ [N: nat] : zero_zero_complex ) ).
% summable_zero
thf(fact_296_summable__zero,axiom,
( summable_real
@ ^ [N: nat] : zero_zero_real ) ).
% summable_zero
thf(fact_297_summable__zero,axiom,
( summable_nat
@ ^ [N: nat] : zero_zero_nat ) ).
% summable_zero
thf(fact_298_summable__zero,axiom,
( summable_int
@ ^ [N: nat] : zero_zero_int ) ).
% summable_zero
thf(fact_299_dbl__simps_I2_J,axiom,
( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% dbl_simps(2)
thf(fact_300_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_301_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_302_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_303_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M ) @ zero_z5237406670263579293d_enat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_304_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_305_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_306_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_307_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_308_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_309_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_310_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_311_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_312_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_313_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_314_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_315_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_316_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_317_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_318_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_319_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_320_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_321_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_322_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_323_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_324_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_325_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_326_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_327_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_328_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_329_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_330_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_331_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_332_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_333_power__zero__numeral,axiom,
! [K: num] :
( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ ( numeral_numeral_nat @ K ) )
= zero_z7100319975126383169nnreal ) ).
% power_zero_numeral
thf(fact_334_power__zero__numeral,axiom,
! [K: num] :
( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( numeral_numeral_nat @ K ) )
= zero_z5237406670263579293d_enat ) ).
% power_zero_numeral
thf(fact_335_summable__divide__iff,axiom,
! [F: nat > real,C: real] :
( ( summable_real
@ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_336_summable__divide__iff,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_divide_iff
thf(fact_337_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_338_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_339_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_340_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_341_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_342_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_343_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_344_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_345_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_346_zero__eq__power2,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_347_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_348_zero__eq__power2,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7100319975126383169nnreal )
= ( A = zero_z7100319975126383169nnreal ) ) ).
% zero_eq_power2
thf(fact_349_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_350_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_351_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_352_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_353_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_354_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_355_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_356_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_357_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_358_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ) ) ).
% complete_real
thf(fact_359_gbinomial__series__aux_Oexhaust,axiom,
! [Abort: $o,Acc: real] :
( ( Abort
=> ( Acc != zero_zero_real ) )
=> ( ~ Abort
| ( Acc != zero_zero_real ) ) ) ).
% gbinomial_series_aux.exhaust
thf(fact_360_summable__const__iff,axiom,
! [C: complex] :
( ( summable_complex
@ ^ [Uu: nat] : C )
= ( C = zero_zero_complex ) ) ).
% summable_const_iff
thf(fact_361_summable__const__iff,axiom,
! [C: real] :
( ( summable_real
@ ^ [Uu: nat] : C )
= ( C = zero_zero_real ) ) ).
% summable_const_iff
thf(fact_362_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_363_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_364_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= one_one_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_365_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_366_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N2 )
= one_on2969667320475766781nnreal ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N2 )
= zero_z7100319975126383169nnreal ) ) ) ).
% power_0_left
thf(fact_367_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
= one_on7984719198319812577d_enat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
= zero_z5237406670263579293d_enat ) ) ) ).
% power_0_left
thf(fact_368_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_369_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_370_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_371_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_372_zero__neq__one,axiom,
zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).
% zero_neq_one
thf(fact_373_zero__neq__one,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_neq_one
thf(fact_374_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_375_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_376_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_377_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_378_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_379_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N2 ) ) ).
% zero_neq_numeral
thf(fact_380_zero__neq__numeral,axiom,
! [N2: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_381_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N2 ) ) ).
% zero_neq_numeral
thf(fact_382_zero__neq__numeral,axiom,
! [N2: num] :
( zero_z7100319975126383169nnreal
!= ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_neq_numeral
thf(fact_383_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N2 ) ) ).
% zero_neq_numeral
thf(fact_384_power__not__zero,axiom,
! [A: real,N2: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N2 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_385_power__not__zero,axiom,
! [A: nat,N2: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_386_power__not__zero,axiom,
! [A: complex,N2: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N2 )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_387_power__not__zero,axiom,
! [A: int,N2: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N2 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_388_power__not__zero,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( A != zero_z7100319975126383169nnreal )
=> ( ( power_6007165696250533058nnreal @ A @ N2 )
!= zero_z7100319975126383169nnreal ) ) ).
% power_not_zero
thf(fact_389_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_390_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_391_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_392_inverse__zero__imp__zero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_393_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_394_nonzero__inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_395_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_396_nonzero__inverse__inverse__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_397_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_398_nonzero__imp__inverse__nonzero,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_399_norm__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) ) ).
% norm_ge_zero
thf(fact_400_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_401_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_402_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_403_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_404_not__one__le__zero,axiom,
~ ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% not_one_le_zero
thf(fact_405_not__one__le__zero,axiom,
~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_le_zero
thf(fact_406_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_407_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_408_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_409_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_410_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_411_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_412_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_413_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_414_zero__less__one__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% zero_less_one_class.zero_le_one
thf(fact_415_zero__less__one__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one_class.zero_le_one
thf(fact_416_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_417_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_418_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_419_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_420_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_le_zero
thf(fact_421_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_422_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_423_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_424_zero__le__numeral,axiom,
! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% zero_le_numeral
thf(fact_425_zero__le__numeral,axiom,
! [N2: num] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_le_numeral
thf(fact_426_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% zero_le_numeral
thf(fact_427_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_le_numeral
thf(fact_428_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_le_numeral
thf(fact_429_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_430_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_431_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_432_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_433_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_434_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_435_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_436_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_437_zero__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_438_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_439_zero__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_440_power__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_441_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_442_power__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_443_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_444_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_445_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_446_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_447_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_448_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_449_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_450_power__le__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_451_power__le__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_452_power__le__one,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_453_inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_454_nonzero__inverse__eq__divide,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_455_nonzero__inverse__eq__divide,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_456_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_457_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_458_Multiseries__Expansion_Ointyness__of__nat,axiom,
! [N2: nat] :
( ( N2 = N2 )
=> ( ( semiri5074537144036343181t_real @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% Multiseries_Expansion.intyness_of_nat
thf(fact_459_nonzero__norm__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_460_nonzero__norm__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_461_power__decreasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_462_power__decreasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_463_power__decreasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_464_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_465_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_466_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_467_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_468_zero__power2,axiom,
( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7100319975126383169nnreal ) ).
% zero_power2
thf(fact_469_zero__power2,axiom,
( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z5237406670263579293d_enat ) ).
% zero_power2
thf(fact_470_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_471_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_472_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_473_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_474_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_475_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_476_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_477_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_478_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_479_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_480_real__of__nat__div4,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_481_summable__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) ) ) ).
% summable_divide
thf(fact_482_summable__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) ) ) ).
% summable_divide
thf(fact_483_Multiseries__Expansion_Ointyness__numeral,axiom,
! [Num: num] :
( ( Num = Num )
=> ( ( numeral_numeral_real @ Num )
= ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num ) ) ) ) ).
% Multiseries_Expansion.intyness_numeral
thf(fact_484_Multiseries__Expansion_Ointyness__simps_I6_J,axiom,
( numeral_numeral_real
= ( ^ [Num2: num] : ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num2 ) ) ) ) ).
% Multiseries_Expansion.intyness_simps(6)
thf(fact_485_Multiseries__Expansion_Ointyness__1,axiom,
( one_one_real
= ( semiri5074537144036343181t_real @ one_one_nat ) ) ).
% Multiseries_Expansion.intyness_1
thf(fact_486_le__zero__eq,axiom,
! [N2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
= ( N2 = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_487_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_488_complex__not__root__unity,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ? [U: complex] :
( ( ( real_V1022390504157884413omplex @ U )
= one_one_real )
& ( ( power_power_complex @ U @ N2 )
!= one_one_complex ) ) ) ).
% complex_not_root_unity
thf(fact_489_norm__imp__pos__and__ge,axiom,
! [X: real,N2: real] :
( ( ( real_V7735802525324610683m_real @ X )
= N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
& ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ) ).
% norm_imp_pos_and_ge
thf(fact_490_norm__imp__pos__and__ge,axiom,
! [X: complex,N2: real] :
( ( ( real_V1022390504157884413omplex @ X )
= N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
& ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ) ).
% norm_imp_pos_and_ge
thf(fact_491_vector__choose__size,axiom,
! [C: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ~ ! [X3: real] :
( ( real_V7735802525324610683m_real @ X3 )
!= C ) ) ).
% vector_choose_size
thf(fact_492_vector__choose__size,axiom,
! [C: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ~ ! [X3: complex] :
( ( real_V1022390504157884413omplex @ X3 )
!= C ) ) ).
% vector_choose_size
thf(fact_493_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_494_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_495_power__numeral,axiom,
! [K: num,L: num] :
( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera1916890842035813515d_enat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_496_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_497_power__numeral,axiom,
! [K: num,L: num] :
( ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera4658534427948366547nnreal @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_498_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_499_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_500_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_501_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_502_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_503_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_504_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_505_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( K
!= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% nonneg_int_cases
thf(fact_506_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_507_exists__complex__root__nonzero,axiom,
! [Z2: complex,N2: nat] :
( ( Z2 != zero_zero_complex )
=> ( ( N2 != zero_zero_nat )
=> ~ ! [W2: complex] :
( ( W2 != zero_zero_complex )
=> ( Z2
!= ( power_power_complex @ W2 @ N2 ) ) ) ) ) ).
% exists_complex_root_nonzero
thf(fact_508_exists__complex__root,axiom,
! [N2: nat,Z2: complex] :
( ( N2 != zero_zero_nat )
=> ~ ! [W2: complex] :
( Z2
!= ( power_power_complex @ W2 @ N2 ) ) ) ).
% exists_complex_root
thf(fact_509_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_510_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_511_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_512_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_513_int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% int_int_eq
thf(fact_514_zdiv__int,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zdiv_int
thf(fact_515_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_516_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_517_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_518_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_519_ile0__eq,axiom,
! [N2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
= ( N2 = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_520_i0__lb,axiom,
! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).
% i0_lb
thf(fact_521_not__exp__less__eq__0__int,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_522_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_523_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_524_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_525_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_526_power__0,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ zero_zero_nat )
= one_on2969667320475766781nnreal ) ).
% power_0
thf(fact_527_power__0,axiom,
! [A: extended_enat] :
( ( power_8040749407984259932d_enat @ A @ zero_zero_nat )
= one_on7984719198319812577d_enat ) ).
% power_0
thf(fact_528_Multiseries__Expansion_Ointyness__0,axiom,
( zero_zero_real
= ( semiri5074537144036343181t_real @ zero_zero_nat ) ) ).
% Multiseries_Expansion.intyness_0
thf(fact_529_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_530_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_531_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_532_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_533_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_534_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_535_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_536_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_537_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_538_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_539_zero__reorient,axiom,
! [X: extended_enat] :
( ( zero_z5237406670263579293d_enat = X )
= ( X = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_540_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_541_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_542_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_543_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_544_one__reorient,axiom,
! [X: extended_enat] :
( ( one_on7984719198319812577d_enat = X )
= ( X = one_on7984719198319812577d_enat ) ) ).
% one_reorient
thf(fact_545_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_546_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_547_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_548_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_549_one__reorient,axiom,
! [X: extend8495563244428889912nnreal] :
( ( one_on2969667320475766781nnreal = X )
= ( X = one_on2969667320475766781nnreal ) ) ).
% one_reorient
thf(fact_550_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_551_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_552_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_553_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_554_int__ops_I3_J,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% int_ops(3)
thf(fact_555_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_556_power__eq__1__iff,axiom,
! [W: real,N2: nat] :
( ( ( power_power_real @ W @ N2 )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W )
= one_one_real )
| ( N2 = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_557_power__eq__1__iff,axiom,
! [W: complex,N2: nat] :
( ( ( power_power_complex @ W @ N2 )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W )
= one_one_real )
| ( N2 = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_558_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_559_inverse__of__nat__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( N2 != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_560_zero__le,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).
% zero_le
thf(fact_561_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_562_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_563_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X4 @ I3 ) @ one_one_nat )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( X4 @ I3 )
= zero_zero_real ) )
=> ( ( L2 @ X4 @ I3 )
= zero_zero_nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( X4 @ I3 )
= one_one_real ) )
=> ( ( L2 @ X4 @ I3 )
= one_one_nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( L2 @ X4 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I3 ) @ ( F @ X4 @ I3 ) ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( L2 @ X4 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_564_power__le__one__iff,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real )
= ( ( N2 = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_565_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_566_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_567_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_568_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_569_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_570_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_571_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_572_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_573_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_574_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_575_bits__1__div__exp,axiom,
! [N2: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_576_bits__1__div__exp,axiom,
! [N2: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_577_one__div__2__pow__eq,axiom,
! [N2: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_578_one__div__2__pow__eq,axiom,
! [N2: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_579_zero__le__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_580_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_581_Polygamma__converges_H,axiom,
! [Z2: real,N2: nat] :
( ( Z2 != zero_zero_real )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( summable_real
@ ^ [K2: nat] : ( inverse_inverse_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ N2 ) ) ) ) ) ).
% Polygamma_converges'
thf(fact_582_Polygamma__converges_H,axiom,
! [Z2: complex,N2: nat] :
( ( Z2 != zero_zero_complex )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( summable_complex
@ ^ [K2: nat] : ( invers8013647133539491842omplex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ N2 ) ) ) ) ) ).
% Polygamma_converges'
thf(fact_583_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_584_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_585_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_586_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_587_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_588_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_589_semiring__norm_I6_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(6)
thf(fact_590_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_591_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_592_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_593_summable__iff__shift,axiom,
! [F: nat > real,K: nat] :
( ( summable_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
= ( summable_real @ F ) ) ).
% summable_iff_shift
thf(fact_594_summable__iff__shift,axiom,
! [F: nat > complex,K: nat] :
( ( summable_complex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
= ( summable_complex @ F ) ) ).
% summable_iff_shift
thf(fact_595_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_596_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_597_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_598_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_599_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_600_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_601_add_Oright__neutral,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
= A ) ).
% add.right_neutral
thf(fact_602_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_603_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_604_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_605_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_606_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_607_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_608_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_609_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_610_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_611_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_612_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_613_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_614_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_615_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_616_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_617_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_618_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_619_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_620_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_621_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_622_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_623_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_624_add__eq__0__iff__both__eq__0,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_625_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_626_zero__eq__add__iff__both__eq__0,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( zero_z5237406670263579293d_enat
= ( plus_p3455044024723400733d_enat @ X @ Y ) )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_627_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_628_add__0,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
= A ) ).
% add_0
thf(fact_629_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_630_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_631_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_632_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_633_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_634_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_635_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_636_add__numeral__left,axiom,
! [V: num,W: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_637_add__numeral__left,axiom,
! [V: num,W: num,Z2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_638_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_639_add__numeral__left,axiom,
! [V: num,W: num,Z2: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_640_add__numeral__left,axiom,
! [V: num,W: num,Z2: complex] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_641_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_642_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_643_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_644_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_645_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_646_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_647_dvd__0__left__iff,axiom,
! [A: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ zero_z5237406670263579293d_enat @ A )
= ( A = zero_z5237406670263579293d_enat ) ) ).
% dvd_0_left_iff
thf(fact_648_dvd__0__left__iff,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
= ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_649_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_650_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_651_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_652_dvd__0__right,axiom,
! [A: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ zero_z5237406670263579293d_enat ) ).
% dvd_0_right
thf(fact_653_dvd__0__right,axiom,
! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_654_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_655_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_656_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_657_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_658_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_659_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_660_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% of_nat_add
thf(fact_661_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_add
thf(fact_662_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri4216267220026989637d_enat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).
% of_nat_add
thf(fact_663_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_664_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_665_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_666_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_667_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_668_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_669_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_670_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_671_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_672_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_673_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_674_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_675_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n1046097342994218471d_enat @ P )
= zero_z5237406670263579293d_enat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_676_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= zero_zero_complex )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_677_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= zero_zero_real )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_678_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_679_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= zero_zero_int )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_680_of__bool__eq_I1_J,axiom,
( ( zero_n1046097342994218471d_enat @ $false )
= zero_z5237406670263579293d_enat ) ).
% of_bool_eq(1)
thf(fact_681_of__bool__eq_I1_J,axiom,
( ( zero_n1201886186963655149omplex @ $false )
= zero_zero_complex ) ).
% of_bool_eq(1)
thf(fact_682_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_683_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_684_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_685_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n1046097342994218471d_enat @ P )
= one_on7984719198319812577d_enat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_686_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= one_one_complex )
= P ) ).
% of_bool_eq_1_iff
thf(fact_687_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= one_one_real )
= P ) ).
% of_bool_eq_1_iff
thf(fact_688_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_689_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_690_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n4168557817388953207nnreal @ P )
= one_on2969667320475766781nnreal )
= P ) ).
% of_bool_eq_1_iff
thf(fact_691_of__bool__eq_I2_J,axiom,
( ( zero_n1046097342994218471d_enat @ $true )
= one_on7984719198319812577d_enat ) ).
% of_bool_eq(2)
thf(fact_692_of__bool__eq_I2_J,axiom,
( ( zero_n1201886186963655149omplex @ $true )
= one_one_complex ) ).
% of_bool_eq(2)
thf(fact_693_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_694_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_695_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_696_of__bool__eq_I2_J,axiom,
( ( zero_n4168557817388953207nnreal @ $true )
= one_on2969667320475766781nnreal ) ).
% of_bool_eq(2)
thf(fact_697_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_698_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% of_nat_of_bool
thf(fact_699_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_nat_of_bool
thf(fact_700_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri6283507881447550617nnreal @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n4168557817388953207nnreal @ P ) ) ).
% of_nat_of_bool
thf(fact_701_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n1201886186963655149omplex @ P ) ) ).
% of_nat_of_bool
thf(fact_702_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri4216267220026989637d_enat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n1046097342994218471d_enat @ P ) ) ).
% of_nat_of_bool
thf(fact_703_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% of_nat_of_bool
thf(fact_704_add__le__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_705_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_706_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_707_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_708_add__le__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_709_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_710_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_711_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_712_le__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel1
thf(fact_713_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_714_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_715_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_716_le__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel2
thf(fact_717_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_718_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_719_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_720_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_721_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_722_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_723_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_724_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_725_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_726_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_727_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_728_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_729_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_730_div__add,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_731_div__add,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_732_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_733_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_734_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_735_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_736_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_737_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_738_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_739_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_740_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_741_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_742_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_743_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_744_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_745_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_746_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_747_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_748_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_749_one__add__one,axiom,
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_750_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_751_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_752_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_753_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_754_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_755_odd__of__bool__self,axiom,
! [P2: $o] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
= P2 ) ).
% odd_of_bool_self
thf(fact_756_odd__of__bool__self,axiom,
! [P2: $o] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
= P2 ) ).
% odd_of_bool_self
thf(fact_757_sum__power2__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_758_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_759_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_760_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_761_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_762_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_763_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_764_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_765_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_766_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_767_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_768_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_769_even__of__nat,axiom,
! [N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% even_of_nat
thf(fact_770_even__of__nat,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% even_of_nat
thf(fact_771_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_772_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_773_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_774_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_775_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_776_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_777_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_778_iadd__is__0,axiom,
! [M: extended_enat,N2: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N2 )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N2 = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_779_of__nat__dvd__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% of_nat_dvd_iff
thf(fact_780_of__nat__dvd__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% of_nat_dvd_iff
thf(fact_781_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_782_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_783_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_784_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_785_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_786_int__plus,axiom,
! [N2: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_787_zadd__int__left,axiom,
! [M: nat,N2: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_788_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_789_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_790_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_791_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_792_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_793_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_794_add_Oleft__commute,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_795_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_796_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_797_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_798_add_Ocommute,axiom,
( plus_p3455044024723400733d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] : ( plus_p3455044024723400733d_enat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_799_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_800_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_801_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_802_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_803_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_804_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_805_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_806_add_Oassoc,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add.assoc
thf(fact_807_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_808_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_809_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_810_group__cancel_Oadd2,axiom,
! [B3: extended_enat,K: extended_enat,B: extended_enat,A: extended_enat] :
( ( B3
= ( plus_p3455044024723400733d_enat @ K @ B ) )
=> ( ( plus_p3455044024723400733d_enat @ A @ B3 )
= ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_811_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_812_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_813_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_814_group__cancel_Oadd1,axiom,
! [A2: extended_enat,K: extended_enat,A: extended_enat,B: extended_enat] :
( ( A2
= ( plus_p3455044024723400733d_enat @ K @ A ) )
=> ( ( plus_p3455044024723400733d_enat @ A2 @ B )
= ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_815_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_816_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_817_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_818_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p3455044024723400733d_enat @ I @ K )
= ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_819_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_820_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_821_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_822_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_823_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_824_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_825_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_826_Multiseries__Expansion_Ointyness__simps_I1_J,axiom,
! [A: nat,B: nat] :
( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).
% Multiseries_Expansion.intyness_simps(1)
thf(fact_827_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_828_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_829_dvd__antisym,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_nat @ M @ N2 )
=> ( ( dvd_dvd_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% dvd_antisym
thf(fact_830_dvd__add__right__iff,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_831_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_832_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_833_dvd__add__left__iff,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_834_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_835_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_836_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_837_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_838_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_839_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_840_dvd__add,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ A @ B )
=> ( ( dvd_dv3785147216227455552d_enat @ A @ C )
=> ( dvd_dv3785147216227455552d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_841_dvd__add,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_842_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_843_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_844_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_845_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_846_summable__offset,axiom,
! [F: nat > real,K: nat] :
( ( summable_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
=> ( summable_real @ F ) ) ).
% summable_offset
thf(fact_847_summable__offset,axiom,
! [F: nat > complex,K: nat] :
( ( summable_complex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
=> ( summable_complex @ F ) ) ).
% summable_offset
thf(fact_848_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_849_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_850_norm__triangle__ineq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_851_norm__triangle__ineq,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_852_norm__triangle__le,axiom,
! [X: real,Y: real,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_853_norm__triangle__le,axiom,
! [X: complex,Y: complex,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_854_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_855_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_856_norm__add__rule__thm,axiom,
! [X1: real,B1: real,X22: real,B22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X1 ) @ B1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X22 ) @ B22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X1 @ X22 ) ) @ ( plus_plus_real @ B1 @ B22 ) ) ) ) ).
% norm_add_rule_thm
thf(fact_857_norm__add__rule__thm,axiom,
! [X1: complex,B1: real,X22: complex,B22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X1 ) @ B1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X22 ) @ B22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X1 @ X22 ) ) @ ( plus_plus_real @ B1 @ B22 ) ) ) ) ).
% norm_add_rule_thm
thf(fact_858_dvd__0__left,axiom,
! [A: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( A = zero_z5237406670263579293d_enat ) ) ).
% dvd_0_left
thf(fact_859_dvd__0__left,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_860_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_861_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_862_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_863_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A3: complex,B2: complex] :
( ( A3 = zero_zero_complex )
=> ( B2 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_864_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A3: real,B2: real] :
( ( A3 = zero_zero_real )
=> ( B2 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_865_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_866_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_867_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_868_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_869_one__dvd,axiom,
! [A: extended_enat] : ( dvd_dv3785147216227455552d_enat @ one_on7984719198319812577d_enat @ A ) ).
% one_dvd
thf(fact_870_one__dvd,axiom,
! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% one_dvd
thf(fact_871_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_872_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_873_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_874_one__dvd,axiom,
! [A: extend8495563244428889912nnreal] : ( dvd_dv1013850698770059486nnreal @ one_on2969667320475766781nnreal @ A ) ).
% one_dvd
thf(fact_875_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_876_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_877_dvd__div__eq__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
=> ( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_878_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_879_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_880_dvd__div__eq__cancel,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
=> ( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_881_dvd__div__eq__iff,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_882_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_883_dvd__div__eq__iff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_884_dvd__div__eq__iff,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_885_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ I @ J )
& ( K = L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_886_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_887_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_888_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_889_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( I = J )
& ( ord_le2932123472753598470d_enat @ K @ L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_890_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_891_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_892_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_893_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ I @ J )
& ( ord_le2932123472753598470d_enat @ K @ L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_894_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_895_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_896_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_897_add__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ C @ D )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_898_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_899_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_900_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_901_add__left__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ C @ A ) @ ( plus_p3455044024723400733d_enat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_902_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_903_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_904_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_905_less__eqE,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ~ ! [C3: extended_enat] :
( B
!= ( plus_p3455044024723400733d_enat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_906_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_907_add__right__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_908_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_909_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_910_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_911_le__iff__add,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
? [C2: extended_enat] :
( B2
= ( plus_p3455044024723400733d_enat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_912_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_913_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_914_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_915_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_916_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_917_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_918_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_919_pth__7_I1_J,axiom,
! [X: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X )
= X ) ).
% pth_7(1)
thf(fact_920_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_921_comm__monoid__add__class_Oadd__0,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_922_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_923_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_924_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_925_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_926_add_Ocomm__neutral,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
= A ) ).
% add.comm_neutral
thf(fact_927_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_928_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_929_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_930_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_931_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_932_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_933_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_934_pth__d,axiom,
! [X: complex] :
( ( plus_plus_complex @ X @ zero_zero_complex )
= X ) ).
% pth_d
thf(fact_935_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_936_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_937_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_938_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_939_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_940_dvd__power__same,axiom,
! [X: real,Y: real,N2: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_941_dvd__power__same,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_942_dvd__power__same,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_943_dvd__power__same,axiom,
! [X: int,Y: int,N2: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_944_dvd__power__same,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N2: nat] :
( ( dvd_dv1013850698770059486nnreal @ X @ Y )
=> ( dvd_dv1013850698770059486nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ ( power_6007165696250533058nnreal @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_945_dvd__power__same,axiom,
! [X: extended_enat,Y: extended_enat,N2: nat] :
( ( dvd_dv3785147216227455552d_enat @ X @ Y )
=> ( dvd_dv3785147216227455552d_enat @ ( power_8040749407984259932d_enat @ X @ N2 ) @ ( power_8040749407984259932d_enat @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_946_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_947_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_948_add__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_949_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_950_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_951_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_952_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_953_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_954_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_955_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_956_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_957_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_958_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_959_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_960_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_961_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
? [K2: nat] :
( N
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_962_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z4: int] :
? [N: nat] :
( Z4
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_963_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_964_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_965_even__addI_I1_J,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% even_addI(1)
thf(fact_966_even__addI_I1_J,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% even_addI(1)
thf(fact_967_odd__addI_I1_J,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_addI(1)
thf(fact_968_odd__addI_I1_J,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_addI(1)
thf(fact_969_odd__addI_I2_J,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_addI(2)
thf(fact_970_odd__addI_I2_J,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_addI(2)
thf(fact_971_summable__add,axiom,
! [F: nat > complex,G: nat > complex] :
( ( summable_complex @ F )
=> ( ( summable_complex @ G )
=> ( summable_complex
@ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% summable_add
thf(fact_972_summable__add,axiom,
! [F: nat > extended_enat,G: nat > extended_enat] :
( ( summab1538256873603986438d_enat @ F )
=> ( ( summab1538256873603986438d_enat @ G )
=> ( summab1538256873603986438d_enat
@ ^ [N: nat] : ( plus_p3455044024723400733d_enat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% summable_add
thf(fact_973_summable__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% summable_add
thf(fact_974_summable__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( summable_nat
@ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% summable_add
thf(fact_975_summable__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( summable_int
@ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% summable_add
thf(fact_976_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_977_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_978_summable__ignore__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_979_summable__ignore__initial__segment,axiom,
! [F: nat > complex,K: nat] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_980_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_981_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_982_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_983_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% of_bool_less_eq_one
thf(fact_984_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_985_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_986_split__of__bool__asm,axiom,
! [P: extend8495563244428889912nnreal > $o,P2: $o] :
( ( P @ ( zero_n4168557817388953207nnreal @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_on2969667320475766781nnreal ) )
| ( ~ P2
& ~ ( P @ zero_z7100319975126383169nnreal ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_987_split__of__bool__asm,axiom,
! [P: extended_enat > $o,P2: $o] :
( ( P @ ( zero_n1046097342994218471d_enat @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_on7984719198319812577d_enat ) )
| ( ~ P2
& ~ ( P @ zero_z5237406670263579293d_enat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_988_split__of__bool__asm,axiom,
! [P: complex > $o,P2: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_one_complex ) )
| ( ~ P2
& ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_989_split__of__bool__asm,axiom,
! [P: real > $o,P2: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_one_real ) )
| ( ~ P2
& ~ ( P @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_990_split__of__bool__asm,axiom,
! [P: nat > $o,P2: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_one_nat ) )
| ( ~ P2
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_991_split__of__bool__asm,axiom,
! [P: int > $o,P2: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_one_int ) )
| ( ~ P2
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_992_split__of__bool,axiom,
! [P: extend8495563244428889912nnreal > $o,P2: $o] :
( ( P @ ( zero_n4168557817388953207nnreal @ P2 ) )
= ( ( P2
=> ( P @ one_on2969667320475766781nnreal ) )
& ( ~ P2
=> ( P @ zero_z7100319975126383169nnreal ) ) ) ) ).
% split_of_bool
thf(fact_993_split__of__bool,axiom,
! [P: extended_enat > $o,P2: $o] :
( ( P @ ( zero_n1046097342994218471d_enat @ P2 ) )
= ( ( P2
=> ( P @ one_on7984719198319812577d_enat ) )
& ( ~ P2
=> ( P @ zero_z5237406670263579293d_enat ) ) ) ) ).
% split_of_bool
thf(fact_994_split__of__bool,axiom,
! [P: complex > $o,P2: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
= ( ( P2
=> ( P @ one_one_complex ) )
& ( ~ P2
=> ( P @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_995_split__of__bool,axiom,
! [P: real > $o,P2: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
= ( ( P2
=> ( P @ one_one_real ) )
& ( ~ P2
=> ( P @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_996_split__of__bool,axiom,
! [P: nat > $o,P2: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( ( P2
=> ( P @ one_one_nat ) )
& ( ~ P2
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_997_split__of__bool,axiom,
! [P: int > $o,P2: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( ( P2
=> ( P @ one_one_int ) )
& ( ~ P2
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_998_of__bool__def,axiom,
( zero_n4168557817388953207nnreal
= ( ^ [P3: $o] : ( if_Ext9135588136721118450nnreal @ P3 @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ) ) ).
% of_bool_def
thf(fact_999_of__bool__def,axiom,
( zero_n1046097342994218471d_enat
= ( ^ [P3: $o] : ( if_Extended_enat @ P3 @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ) ) ).
% of_bool_def
thf(fact_1000_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_1001_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_1002_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_1003_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_1004_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1005_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1006_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1007_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1008_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1009_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1010_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1011_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1012_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1013_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1014_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1015_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1016_add__decreasing,axiom,
! [A: extended_enat,C: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1017_add__decreasing,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ C @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1018_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1019_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1020_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1021_add__increasing,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1022_add__increasing,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ B @ C )
=> ( ord_less_eq_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1023_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1024_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1025_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1026_add__decreasing2,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1027_add__decreasing2,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1028_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1029_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1030_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1031_add__increasing2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1032_add__increasing2,axiom,
! [C: complex,B: complex,A: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ( ord_less_eq_complex @ B @ A )
=> ( ord_less_eq_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1033_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1034_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1035_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1036_add__nonneg__nonneg,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1037_add__nonneg__nonneg,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1038_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1039_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1040_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1041_add__nonpos__nonpos,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1042_add__nonpos__nonpos,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_nonpos_nonpos
thf(fact_1043_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1044_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1045_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1046_add__nonneg__eq__0__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ Y )
=> ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1047_add__nonneg__eq__0__iff,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ Y )
=> ( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1048_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1049_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1050_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1051_add__nonpos__eq__0__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ Y @ zero_z5237406670263579293d_enat )
=> ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1052_add__nonpos__eq__0__iff,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ Y @ zero_zero_complex )
=> ( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1053_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1054_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1055_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1056_div__power,axiom,
! [B: nat,A: nat,N2: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
= ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% div_power
thf(fact_1057_div__power,axiom,
! [B: int,A: int,N2: nat] :
( ( dvd_dvd_int @ B @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
= ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% div_power
thf(fact_1058_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_1059_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_1060_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: complex] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_1061_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_1062_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dv1013850698770059486nnreal @ ( power_6007165696250533058nnreal @ A @ M ) @ ( power_6007165696250533058nnreal @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_1063_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: extended_enat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dv3785147216227455552d_enat @ ( power_8040749407984259932d_enat @ A @ M ) @ ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_1064_power__le__dvd,axiom,
! [A: real,N2: nat,B: real,M: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_1065_power__le__dvd,axiom,
! [A: nat,N2: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_1066_power__le__dvd,axiom,
! [A: complex,N2: nat,B: complex,M: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_1067_power__le__dvd,axiom,
! [A: int,N2: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_1068_power__le__dvd,axiom,
! [A: extend8495563244428889912nnreal,N2: nat,B: extend8495563244428889912nnreal,M: nat] :
( ( dvd_dv1013850698770059486nnreal @ ( power_6007165696250533058nnreal @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dv1013850698770059486nnreal @ ( power_6007165696250533058nnreal @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_1069_power__le__dvd,axiom,
! [A: extended_enat,N2: nat,B: extended_enat,M: nat] :
( ( dvd_dv3785147216227455552d_enat @ ( power_8040749407984259932d_enat @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dv3785147216227455552d_enat @ ( power_8040749407984259932d_enat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_1070_dvd__power__le,axiom,
! [X: real,Y: real,N2: nat,M: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_1071_dvd__power__le,axiom,
! [X: nat,Y: nat,N2: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_1072_dvd__power__le,axiom,
! [X: complex,Y: complex,N2: nat,M: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_1073_dvd__power__le,axiom,
! [X: int,Y: int,N2: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_1074_dvd__power__le,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N2: nat,M: nat] :
( ( dvd_dv1013850698770059486nnreal @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dv1013850698770059486nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ ( power_6007165696250533058nnreal @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_1075_dvd__power__le,axiom,
! [X: extended_enat,Y: extended_enat,N2: nat,M: nat] :
( ( dvd_dv3785147216227455552d_enat @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dv3785147216227455552d_enat @ ( power_8040749407984259932d_enat @ X @ N2 ) @ ( power_8040749407984259932d_enat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_1076_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_1077_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_1078_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_1079_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_1080_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ X ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ X ) @ one_on2969667320475766781nnreal ) ) ).
% one_plus_numeral_commute
thf(fact_1081_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% one_plus_numeral_commute
thf(fact_1082_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_1083_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_1084_numeral__Bit0,axiom,
! [N2: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_1085_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_1086_numeral__Bit0,axiom,
! [N2: num] :
( ( numera4658534427948366547nnreal @ ( bit0 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ ( numera4658534427948366547nnreal @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_1087_numeral__Bit0,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_1088_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_1089_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_1090_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_1091_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_1092_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera4658534427948366547nnreal @ ( bit0 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ ( numera4658534427948366547nnreal @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_1093_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_1094_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1095_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1096_odd__Numeral1,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ one ) ) ).
% odd_Numeral1
thf(fact_1097_real__of__nat__div,axiom,
! [D: nat,N2: nat] :
( ( dvd_dvd_nat @ D @ N2 )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_1098_numeral__eq__of__nat,axiom,
( numera4658534427948366547nnreal
= ( ^ [A3: num] : ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ A3 ) ) ) ) ).
% numeral_eq_of_nat
thf(fact_1099_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_1100_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% dvd_power_iff_le
thf(fact_1101_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N5: nat] :
( ( P @ N5 )
=> ( P @ ( plus_plus_nat @ N5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_1102_triangle__lemma,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ Z2 ) ) ) ) ) ) ).
% triangle_lemma
thf(fact_1103_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_1104_nat__add__1__add__1,axiom,
! [N2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_1105_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1106_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
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_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1109_int__dvd__int__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% int_dvd_int_iff
thf(fact_1110_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1111_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1112_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1113_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1114_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1115_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1116_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1117_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1118_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1119_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1120_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1121_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1122_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_1123_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1124_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1125_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1126_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1127_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( P @ M3 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1128_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1129_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1130_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1131_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1132_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1133_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1134_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1135_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1136_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1137_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1138_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1139_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1140_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1141_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1142_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1143_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_1144_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1145_zdvd__antisym__nonneg,axiom,
! [M: int,N2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N2 )
=> ( ( dvd_dvd_int @ M @ N2 )
=> ( ( dvd_dvd_int @ N2 @ M )
=> ( M = N2 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1146_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1147_nat__dvd__not__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1148_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1149_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1150_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1151_kuhn__lemma,axiom,
! [P2: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X3: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I3 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( Label @ X3 @ I2 )
= zero_zero_nat )
| ( ( Label @ X3 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I3 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X3 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X3 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I3 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X3 @ I2 )
= P2 )
=> ( ( Label @ X3 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_nat @ ( Q2 @ I3 ) @ P2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ? [R3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( R3 @ J3 ) )
& ( ord_less_eq_nat @ ( R3 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( S3 @ J3 ) )
& ( ord_less_eq_nat @ ( S3 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I3 )
!= ( Label @ S3 @ I3 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1152_dvd__imp__le,axiom,
! [K: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_1153_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1154_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1155_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1156_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N2 )
= M )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1157_less__exp,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% less_exp
thf(fact_1158_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% power_dvd_imp_le
thf(fact_1159_odd__pos,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% odd_pos
thf(fact_1160_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N5: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N5 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1161_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N5: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N5 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1162_log__induct,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 )
=> ( ( P @ ( divide_divide_nat @ N5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( P @ N5 ) ) )
=> ( P @ N2 ) ) ) ) ).
% log_induct
thf(fact_1163_i0__less,axiom,
! [N2: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
= ( N2 != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1164_semiring__norm_I78_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(78)
thf(fact_1165_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1166_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1167_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1168_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1169_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1170_enat__ord__number_I2_J,axiom,
! [M: num,N2: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% enat_ord_number(2)
thf(fact_1171_one__less__numeral,axiom,
! [N2: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral
thf(fact_1172_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_1173_numeral__less__real__of__nat__iff,axiom,
! [W: num,N2: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_1174_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1175_zdvd__not__zless,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N2 )
=> ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1176_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1177_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1178_not__iless0,axiom,
! [N2: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1179_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1180_zdvd__imp__le,axiom,
! [Z2: int,N2: int] :
( ( dvd_dvd_int @ Z2 @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ord_less_eq_int @ Z2 @ N2 ) ) ) ).
% zdvd_imp_le
thf(fact_1181_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N5: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N5 ) ) ) ).
% real_arch_pow
thf(fact_1182_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1183_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1184_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1185_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1186_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1187_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1188_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N5: nat] : ( ord_less_real @ ( power_power_real @ X @ N5 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1189_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1190_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1191_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1192_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1193_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1194_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1195_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1196_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1197_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1198_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1199_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1200_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1201_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1202_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1203_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1204_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1205_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1206_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_1207_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_1208_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_1209_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
& ( dvd_dvd_nat @ B2 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1210_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_1211_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_1212_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1213_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1214_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1215_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1216_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
| ( A3 = B2 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1217_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1218_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1219_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1220_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1221_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N5 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% pos_int_cases
thf(fact_1222_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
& ( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1223_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1224_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [N: nat] :
( ( N != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_1225_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D2: real,E2: real] :
( ( ord_less_real @ D2 @ E2 )
=> ( ( P @ D2 )
=> ( P @ E2 ) ) )
=> ( ! [N5: nat] :
( ( N5 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N5 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_1226_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1227_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K3 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1228_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1229_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1230_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1231_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1232_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1233_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1234_dvd__pos__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_nat @ M @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1235_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_1236_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1237_one__divide__one__divide__ennreal,axiom,
! [C: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ one_on2969667320475766781nnreal @ ( divide4826598186094686858nnreal @ one_on2969667320475766781nnreal @ C ) )
= C ) ).
% one_divide_one_divide_ennreal
thf(fact_1238_ennreal__inverse__1,axiom,
( ( invers7556275967461373580nnreal @ one_on2969667320475766781nnreal )
= one_on2969667320475766781nnreal ) ).
% ennreal_inverse_1
thf(fact_1239_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_1240_enat__less__induct,axiom,
! [P: extended_enat > $o,N2: extended_enat] :
( ! [N5: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_le72135733267957522d_enat @ M3 @ N5 )
=> ( P @ M3 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% enat_less_induct
thf(fact_1241_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N2 )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N2 )
= A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1242_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1243_ennreal__zero__divide,axiom,
! [X: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ zero_z7100319975126383169nnreal @ X )
= zero_z7100319975126383169nnreal ) ).
% ennreal_zero_divide
thf(fact_1244_add__divide__distrib__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C )
= ( plus_p1859984266308609217nnreal @ ( divide4826598186094686858nnreal @ A @ C ) @ ( divide4826598186094686858nnreal @ B @ C ) ) ) ).
% add_divide_distrib_ennreal
thf(fact_1245_ennreal__inverse__power,axiom,
! [X: extend8495563244428889912nnreal,N2: nat] :
( ( invers7556275967461373580nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) )
= ( power_6007165696250533058nnreal @ ( invers7556275967461373580nnreal @ X ) @ N2 ) ) ).
% ennreal_inverse_power
thf(fact_1246_power__divide__distrib__ennreal,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N2: nat] :
( ( power_6007165696250533058nnreal @ ( divide4826598186094686858nnreal @ X @ Y ) @ N2 )
= ( divide4826598186094686858nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ ( power_6007165696250533058nnreal @ Y @ N2 ) ) ) ).
% power_divide_distrib_ennreal
thf(fact_1247_divide__right__mono__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ ( divide4826598186094686858nnreal @ A @ C ) @ ( divide4826598186094686858nnreal @ B @ C ) ) ) ).
% divide_right_mono_ennreal
thf(fact_1248_real__arch__invD,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N5: nat] :
( ( N5 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N5 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N5 ) ) @ E ) ) ) ).
% real_arch_invD
thf(fact_1249_power__mono__ennreal,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N2: nat] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ord_le3935885782089961368nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ ( power_6007165696250533058nnreal @ Y @ N2 ) ) ) ).
% power_mono_ennreal
thf(fact_1250_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1251_Multiseries__Expansion__Bounds_Oneg__imp__inverse__neg,axiom,
! [F: real > real,X4: real] :
( ( ord_less_real @ ( F @ X4 ) @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ ( F @ X4 ) ) @ zero_zero_real ) ) ).
% Multiseries_Expansion_Bounds.neg_imp_inverse_neg
thf(fact_1252_Multiseries__Expansion__Bounds_Oeq__zero__imp__nonneg,axiom,
! [X: real] :
( ( X = zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% Multiseries_Expansion_Bounds.eq_zero_imp_nonneg
thf(fact_1253_Multiseries__Expansion__Bounds_Opos__imp__inverse__pos,axiom,
! [F: real > real,X4: real] :
( ( ord_less_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( F @ X4 ) ) ) ) ).
% Multiseries_Expansion_Bounds.pos_imp_inverse_pos
thf(fact_1254_set__decode__0,axiom,
! [X: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_1255_seq__mono__lemma,axiom,
! [M: nat,D: nat > real,E: nat > real] :
( ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ord_less_real @ ( D @ N5 ) @ ( E @ N5 ) ) )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ord_less_eq_real @ ( E @ N5 ) @ ( E @ M ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M @ N6 )
=> ( ord_less_real @ ( D @ N6 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1256_subset__decode__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% subset_decode_imp_le
thf(fact_1257_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect_nat
@ ^ [N: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% set_decode_def
thf(fact_1258_conj__le__cong,axiom,
! [X: int,X5: int,P: $o,P4: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1259_imp__le__cong,axiom,
! [X: int,X5: int,P: $o,P4: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1260_Bernoulli__inequality__even,axiom,
! [N2: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% Bernoulli_inequality_even
thf(fact_1261_mod__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( modulo_modulo_nat @ M @ N2 )
= M ) ) ).
% mod_less
thf(fact_1262_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_1263_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1264_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1265_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
% Helper facts (13)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( summable_real
@ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
%------------------------------------------------------------------------------