TPTP Problem File: SLH0982^1.p
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%------------------------------------------------------------------------------
% 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 : Frequency_Moments/0085_Frequency_Moment_0/prob_01218_055972__19952594_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1124 ( 747 unt; 111 typ; 0 def)
% Number of atoms : 2027 (1689 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 7626 ( 256 ~; 66 |; 88 &;6817 @)
% ( 0 <=>; 399 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 4 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 352 ( 352 >; 0 *; 0 +; 0 <<)
% Number of symbols : 95 ( 92 usr; 18 con; 0-3 aty)
% Number of variables : 2236 ( 200 ^;2025 !; 11 ?;2236 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:17:18.782
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J_Mt__Real__Oreal_J_J,type,
set_Pr1128732697603872439t_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J_J,type,
filter3199273883467263174at_rat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
produc5691113562410904374at_rat: $tType ).
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
numera4273646738625120315l_num1: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
filter8908148590052480407at_rat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
numera6367994245245682809l_num1: $tType ).
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2417102609627094330l_num1: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
set_nat_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Rat__Orat_J,type,
filter_rat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (92)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
at_top_nat: filter_nat ).
thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
at_top_real: filter_real ).
thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J,type,
prod_f1623372399986984716at_rat: filter_nat > filter8908148590052480407at_rat > filter3199273883467263174at_rat ).
thf(sy_c_Filter_Oprod__filter_001t__Rat__Orat_001t__Rat__Orat,type,
prod_filter_rat_rat: filter_rat > filter_rat > filter8908148590052480407at_rat ).
thf(sy_c_Float_Odiv__twopow,type,
div_twopow: int > nat > int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
one_on7795324986448017462l_num1: numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
one_on7819281148064737470l_num1: numera6367994245245682809l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
plus_p1441664204671982194l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
plus_p2313304076027620419l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
times_2938166955517408246l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
times_8498157372700349887l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
zero_z2241845390563828978l_num1: numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
zero_z5982384998485459395l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_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__Rat__Orat,type,
if_rat: $o > rat > rat > rat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Landau__Symbols_Obigo_001t__Nat__Onat_001t__Real__Oreal,type,
landau_bigo_nat_real: filter_nat > ( nat > real ) > set_nat_real ).
thf(sy_c_Landau__Symbols_Obigo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J_001t__Real__Oreal,type,
landau6322959426088225955t_real: filter3199273883467263174at_rat > ( produc5691113562410904374at_rat > real ) > set_Pr1128732697603872439t_real ).
thf(sy_c_Landau__Symbols_Obigo_001t__Real__Oreal_001t__Real__Oreal,type,
landau308303187242894617l_real: filter_real > ( real > real ) > set_real_real ).
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__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
semiri5667362542588693146l_num1: nat > numera4273646738625120315l_num1 ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
semiri1795386414920522267l_num1: nat > numera2417102609627094330l_num1 ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
semiri681578069525770553at_rat: nat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
numera7754357348821619680l_num1: num > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2161328050825114965l_num1: num > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
numera6112219686443703444l_num1: num > numera6367994245245682809l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
numeral_numeral_rat: num > rat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
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__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
power_1002146276965246001l_num1: numera4273646738625120315l_num1 > nat > numera4273646738625120315l_num1 ).
thf(sy_c_Power_Opower__class_Opower_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
power_7402600760894073284l_num1: numera2417102609627094330l_num1 > nat > numera2417102609627094330l_num1 ).
thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
power_power_rat: rat > nat > rat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Rat__Orat,type,
field_2639924705303425560at_rat: rat > rat ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
field_7254667332652039916t_real: rat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__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_Set_OCollect_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
collect_nat_real: ( ( nat > real ) > $o ) > set_nat_real ).
thf(sy_c_Set_OCollect_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J_Mt__Real__Oreal_J,type,
collec8488528251386215510t_real: ( ( produc5691113562410904374at_rat > real ) > $o ) > set_Pr1128732697603872439t_real ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
collect_real_real: ( ( real > real ) > $o ) > set_real_real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Rat__Orat,type,
topolo4023969691036296984in_rat: rat > set_rat > filter_rat ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
member_nat_real: ( nat > real ) > set_nat_real > $o ).
thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J_Mt__Real__Oreal_J,type,
member1610887461201275416t_real: ( produc5691113562410904374at_rat > real ) > set_Pr1128732697603872439t_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v__092_060delta_062__of____,type,
delta_of: produc5691113562410904374at_rat > rat ).
thf(sy_v__092_060epsilon_062__of____,type,
epsilon_of: produc5691113562410904374at_rat > rat ).
thf(sy_v_g____,type,
g: produc5691113562410904374at_rat > real ).
thf(sy_v_n__of____,type,
n_of: produc5691113562410904374at_rat > nat ).
thf(sy_v_s__of____,type,
s_of: produc5691113562410904374at_rat > nat ).
thf(sy_v_t__of____,type,
t_of: produc5691113562410904374at_rat > nat ).
% Relevant facts (1003)
thf(fact_0__C7_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) ) ).
% "7"
thf(fact_1__C2_C,axiom,
( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) ) ) ) ) ).
% "2"
thf(fact_2__C4_C,axiom,
( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% "4"
thf(fact_3__C3_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( plus_plus_real @ ( ln_ln_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) @ ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) ) ) ) ) ) ).
% "3"
thf(fact_4__C5_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( semiri5074537144036343181t_real @ ( t_of @ X ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% "5"
thf(fact_5__C8_C,axiom,
( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( plus_plus_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( ln_ln_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) @ ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) ) ) ) ) ) ) ) ).
% "8"
thf(fact_6__C10_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) ) @ g ) ) ).
% "10"
thf(fact_7__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ireal__of__rat_A_I_092_060epsilon_062__of_Ax_J_J_J_A_092_060in_062_AO_091sequentially_A_092_060times_062_092_060_094sub_062F_Aat__right_A0_A_092_060times_062_092_060_094sub_062F_Aat__right_A0_093_I_092_060lambda_062x_O_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060epsilon_062__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X ) ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X ) ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (real_of_rat (\<epsilon>_of x))) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (1 / real_of_rat (\<epsilon>_of x)))\<close>
thf(fact_8__C1_C,axiom,
( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X ) ) ) ) ) ) ).
% "1"
thf(fact_9_sum__power2__eq__zero__iff,axiom,
! [X2: rat,Y: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X2 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_10_sum__power2__eq__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_11_sum__power2__eq__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_12_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_13_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_14_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_15_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_16_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_17_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_18_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_19_zero__eq__power2,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% zero_eq_power2
thf(fact_20_arith__special_I3_J,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% arith_special(3)
thf(fact_21_arith__special_I3_J,axiom,
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% arith_special(3)
thf(fact_22_arith__special_I3_J,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% arith_special(3)
thf(fact_23_arith__special_I3_J,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% arith_special(3)
thf(fact_24_arith__special_I3_J,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% arith_special(3)
thf(fact_25_arith__special_I3_J,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% arith_special(3)
thf(fact_26_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N )
= ( semiri681578069525770553at_rat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_27_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_28_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_29_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_30_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri681578069525770553at_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_31_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_32_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_33_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_34_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_35_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_36_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_37_arithmetic__simps_I5_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% arithmetic_simps(5)
thf(fact_38_eq__num__simps_I6_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% eq_num_simps(6)
thf(fact_39_num_Oinject_I1_J,axiom,
! [X22: num,Y2: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y2 ) )
= ( X22 = Y2 ) ) ).
% num.inject(1)
thf(fact_40_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_41_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_42_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_43_power__one__right,axiom,
! [A: rat] :
( ( power_power_rat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_44_arithmetic__simps_I62_J,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% arithmetic_simps(62)
thf(fact_45_arithmetic__simps_I62_J,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% arithmetic_simps(62)
thf(fact_46_arithmetic__simps_I62_J,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_47_arithmetic__simps_I62_J,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% arithmetic_simps(62)
thf(fact_48_arithmetic__simps_I63_J,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% arithmetic_simps(63)
thf(fact_49_arithmetic__simps_I63_J,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% arithmetic_simps(63)
thf(fact_50_arithmetic__simps_I63_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_51_arithmetic__simps_I63_J,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% arithmetic_simps(63)
thf(fact_52_arithmetic__simps_I49_J,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% arithmetic_simps(49)
thf(fact_53_arithmetic__simps_I49_J,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% arithmetic_simps(49)
thf(fact_54_arithmetic__simps_I49_J,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% arithmetic_simps(49)
thf(fact_55_arithmetic__simps_I49_J,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% arithmetic_simps(49)
thf(fact_56_arithmetic__simps_I50_J,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% arithmetic_simps(50)
thf(fact_57_arithmetic__simps_I50_J,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% arithmetic_simps(50)
thf(fact_58_arithmetic__simps_I50_J,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% arithmetic_simps(50)
thf(fact_59_arithmetic__simps_I50_J,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% arithmetic_simps(50)
thf(fact_60_arithmetic__simps_I78_J,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_61_arithmetic__simps_I78_J,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_62_arithmetic__simps_I78_J,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_63_arithmetic__simps_I78_J,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_64_arithmetic__simps_I78_J,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_65_arithmetic__simps_I79_J,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% arithmetic_simps(79)
thf(fact_66_arithmetic__simps_I79_J,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% arithmetic_simps(79)
thf(fact_67_arithmetic__simps_I79_J,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% arithmetic_simps(79)
thf(fact_68_arithmetic__simps_I79_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% arithmetic_simps(79)
thf(fact_69_arithmetic__simps_I79_J,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% arithmetic_simps(79)
thf(fact_70_mem__Collect__eq,axiom,
! [A: produc5691113562410904374at_rat > real,P: ( produc5691113562410904374at_rat > real ) > $o] :
( ( member1610887461201275416t_real @ A @ ( collec8488528251386215510t_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: real > real,P: ( real > real ) > $o] :
( ( member_real_real @ A @ ( collect_real_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ A @ ( collect_nat_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_Pr1128732697603872439t_real] :
( ( collec8488528251386215510t_real
@ ^ [X: produc5691113562410904374at_rat > real] : ( member1610887461201275416t_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A2: set_real_real] :
( ( collect_real_real
@ ^ [X: real > real] : ( member_real_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A2: set_nat_real] :
( ( collect_nat_real
@ ^ [X: nat > real] : ( member_nat_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_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_79_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ W ) @ Z ) )
= ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_80_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z ) )
= ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_81_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_82_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_83_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_84_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_85_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_86_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_87_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_88_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_89_add__numeral__left,axiom,
! [V: num,W: num,Z: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_90_add__numeral__left,axiom,
! [V: num,W: num,Z: numera2417102609627094330l_num1] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ W ) @ Z ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_91_add__numeral__left,axiom,
! [V: num,W: num,Z: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_92_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_93_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_94_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_95_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_96_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_97_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_98_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_99_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_100_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_101_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_102_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_103_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_104_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_105_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% Num.of_nat_simps(1)
thf(fact_106_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% Num.of_nat_simps(1)
thf(fact_107_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% Num.of_nat_simps(1)
thf(fact_108_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% Num.of_nat_simps(1)
thf(fact_109_arithmetic__simps_I1_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% arithmetic_simps(1)
thf(fact_110_eq__num__simps_I2_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% eq_num_simps(2)
thf(fact_111_eq__num__simps_I4_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% eq_num_simps(4)
thf(fact_112_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_113_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_114_power__one,axiom,
! [N: nat] :
( ( power_7402600760894073284l_num1 @ one_on3868389512446148991l_num1 @ N )
= one_on3868389512446148991l_num1 ) ).
% power_one
thf(fact_115_power__one,axiom,
! [N: nat] :
( ( power_1002146276965246001l_num1 @ one_on7795324986448017462l_num1 @ N )
= one_on7795324986448017462l_num1 ) ).
% power_one
thf(fact_116_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_117_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_118_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_119_power__one,axiom,
! [N: nat] :
( ( power_power_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_one
thf(fact_120_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_121_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_122_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_123_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1795386414920522267l_num1 @ one_one_nat )
= one_on3868389512446148991l_num1 ) ).
% Num.of_nat_simps(2)
thf(fact_124_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri5667362542588693146l_num1 @ one_one_nat )
= one_on7795324986448017462l_num1 ) ).
% Num.of_nat_simps(2)
thf(fact_125_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% Num.of_nat_simps(2)
thf(fact_126_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% Num.of_nat_simps(2)
thf(fact_127_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% Num.of_nat_simps(2)
thf(fact_128_Num_Oof__nat__simps_I4_J,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% Num.of_nat_simps(4)
thf(fact_129_Num_Oof__nat__simps_I4_J,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% Num.of_nat_simps(4)
thf(fact_130_Num_Oof__nat__simps_I4_J,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% Num.of_nat_simps(4)
thf(fact_131_Num_Oof__nat__simps_I4_J,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% Num.of_nat_simps(4)
thf(fact_132_sum__squares__eq__zero__iff,axiom,
! [X2: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X2 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_133_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_134_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_135_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_136_distrib__right__numeral,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,V: num] :
( ( times_8498157372700349887l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ V ) ) @ ( times_8498157372700349887l_num1 @ B @ ( numera2161328050825114965l_num1 @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_137_distrib__right__numeral,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,V: num] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ B @ ( numera7754357348821619680l_num1 @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_138_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_139_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_140_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_141_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_142_distrib__left__numeral,axiom,
! [V: num,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( plus_p2313304076027620419l_num1 @ B @ C ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ B ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_143_distrib__left__numeral,axiom,
! [V: num,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ B @ C ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_144_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_145_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_146_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_147_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_148_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_149_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_150_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_151_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_152_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_153_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_154_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_155_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_156_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_157_power__add__numeral2,axiom,
! [A: rat,M: num,N: num,B: rat] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_158_power__add__numeral2,axiom,
! [A: real,M: num,N: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_159_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_160_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_161_power__add__numeral,axiom,
! [A: rat,M: num,N: num] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_162_power__add__numeral,axiom,
! [A: real,M: num,N: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_163_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_164_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_165_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1795386414920522267l_num1 @ ( numeral_numeral_nat @ N ) )
= ( numera2161328050825114965l_num1 @ N ) ) ).
% of_nat_numeral
thf(fact_166_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5667362542588693146l_num1 @ ( numeral_numeral_nat @ N ) )
= ( numera7754357348821619680l_num1 @ N ) ) ).
% of_nat_numeral
thf(fact_167_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_168_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_169_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_170_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X2 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_171_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri681578069525770553at_rat @ X2 )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_172_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X2 )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_173_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X2 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_174_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_175_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
= ( semiri681578069525770553at_rat @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_176_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_177_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_178_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_179_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_180_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_181_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_182_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_183_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_184_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_185_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_186_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_187_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_188_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_189_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_190_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_191_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_192_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_193_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_194_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_195_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_196_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_197_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_198_g__def,axiom,
( g
= ( ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X ) ) ) ) @ ( plus_plus_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( ln_ln_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) @ ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) ) ) ) ) ) ) ) ) ).
% g_def
thf(fact_199__C6_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( semiri5074537144036343181t_real @ ( s_of @ X ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X ) ) ) ) ) ) ).
% "6"
thf(fact_200__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ireal_A_Is__of_Ax_J_A_L_A1_J_J_A_092_060in_062_AO_091sequentially_A_092_060times_062_092_060_094sub_062F_Aat__right_A0_A_092_060times_062_092_060_094sub_062F_Aat__right_A0_093_I_092_060lambda_062x_O_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060epsilon_062__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( s_of @ X ) ) @ one_one_real ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X ) ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (real (s_of x) + 1)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (1 / real_of_rat (\<epsilon>_of x)))\<close>
thf(fact_201_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_202_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_203_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_204_power__add,axiom,
! [A: rat,M: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_205_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_206_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_207_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_208_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_209_power__0,axiom,
! [A: numera2417102609627094330l_num1] :
( ( power_7402600760894073284l_num1 @ A @ zero_zero_nat )
= one_on3868389512446148991l_num1 ) ).
% power_0
thf(fact_210_power__0,axiom,
! [A: numera4273646738625120315l_num1] :
( ( power_1002146276965246001l_num1 @ A @ zero_zero_nat )
= one_on7795324986448017462l_num1 ) ).
% power_0
thf(fact_211_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_212_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_213_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_214_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_215_more__arith__simps_I11_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_216_more__arith__simps_I11_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_217_more__arith__simps_I11_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_218_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_219_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_220_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_221_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_222_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_223_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_224_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_225_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_226_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_227_eq__numeral__extra_I1_J,axiom,
zero_z5982384998485459395l_num1 != one_on3868389512446148991l_num1 ).
% eq_numeral_extra(1)
thf(fact_228_eq__numeral__extra_I1_J,axiom,
zero_z2241845390563828978l_num1 != one_on7795324986448017462l_num1 ).
% eq_numeral_extra(1)
thf(fact_229_eq__numeral__extra_I1_J,axiom,
zero_zero_rat != one_one_rat ).
% eq_numeral_extra(1)
thf(fact_230_eq__numeral__extra_I1_J,axiom,
zero_zero_nat != one_one_nat ).
% eq_numeral_extra(1)
thf(fact_231_eq__numeral__extra_I1_J,axiom,
zero_zero_real != one_one_real ).
% eq_numeral_extra(1)
thf(fact_232_eq__numeral__extra_I1_J,axiom,
zero_zero_int != one_one_int ).
% eq_numeral_extra(1)
thf(fact_233_pth__d,axiom,
! [X2: real] :
( ( plus_plus_real @ X2 @ zero_zero_real )
= X2 ) ).
% pth_d
thf(fact_234_pth__7_I1_J,axiom,
! [X2: real] :
( ( plus_plus_real @ zero_zero_real @ X2 )
= X2 ) ).
% pth_7(1)
thf(fact_235_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_236_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_237_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_238_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_239_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_240_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_241_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_242_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_243_num_Odistinct_I1_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% num.distinct(1)
thf(fact_244_power__commuting__commutes,axiom,
! [X2: rat,Y: rat,N: nat] :
( ( ( times_times_rat @ X2 @ Y )
= ( times_times_rat @ Y @ X2 ) )
=> ( ( times_times_rat @ ( power_power_rat @ X2 @ N ) @ Y )
= ( times_times_rat @ Y @ ( power_power_rat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_245_power__commuting__commutes,axiom,
! [X2: real,Y: real,N: nat] :
( ( ( times_times_real @ X2 @ Y )
= ( times_times_real @ Y @ X2 ) )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_246_power__commuting__commutes,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y )
= ( times_times_nat @ Y @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_247_power__commuting__commutes,axiom,
! [X2: int,Y: int,N: nat] :
( ( ( times_times_int @ X2 @ Y )
= ( times_times_int @ Y @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_248_power__mult__distrib,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_249_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_250_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_251_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_252_power__commutes,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_commutes
thf(fact_253_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_254_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_255_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_256_power__divide,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
= ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_divide
thf(fact_257_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_258_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_259_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_260_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat ) ) ).
% one_plus_numeral_commute
thf(fact_261_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ X2 ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ X2 ) @ one_on3868389512446148991l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_262_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ X2 ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X2 ) @ one_on7795324986448017462l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_263_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_264_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_265_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_266_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_7402600760894073284l_num1 @ zero_z5982384998485459395l_num1 @ N )
= one_on3868389512446148991l_num1 ) )
& ( ( N != zero_zero_nat )
=> ( ( power_7402600760894073284l_num1 @ zero_z5982384998485459395l_num1 @ N )
= zero_z5982384998485459395l_num1 ) ) ) ).
% power_0_left
thf(fact_267_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_1002146276965246001l_num1 @ zero_z2241845390563828978l_num1 @ N )
= one_on7795324986448017462l_num1 ) )
& ( ( N != zero_zero_nat )
=> ( ( power_1002146276965246001l_num1 @ zero_z2241845390563828978l_num1 @ N )
= zero_z2241845390563828978l_num1 ) ) ) ).
% power_0_left
thf(fact_268_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_269_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_270_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_271_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_272_mult__numeral__1__right,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_273_mult__numeral__1__right,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_274_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_275_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_276_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_277_mult__numeral__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_278_mult__numeral__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_279_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_280_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_281_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_282_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_283_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_284_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_285_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_286_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_287_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_Bit0
thf(fact_288_numeral__Bit0,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit0 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) ) ).
% numeral_Bit0
thf(fact_289_numeral__Bit0,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ).
% numeral_Bit0
thf(fact_290_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_291_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_292_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_293_left__right__inverse__power,axiom,
! [X2: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1,N: nat] :
( ( ( times_8498157372700349887l_num1 @ X2 @ Y )
= one_on3868389512446148991l_num1 )
=> ( ( times_8498157372700349887l_num1 @ ( power_7402600760894073284l_num1 @ X2 @ N ) @ ( power_7402600760894073284l_num1 @ Y @ N ) )
= one_on3868389512446148991l_num1 ) ) ).
% left_right_inverse_power
thf(fact_294_left__right__inverse__power,axiom,
! [X2: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1,N: nat] :
( ( ( times_2938166955517408246l_num1 @ X2 @ Y )
= one_on7795324986448017462l_num1 )
=> ( ( times_2938166955517408246l_num1 @ ( power_1002146276965246001l_num1 @ X2 @ N ) @ ( power_1002146276965246001l_num1 @ Y @ N ) )
= one_on7795324986448017462l_num1 ) ) ).
% left_right_inverse_power
thf(fact_295_left__right__inverse__power,axiom,
! [X2: rat,Y: rat,N: nat] :
( ( ( times_times_rat @ X2 @ Y )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y @ N ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_296_left__right__inverse__power,axiom,
! [X2: real,Y: real,N: nat] :
( ( ( times_times_real @ X2 @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_297_left__right__inverse__power,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_298_left__right__inverse__power,axiom,
! [X2: int,Y: int,N: nat] :
( ( ( times_times_int @ X2 @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_299_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_300_power__one__over,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_one_over
thf(fact_301_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_302_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_303_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_304_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_305_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit0 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) ) ).
% numeral_code(2)
thf(fact_306_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ).
% numeral_code(2)
thf(fact_307_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_308_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_309_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_310_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( numeral_numeral_rat @ W )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_311_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_312_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( numeral_numeral_rat @ W ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_313_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_314_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_315_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_316_power__numeral__even,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_317_power__numeral__even,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_318_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_319_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_320_sum__sqs__eq,axiom,
! [X2: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ X2 @ ( times_times_rat @ Y @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( Y = X2 ) ) ).
% sum_sqs_eq
thf(fact_321_sum__sqs__eq,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ X2 @ ( times_times_real @ Y @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( Y = X2 ) ) ).
% sum_sqs_eq
thf(fact_322_sum__sqs__eq,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ X2 @ ( times_times_int @ Y @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( Y = X2 ) ) ).
% sum_sqs_eq
thf(fact_323_left__add__twice,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_324_left__add__twice,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( plus_p2313304076027620419l_num1 @ A @ ( plus_p2313304076027620419l_num1 @ A @ B ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_325_left__add__twice,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ A @ B ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_326_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_327_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_328_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_329_mult__2__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_330_mult__2__right,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ Z @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) )
= ( plus_p2313304076027620419l_num1 @ Z @ Z ) ) ).
% mult_2_right
thf(fact_331_mult__2__right,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ Z @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2_right
thf(fact_332_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_333_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_334_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_335_mult__2,axiom,
! [Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2
thf(fact_336_mult__2,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p2313304076027620419l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_337_mult__2,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_338_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_339_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_340_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_341_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_342_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_343_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_344_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_345_power2__eq__square,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ A @ A ) ) ).
% power2_eq_square
thf(fact_346_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_347_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_348_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_349_power4__eq__xxxx,axiom,
! [X2: rat] :
( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_350_power4__eq__xxxx,axiom,
! [X2: real] :
( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_351_power4__eq__xxxx,axiom,
! [X2: nat] :
( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_352_power4__eq__xxxx,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_353_one__power2,axiom,
( ( power_7402600760894073284l_num1 @ one_on3868389512446148991l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on3868389512446148991l_num1 ) ).
% one_power2
thf(fact_354_one__power2,axiom,
( ( power_1002146276965246001l_num1 @ one_on7795324986448017462l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on7795324986448017462l_num1 ) ).
% one_power2
thf(fact_355_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_356_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_357_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_358_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_359_power2__sum,axiom,
! [X2: rat,Y: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y ) ) ) ).
% power2_sum
thf(fact_360_power2__sum,axiom,
! [X2: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( power_7402600760894073284l_num1 @ ( plus_p2313304076027620419l_num1 @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ ( power_7402600760894073284l_num1 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7402600760894073284l_num1 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_8498157372700349887l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ X2 ) @ Y ) ) ) ).
% power2_sum
thf(fact_361_power2__sum,axiom,
! [X2: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( power_1002146276965246001l_num1 @ ( plus_p1441664204671982194l_num1 @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( power_1002146276965246001l_num1 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_1002146276965246001l_num1 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_2938166955517408246l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ X2 ) @ Y ) ) ) ).
% power2_sum
thf(fact_362_power2__sum,axiom,
! [X2: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y ) ) ) ).
% power2_sum
thf(fact_363_power2__sum,axiom,
! [X2: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y ) ) ) ).
% power2_sum
thf(fact_364_power2__sum,axiom,
! [X2: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y ) ) ) ).
% power2_sum
thf(fact_365__092_060open_062_I_092_060lambda_062x_O_A80_A_K_A_I1_A_P_A_Ireal__of__rat_A_I_092_060delta_062__of_Ax_J_J_092_060_094sup_0622_J_J_A_092_060in_062_AO_091sequentially_A_092_060times_062_092_060_094sub_062F_Aat__right_A0_A_092_060times_062_092_060_094sub_062F_Aat__right_A0_093_I_092_060lambda_062x_O_A1_A_P_A_Ireal__of__rat_A_I_092_060delta_062__of_Ax_J_J_092_060_094sup_0622_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% \<open>(\<lambda>x. 80 * (1 / (real_of_rat (\<delta>_of x))\<^sup>2)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. 1 / (real_of_rat (\<delta>_of x))\<^sup>2)\<close>
thf(fact_366_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_367_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_368_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_369_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_370_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_371_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_372_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_373_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_374_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_375_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_376_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_377_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_378_nat__neq__4k1,axiom,
! [M: nat,K: nat,N: nat] :
( ( semiri681578069525770553at_rat @ M )
!= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% nat_neq_4k1
thf(fact_379_nat__neq__4k1,axiom,
! [M: nat,K: nat,N: nat] :
( ( semiri5074537144036343181t_real @ M )
!= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% nat_neq_4k1
thf(fact_380_landau__o_Obig_Ocdiv__in__iff_H,axiom,
! [C: real,F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F @ X ) @ C )
@ ( landau6322959426088225955t_real @ F2 @ G ) )
= ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cdiv_in_iff'
thf(fact_381_landau__o_Obig_Ocdiv__in__iff_H,axiom,
! [C: real,F: real > real,F2: filter_real,G: real > real] :
( ( C != zero_zero_real )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ C )
@ ( landau308303187242894617l_real @ F2 @ G ) )
= ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cdiv_in_iff'
thf(fact_382_landau__o_Obig_Ocdiv__in__iff_H,axiom,
! [C: real,F: nat > real,F2: filter_nat,G: nat > real] :
( ( C != zero_zero_real )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F @ X ) @ C )
@ ( landau_bigo_nat_real @ F2 @ G ) )
= ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cdiv_in_iff'
thf(fact_383_landau__o_Obig_Ocdiv,axiom,
! [C: real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F @ X ) @ C ) )
= ( landau6322959426088225955t_real @ F2 @ F ) ) ) ).
% landau_o.big.cdiv
thf(fact_384_landau__o_Obig_Ocdiv,axiom,
! [C: real,F2: filter_real,F: real > real] :
( ( C != zero_zero_real )
=> ( ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ C ) )
= ( landau308303187242894617l_real @ F2 @ F ) ) ) ).
% landau_o.big.cdiv
thf(fact_385_landau__o_Obig_Ocdiv,axiom,
! [C: real,F2: filter_nat,F: nat > real] :
( ( C != zero_zero_real )
=> ( ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( F @ X ) @ C ) )
= ( landau_bigo_nat_real @ F2 @ F ) ) ) ).
% landau_o.big.cdiv
thf(fact_386_cmult__in__bigo__iff_H,axiom,
! [F: produc5691113562410904374at_rat > real,C: real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ C )
@ ( landau6322959426088225955t_real @ F2 @ G ) )
= ( ( C = zero_zero_real )
| ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% cmult_in_bigo_iff'
thf(fact_387_cmult__in__bigo__iff_H,axiom,
! [F: real > real,C: real,F2: filter_real,G: real > real] :
( ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C )
@ ( landau308303187242894617l_real @ F2 @ G ) )
= ( ( C = zero_zero_real )
| ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% cmult_in_bigo_iff'
thf(fact_388_cmult__in__bigo__iff_H,axiom,
! [F: nat > real,C: real,F2: filter_nat,G: nat > real] :
( ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ C )
@ ( landau_bigo_nat_real @ F2 @ G ) )
= ( ( C = zero_zero_real )
| ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% cmult_in_bigo_iff'
thf(fact_389_cmult__in__bigo__iff,axiom,
! [C: real,F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ C @ ( F @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ G ) )
= ( ( C = zero_zero_real )
| ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% cmult_in_bigo_iff
thf(fact_390_cmult__in__bigo__iff,axiom,
! [C: real,F: real > real,F2: filter_real,G: real > real] :
( ( member_real_real
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) )
@ ( landau308303187242894617l_real @ F2 @ G ) )
= ( ( C = zero_zero_real )
| ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% cmult_in_bigo_iff
thf(fact_391_cmult__in__bigo__iff,axiom,
! [C: real,F: nat > real,F2: filter_nat,G: nat > real] :
( ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ C @ ( F @ X ) )
@ ( landau_bigo_nat_real @ F2 @ G ) )
= ( ( C = zero_zero_real )
| ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% cmult_in_bigo_iff
thf(fact_392_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_393_num_Osimps_I2_J,axiom,
! [X32: num,Y3: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y3 ) )
= ( X32 = Y3 ) ) ).
% num.simps(2)
thf(fact_394_rel__simps_I23_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% rel_simps(23)
thf(fact_395_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_396_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_397_divide__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( divide_divide_rat @ C @ A )
= ( divide_divide_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_398_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_399_divide__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_400_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_401_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_402_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_403_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_404_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_405_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_406_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_407_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_408_rel__simps_I22_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% rel_simps(22)
thf(fact_409_rel__simps_I21_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% rel_simps(21)
thf(fact_410_rel__simps_I19_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% rel_simps(19)
thf(fact_411_rel__simps_I17_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% rel_simps(17)
thf(fact_412_arith__simps_I12_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% arith_simps(12)
thf(fact_413_arith__simps_I11_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% arith_simps(11)
thf(fact_414_arith__simps_I10_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% arith_simps(10)
thf(fact_415_landau__symbols__if__at__top__eq_I1_J,axiom,
! [A: real,F: real > real,G: real > real] :
( ( landau308303187242894617l_real @ at_top_real
@ ^ [X: real] : ( if_real @ ( X = A ) @ ( F @ X ) @ ( G @ X ) ) )
= ( landau308303187242894617l_real @ at_top_real @ G ) ) ).
% landau_symbols_if_at_top_eq(1)
thf(fact_416_landau__symbols__if__at__top__eq_I1_J,axiom,
! [A: nat,F: nat > real,G: nat > real] :
( ( landau_bigo_nat_real @ at_top_nat
@ ^ [X: nat] : ( if_real @ ( X = A ) @ ( F @ X ) @ ( G @ X ) ) )
= ( landau_bigo_nat_real @ at_top_nat @ G ) ) ).
% landau_symbols_if_at_top_eq(1)
thf(fact_417_mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_418_mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_419_mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_420_mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_421_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_422_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_423_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_424_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_425_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_426_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_427_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_428_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_429_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_430_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_431_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_432_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_433_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_434_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_435_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_436_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_437_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_438_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_439_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_440_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_441_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_442_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_443_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_444_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_445_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_446_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_447_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_448_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_449_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_450_arithmetic__simps_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% arithmetic_simps(6)
thf(fact_451_arithmetic__simps_I8_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% arithmetic_simps(8)
thf(fact_452_arithmetic__simps_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% arithmetic_simps(13)
thf(fact_453_arithmetic__simps_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% arithmetic_simps(14)
thf(fact_454_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_455_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_456_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_457_power__mult__numeral,axiom,
! [A: rat,M: num,N: num] :
( ( power_power_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_458_landau__o_Obig_Ocmult,axiom,
! [C: real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ C @ ( F @ X ) ) )
= ( landau6322959426088225955t_real @ F2 @ F ) ) ) ).
% landau_o.big.cmult
thf(fact_459_landau__o_Obig_Ocmult,axiom,
! [C: real,F2: filter_real,F: real > real] :
( ( C != zero_zero_real )
=> ( ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) ) )
= ( landau308303187242894617l_real @ F2 @ F ) ) ) ).
% landau_o.big.cmult
thf(fact_460_landau__o_Obig_Ocmult,axiom,
! [C: real,F2: filter_nat,F: nat > real] :
( ( C != zero_zero_real )
=> ( ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ C @ ( F @ X ) ) )
= ( landau_bigo_nat_real @ F2 @ F ) ) ) ).
% landau_o.big.cmult
thf(fact_461_landau__o_Obig_Ocmult_H,axiom,
! [C: real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ C ) )
= ( landau6322959426088225955t_real @ F2 @ F ) ) ) ).
% landau_o.big.cmult'
thf(fact_462_landau__o_Obig_Ocmult_H,axiom,
! [C: real,F2: filter_real,F: real > real] :
( ( C != zero_zero_real )
=> ( ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C ) )
= ( landau308303187242894617l_real @ F2 @ F ) ) ) ).
% landau_o.big.cmult'
thf(fact_463_landau__o_Obig_Ocmult_H,axiom,
! [C: real,F2: filter_nat,F: nat > real] :
( ( C != zero_zero_real )
=> ( ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ C ) )
= ( landau_bigo_nat_real @ F2 @ F ) ) ) ).
% landau_o.big.cmult'
thf(fact_464_landau__o_Obig_Ocmult__in__iff,axiom,
! [C: real,F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ C @ ( F @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ G ) )
= ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cmult_in_iff
thf(fact_465_landau__o_Obig_Ocmult__in__iff,axiom,
! [C: real,F: real > real,F2: filter_real,G: real > real] :
( ( C != zero_zero_real )
=> ( ( member_real_real
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) )
@ ( landau308303187242894617l_real @ F2 @ G ) )
= ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cmult_in_iff
thf(fact_466_landau__o_Obig_Ocmult__in__iff,axiom,
! [C: real,F: nat > real,F2: filter_nat,G: nat > real] :
( ( C != zero_zero_real )
=> ( ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ C @ ( F @ X ) )
@ ( landau_bigo_nat_real @ F2 @ G ) )
= ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cmult_in_iff
thf(fact_467_landau__o_Obig_Ocmult__in__iff_H,axiom,
! [C: real,F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ C )
@ ( landau6322959426088225955t_real @ F2 @ G ) )
= ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cmult_in_iff'
thf(fact_468_landau__o_Obig_Ocmult__in__iff_H,axiom,
! [C: real,F: real > real,F2: filter_real,G: real > real] :
( ( C != zero_zero_real )
=> ( ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C )
@ ( landau308303187242894617l_real @ F2 @ G ) )
= ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cmult_in_iff'
thf(fact_469_landau__o_Obig_Ocmult__in__iff_H,axiom,
! [C: real,F: nat > real,F2: filter_nat,G: nat > real] :
( ( C != zero_zero_real )
=> ( ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ C )
@ ( landau_bigo_nat_real @ F2 @ G ) )
= ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_o.big.cmult_in_iff'
thf(fact_470_arith__simps_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% arith_simps(9)
thf(fact_471_arith__simps_I7_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% arith_simps(7)
thf(fact_472_arith__simps_I4_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% arith_simps(4)
thf(fact_473_arith__simps_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% arith_simps(3)
thf(fact_474_arith__simps_I2_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% arith_simps(2)
thf(fact_475_arithmetic__simps_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% arithmetic_simps(15)
thf(fact_476_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_477_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_478_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_479_num_Osimps_I8_J,axiom,
! [X32: num,X22: num] :
( ( bit1 @ X32 )
!= ( bit0 @ X22 ) ) ).
% num.simps(8)
thf(fact_480_num_Osimps_I6_J,axiom,
! [X32: num] :
( ( bit1 @ X32 )
!= one ) ).
% num.simps(6)
thf(fact_481_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_482_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_483_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_484_power__mult,axiom,
! [A: rat,M: nat,N: nat] :
( ( power_power_rat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_rat @ ( power_power_rat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_485_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_486_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_487_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_488_numeral_Osimps_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral.simps(3)
thf(fact_489_numeral_Osimps_I3_J,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit1 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) @ one_on3868389512446148991l_num1 ) ) ).
% numeral.simps(3)
thf(fact_490_numeral_Osimps_I3_J,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit1 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral.simps(3)
thf(fact_491_numeral_Osimps_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral.simps(3)
thf(fact_492_numeral_Osimps_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral.simps(3)
thf(fact_493_numeral_Osimps_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral.simps(3)
thf(fact_494_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_495_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit1 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) @ one_on3868389512446148991l_num1 ) ) ).
% numeral_code(3)
thf(fact_496_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit1 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_code(3)
thf(fact_497_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_498_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_499_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_500_power__numeral__odd,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_501_power__numeral__odd,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_502_power__numeral__odd,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_503_power__numeral__odd,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_504_landau__trans_I31_J,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( ( member1610887461201275416t_real @ G @ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ H ) ) ) ) ).
% landau_trans(31)
thf(fact_505_landau__trans_I31_J,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( ( member_real_real @ G @ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ H ) ) ) ) ).
% landau_trans(31)
thf(fact_506_landau__trans_I31_J,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( ( member_nat_real @ G @ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ H ) ) ) ) ).
% landau_trans(31)
thf(fact_507_landau__o_Obig__refl,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat] : ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ F ) ) ).
% landau_o.big_refl
thf(fact_508_landau__o_Obig__refl,axiom,
! [F: real > real,F2: filter_real] : ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ F ) ) ).
% landau_o.big_refl
thf(fact_509_landau__o_Obig__refl,axiom,
! [F: nat > real,F2: filter_nat] : ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ F ) ) ).
% landau_o.big_refl
thf(fact_510_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_511_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_512_power3__eq__cube,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_513_power3__eq__cube,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_514_power3__eq__cube,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_515_power3__eq__cube,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_516_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_517_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_518_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_519_power__even__eq,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_520_landau__trans__lift_I6_J,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,T: produc5691113562410904374at_rat > real > real,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( G @ X ) ) ) )
=> ( ( member1610887461201275416t_real @ G @ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( ! [G2: produc5691113562410904374at_rat > real,H2: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ G2 @ ( landau6322959426088225955t_real @ F2 @ H2 ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( G2 @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( H2 @ X ) ) ) ) )
=> ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( H @ X ) ) ) ) ) ) ) ).
% landau_trans_lift(6)
thf(fact_521_landau__trans__lift_I6_J,axiom,
! [F: real > real,F2: filter_real,T: real > real > real,G: real > real,H: real > real] :
( ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( T @ X @ ( G @ X ) ) ) )
=> ( ( member_real_real @ G @ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( ! [G2: real > real,H2: real > real] :
( ( member_real_real @ G2 @ ( landau308303187242894617l_real @ F2 @ H2 ) )
=> ( member_real_real
@ ^ [X: real] : ( T @ X @ ( G2 @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( T @ X @ ( H2 @ X ) ) ) ) )
=> ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( T @ X @ ( H @ X ) ) ) ) ) ) ) ).
% landau_trans_lift(6)
thf(fact_522_landau__trans__lift_I6_J,axiom,
! [F: nat > real,F2: filter_nat,T: nat > real > real,G: nat > real,H: nat > real] :
( ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( T @ X @ ( G @ X ) ) ) )
=> ( ( member_nat_real @ G @ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( ! [G2: nat > real,H2: nat > real] :
( ( member_nat_real @ G2 @ ( landau_bigo_nat_real @ F2 @ H2 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( T @ X @ ( G2 @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( T @ X @ ( H2 @ X ) ) ) ) )
=> ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( T @ X @ ( H @ X ) ) ) ) ) ) ) ).
% landau_trans_lift(6)
thf(fact_523_landau__trans__lift_I1_J,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,T: produc5691113562410904374at_rat > real > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( G @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( ! [F3: produc5691113562410904374at_rat > real,G2: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F3 @ ( landau6322959426088225955t_real @ F2 @ G2 ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( F3 @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( G2 @ X ) ) ) ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( T @ X @ ( F @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ H ) ) ) ) ) ).
% landau_trans_lift(1)
thf(fact_524_landau__trans__lift_I1_J,axiom,
! [F: real > real,F2: filter_real,G: real > real,T: real > real > real,H: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( ( member_real_real
@ ^ [X: real] : ( T @ X @ ( G @ X ) )
@ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( ! [F3: real > real,G2: real > real] :
( ( member_real_real @ F3 @ ( landau308303187242894617l_real @ F2 @ G2 ) )
=> ( member_real_real
@ ^ [X: real] : ( T @ X @ ( F3 @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( T @ X @ ( G2 @ X ) ) ) ) )
=> ( member_real_real
@ ^ [X: real] : ( T @ X @ ( F @ X ) )
@ ( landau308303187242894617l_real @ F2 @ H ) ) ) ) ) ).
% landau_trans_lift(1)
thf(fact_525_landau__trans__lift_I1_J,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,T: nat > real > real,H: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( ( member_nat_real
@ ^ [X: nat] : ( T @ X @ ( G @ X ) )
@ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( ! [F3: nat > real,G2: nat > real] :
( ( member_nat_real @ F3 @ ( landau_bigo_nat_real @ F2 @ G2 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( T @ X @ ( F3 @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( T @ X @ ( G2 @ X ) ) ) ) )
=> ( member_nat_real
@ ^ [X: nat] : ( T @ X @ ( F @ X ) )
@ ( landau_bigo_nat_real @ F2 @ H ) ) ) ) ) ).
% landau_trans_lift(1)
thf(fact_526_nat__neq__4k3,axiom,
! [M: nat,K: nat,N: nat] :
( ( semiri681578069525770553at_rat @ M )
!= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K ) ) @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% nat_neq_4k3
thf(fact_527_nat__neq__4k3,axiom,
! [M: nat,K: nat,N: nat] :
( ( semiri5074537144036343181t_real @ M )
!= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K ) ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% nat_neq_4k3
thf(fact_528_neq__4k1__k43,axiom,
! [M: nat,N: nat,M2: nat,N3: nat] :
( ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) )
!= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% neq_4k1_k43
thf(fact_529_times__divide__times__eq,axiom,
! [X2: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_530_divide__divide__times__eq,axiom,
! [X2: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_531_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_532_add__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_533_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_534_zero__in__bigo,axiom,
! [F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real] :
( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : zero_zero_real
@ ( landau6322959426088225955t_real @ F2 @ F ) ) ).
% zero_in_bigo
thf(fact_535_zero__in__bigo,axiom,
! [F2: filter_real,F: real > real] :
( member_real_real
@ ^ [Uu: real] : zero_zero_real
@ ( landau308303187242894617l_real @ F2 @ F ) ) ).
% zero_in_bigo
thf(fact_536_zero__in__bigo,axiom,
! [F2: filter_nat,F: nat > real] :
( member_nat_real
@ ^ [Uu: nat] : zero_zero_real
@ ( landau_bigo_nat_real @ F2 @ F ) ) ).
% zero_in_bigo
thf(fact_537_landau__o_Omult_I1_J,axiom,
! [F1: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G1: produc5691113562410904374at_rat > real,F22: produc5691113562410904374at_rat > real,G22: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F1 @ ( landau6322959426088225955t_real @ F2 @ G1 ) )
=> ( ( member1610887461201275416t_real @ F22 @ ( landau6322959426088225955t_real @ F2 @ G22 ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F1 @ X ) @ ( F22 @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( G1 @ X ) @ ( G22 @ X ) ) ) ) ) ) ).
% landau_o.mult(1)
thf(fact_538_landau__o_Omult_I1_J,axiom,
! [F1: real > real,F2: filter_real,G1: real > real,F22: real > real,G22: real > real] :
( ( member_real_real @ F1 @ ( landau308303187242894617l_real @ F2 @ G1 ) )
=> ( ( member_real_real @ F22 @ ( landau308303187242894617l_real @ F2 @ G22 ) )
=> ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F1 @ X ) @ ( F22 @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( G1 @ X ) @ ( G22 @ X ) ) ) ) ) ) ).
% landau_o.mult(1)
thf(fact_539_landau__o_Omult_I1_J,axiom,
! [F1: nat > real,F2: filter_nat,G1: nat > real,F22: nat > real,G22: nat > real] :
( ( member_nat_real @ F1 @ ( landau_bigo_nat_real @ F2 @ G1 ) )
=> ( ( member_nat_real @ F22 @ ( landau_bigo_nat_real @ F2 @ G22 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F1 @ X ) @ ( F22 @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( G1 @ X ) @ ( G22 @ X ) ) ) ) ) ) ).
% landau_o.mult(1)
thf(fact_540_landau__o_Obig_Omult__left,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( H @ X ) @ ( G @ X ) ) ) ) ) ).
% landau_o.big.mult_left
thf(fact_541_landau__o_Obig_Omult__left,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( H @ X ) @ ( G @ X ) ) ) ) ) ).
% landau_o.big.mult_left
thf(fact_542_landau__o_Obig_Omult__left,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( H @ X ) @ ( G @ X ) ) ) ) ) ).
% landau_o.big.mult_left
thf(fact_543_landau__o_Obig_Omult__right,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ).
% landau_o.big.mult_right
thf(fact_544_landau__o_Obig_Omult__right,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ).
% landau_o.big.mult_right
thf(fact_545_landau__o_Obig_Omult__right,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ).
% landau_o.big.mult_right
thf(fact_546_bigo__const,axiom,
! [C: real,F2: filter3199273883467263174at_rat] :
( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : C
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) ) ).
% bigo_const
thf(fact_547_bigo__const,axiom,
! [C: real,F2: filter_real] :
( member_real_real
@ ^ [Uu: real] : C
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) ) ).
% bigo_const
thf(fact_548_bigo__const,axiom,
! [C: real,F2: filter_nat] :
( member_nat_real
@ ^ [Uu: nat] : C
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) ) ).
% bigo_const
thf(fact_549_sum__in__bigo_I1_J,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,H: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( ( member1610887461201275416t_real @ G @ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ H ) ) ) ) ).
% sum_in_bigo(1)
thf(fact_550_sum__in__bigo_I1_J,axiom,
! [F: real > real,F2: filter_real,H: real > real,G: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( ( member_real_real @ G @ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( member_real_real
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
@ ( landau308303187242894617l_real @ F2 @ H ) ) ) ) ).
% sum_in_bigo(1)
thf(fact_551_sum__in__bigo_I1_J,axiom,
! [F: nat > real,F2: filter_nat,H: nat > real,G: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( ( member_nat_real @ G @ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( member_nat_real
@ ^ [X: nat] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
@ ( landau_bigo_nat_real @ F2 @ H ) ) ) ) ).
% sum_in_bigo(1)
thf(fact_552_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X2: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X2 @ Y )
= ( divide_divide_rat @ W @ Z ) )
= ( ( times_times_rat @ X2 @ Z )
= ( times_times_rat @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_553_frac__eq__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X2 @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_554_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C )
= A )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_555_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_556_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_557_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_558_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C ) )
=> ( ( divide_divide_rat @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_559_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_560_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= B )
=> ( A
= ( divide_divide_rat @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_561_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_562_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C )
= A )
= ( B
= ( times_times_rat @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_563_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_564_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( times_times_rat @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_565_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_566_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_567_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_568_landau__o_Obig_Oconst,axiom,
! [C: real,F2: filter3199273883467263174at_rat] :
( ( C != zero_zero_real )
=> ( ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : C )
= ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) ) ) ).
% landau_o.big.const
thf(fact_569_landau__o_Obig_Oconst,axiom,
! [C: real,F2: filter_real] :
( ( C != zero_zero_real )
=> ( ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : C )
= ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) ) ) ).
% landau_o.big.const
thf(fact_570_landau__o_Obig_Oconst,axiom,
! [C: real,F2: filter_nat] :
( ( C != zero_zero_real )
=> ( ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : C )
= ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) ) ) ).
% landau_o.big.const
thf(fact_571_landau__o_Obig_Oconst__in__iff,axiom,
! [C: real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real] :
( ( C != zero_zero_real )
=> ( ( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : C
@ ( landau6322959426088225955t_real @ F2 @ F ) )
= ( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ F2 @ F ) ) ) ) ).
% landau_o.big.const_in_iff
thf(fact_572_landau__o_Obig_Oconst__in__iff,axiom,
! [C: real,F2: filter_real,F: real > real] :
( ( C != zero_zero_real )
=> ( ( member_real_real
@ ^ [Uu: real] : C
@ ( landau308303187242894617l_real @ F2 @ F ) )
= ( member_real_real
@ ^ [Uu: real] : one_one_real
@ ( landau308303187242894617l_real @ F2 @ F ) ) ) ) ).
% landau_o.big.const_in_iff
thf(fact_573_landau__o_Obig_Oconst__in__iff,axiom,
! [C: real,F2: filter_nat,F: nat > real] :
( ( C != zero_zero_real )
=> ( ( member_nat_real
@ ^ [Uu: nat] : C
@ ( landau_bigo_nat_real @ F2 @ F ) )
= ( member_nat_real
@ ^ [Uu: nat] : one_one_real
@ ( landau_bigo_nat_real @ F2 @ F ) ) ) ) ).
% landau_o.big.const_in_iff
thf(fact_574_landau__mult__1__trans_I8_J,axiom,
! [H: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ H
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) )
=> ( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% landau_mult_1_trans(8)
thf(fact_575_landau__mult__1__trans_I8_J,axiom,
! [H: real > real,F2: filter_real,F: real > real,G: real > real] :
( ( member_real_real @ H
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) )
=> ( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% landau_mult_1_trans(8)
thf(fact_576_landau__mult__1__trans_I8_J,axiom,
! [H: nat > real,F2: filter_nat,F: nat > real,G: nat > real] :
( ( member_nat_real @ H
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) )
=> ( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_mult_1_trans(8)
thf(fact_577_landau__mult__1__trans_I7_J,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( ( member1610887461201275416t_real @ H
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% landau_mult_1_trans(7)
thf(fact_578_landau__mult__1__trans_I7_J,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( ( member_real_real @ H
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) )
=> ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% landau_mult_1_trans(7)
thf(fact_579_landau__mult__1__trans_I7_J,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( ( member_nat_real @ H
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) )
=> ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_mult_1_trans(7)
thf(fact_580_landau__mult__1__trans_I2_J,axiom,
! [F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,F: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_mult_1_trans(2)
thf(fact_581_landau__mult__1__trans_I2_J,axiom,
! [F2: filter_real,G: real > real,F: real > real,H: real > real] :
( ( member_real_real
@ ^ [Uu: real] : one_one_real
@ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_mult_1_trans(2)
thf(fact_582_landau__mult__1__trans_I2_J,axiom,
! [F2: filter_nat,G: nat > real,F: nat > real,H: nat > real] :
( ( member_nat_real
@ ^ [Uu: nat] : one_one_real
@ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_mult_1_trans(2)
thf(fact_583_landau__mult__1__trans_I1_J,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( ( member1610887461201275416t_real
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real
@ ( landau6322959426088225955t_real @ F2 @ H ) )
=> ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_mult_1_trans(1)
thf(fact_584_landau__mult__1__trans_I1_J,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( ( member_real_real
@ ^ [Uu: real] : one_one_real
@ ( landau308303187242894617l_real @ F2 @ H ) )
=> ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_mult_1_trans(1)
thf(fact_585_landau__mult__1__trans_I1_J,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( ( member_nat_real
@ ^ [Uu: nat] : one_one_real
@ ( landau_bigo_nat_real @ F2 @ H ) )
=> ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_mult_1_trans(1)
thf(fact_586_landau__o_Obig_Omult__in__1,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) )
=> ( ( member1610887461201275416t_real @ G
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : one_one_real ) ) ) ) ).
% landau_o.big.mult_in_1
thf(fact_587_landau__o_Obig_Omult__in__1,axiom,
! [F: real > real,F2: filter_real,G: real > real] :
( ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) )
=> ( ( member_real_real @ G
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) )
=> ( member_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : one_one_real ) ) ) ) ).
% landau_o.big.mult_in_1
thf(fact_588_landau__o_Obig_Omult__in__1,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real] :
( ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) )
=> ( ( member_nat_real @ G
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) )
=> ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : one_one_real ) ) ) ) ).
% landau_o.big.mult_in_1
thf(fact_589_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_590_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_591_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_592_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_593_add__frac__eq,axiom,
! [Y: rat,Z: rat,X2: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_594_add__frac__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_595_add__frac__num,axiom,
! [Y: rat,X2: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y ) @ Z )
= ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_596_add__frac__num,axiom,
! [Y: real,X2: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_597_add__num__frac,axiom,
! [Y: rat,Z: rat,X2: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_598_add__num__frac,axiom,
! [Y: real,Z: real,X2: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_599_add__divide__eq__iff,axiom,
! [Z: rat,X2: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_600_add__divide__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_601_divide__add__eq__iff,axiom,
! [Z: rat,X2: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_602_divide__add__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_603_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_604_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_605_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_606_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_607__C9_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( s_of @ X ) ) @ one_one_real ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) ) @ g ) ) ).
% "9"
thf(fact_608_of__rat__0,axiom,
( ( field_2639924705303425560at_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% of_rat_0
thf(fact_609_of__rat__0,axiom,
( ( field_7254667332652039916t_real @ zero_zero_rat )
= zero_zero_real ) ).
% of_rat_0
thf(fact_610_of__rat__eq__iff,axiom,
! [A: rat,B: rat] :
( ( ( field_7254667332652039916t_real @ A )
= ( field_7254667332652039916t_real @ B ) )
= ( A = B ) ) ).
% of_rat_eq_iff
thf(fact_611_one__eq__of__rat__iff,axiom,
! [A: rat] :
( ( one_one_real
= ( field_7254667332652039916t_real @ A ) )
= ( one_one_rat = A ) ) ).
% one_eq_of_rat_iff
thf(fact_612_of__rat__eq__1__iff,axiom,
! [A: rat] :
( ( ( field_7254667332652039916t_real @ A )
= one_one_real )
= ( A = one_one_rat ) ) ).
% of_rat_eq_1_iff
thf(fact_613_of__rat__1,axiom,
( ( field_7254667332652039916t_real @ one_one_rat )
= one_one_real ) ).
% of_rat_1
thf(fact_614_of__rat__numeral__eq,axiom,
! [W: num] :
( ( field_7254667332652039916t_real @ ( numeral_numeral_rat @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% of_rat_numeral_eq
thf(fact_615_of__rat__of__nat__eq,axiom,
! [N: nat] :
( ( field_7254667332652039916t_real @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_rat_of_nat_eq
thf(fact_616_bigo__real__nat__transfer,axiom,
! [F: real > real,G: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ at_top_real @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( F @ ( semiri5074537144036343181t_real @ X ) )
@ ( landau_bigo_nat_real @ at_top_nat
@ ^ [X: nat] : ( G @ ( semiri5074537144036343181t_real @ X ) ) ) ) ) ).
% bigo_real_nat_transfer
thf(fact_617_zero__eq__of__rat__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( field_2639924705303425560at_rat @ A ) )
= ( zero_zero_rat = A ) ) ).
% zero_eq_of_rat_iff
thf(fact_618_zero__eq__of__rat__iff,axiom,
! [A: rat] :
( ( zero_zero_real
= ( field_7254667332652039916t_real @ A ) )
= ( zero_zero_rat = A ) ) ).
% zero_eq_of_rat_iff
thf(fact_619_of__rat__eq__0__iff,axiom,
! [A: rat] :
( ( ( field_2639924705303425560at_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% of_rat_eq_0_iff
thf(fact_620_of__rat__eq__0__iff,axiom,
! [A: rat] :
( ( ( field_7254667332652039916t_real @ A )
= zero_zero_real )
= ( A = zero_zero_rat ) ) ).
% of_rat_eq_0_iff
thf(fact_621_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_622_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_623_of__rat__mult,axiom,
! [A: rat,B: rat] :
( ( field_7254667332652039916t_real @ ( times_times_rat @ A @ B ) )
= ( times_times_real @ ( field_7254667332652039916t_real @ A ) @ ( field_7254667332652039916t_real @ B ) ) ) ).
% of_rat_mult
thf(fact_624_of__rat__add,axiom,
! [A: rat,B: rat] :
( ( field_2639924705303425560at_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( field_2639924705303425560at_rat @ A ) @ ( field_2639924705303425560at_rat @ B ) ) ) ).
% of_rat_add
thf(fact_625_of__rat__add,axiom,
! [A: rat,B: rat] :
( ( field_7254667332652039916t_real @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_real @ ( field_7254667332652039916t_real @ A ) @ ( field_7254667332652039916t_real @ B ) ) ) ).
% of_rat_add
thf(fact_626_of__rat__divide,axiom,
! [A: rat,B: rat] :
( ( field_7254667332652039916t_real @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_real @ ( field_7254667332652039916t_real @ A ) @ ( field_7254667332652039916t_real @ B ) ) ) ).
% of_rat_divide
thf(fact_627_of__rat__power,axiom,
! [A: rat,N: nat] :
( ( field_2639924705303425560at_rat @ ( power_power_rat @ A @ N ) )
= ( power_power_rat @ ( field_2639924705303425560at_rat @ A ) @ N ) ) ).
% of_rat_power
thf(fact_628_of__rat__power,axiom,
! [A: rat,N: nat] :
( ( field_7254667332652039916t_real @ ( power_power_rat @ A @ N ) )
= ( power_power_real @ ( field_7254667332652039916t_real @ A ) @ N ) ) ).
% of_rat_power
thf(fact_629_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_630_nonzero__of__rat__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( field_7254667332652039916t_real @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_real @ ( field_7254667332652039916t_real @ A ) @ ( field_7254667332652039916t_real @ B ) ) ) ) ).
% nonzero_of_rat_divide
thf(fact_631_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_632_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_633_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri1316708129612266289at_nat @ X2 )
= one_one_nat )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_634_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri5074537144036343181t_real @ X2 )
= one_one_real )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_635_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri1314217659103216013at_int @ X2 )
= one_one_int )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_636_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_637_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_638_double__eq__0__iff,axiom,
! [A: rat] :
( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_639_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_640_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_641_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_642_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_643_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_644_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_645_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_646_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_647_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_648_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_649_Transcendental_Olog__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
% Transcendental.log_one
thf(fact_650_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_651_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_652_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_653_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_654_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri681578069525770553at_rat @ M )
= zero_zero_rat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_655_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_656_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_657_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_658_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_659_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_660_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_661_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_662_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_663_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_664_nat__arith_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 ) ) ) ) ).
% nat_arith.add1
thf(fact_665_nat__arith_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 ) ) ) ) ).
% nat_arith.add1
thf(fact_666_nat__arith_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 ) ) ) ) ).
% nat_arith.add1
thf(fact_667_nat__arith_Oadd1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_668_nat__arith_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_669_nat__arith_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_670_nat__arith_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_671_nat__arith_Oadd2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B2 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_672_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_673_nat__arith_Orule0,axiom,
! [A: rat] :
( A
= ( plus_plus_rat @ A @ zero_zero_rat ) ) ).
% nat_arith.rule0
thf(fact_674_nat__arith_Orule0,axiom,
! [A: nat] :
( A
= ( plus_plus_nat @ A @ zero_zero_nat ) ) ).
% nat_arith.rule0
thf(fact_675_nat__arith_Orule0,axiom,
! [A: real] :
( A
= ( plus_plus_real @ A @ zero_zero_real ) ) ).
% nat_arith.rule0
thf(fact_676_nat__arith_Orule0,axiom,
! [A: int] :
( A
= ( plus_plus_int @ A @ zero_zero_int ) ) ).
% nat_arith.rule0
thf(fact_677_nat__distrib_I2_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% nat_distrib(2)
thf(fact_678_nat__distrib_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% nat_distrib(2)
thf(fact_679_nat__distrib_I2_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% nat_distrib(2)
thf(fact_680_nat__distrib_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% nat_distrib(2)
thf(fact_681_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_682_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_683_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_684_plus__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.simps(1)
thf(fact_685_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_686_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_687_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_688_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_689_nat__distrib_I1_J,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% nat_distrib(1)
thf(fact_690_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_691_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_692_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_693_log__def,axiom,
( log
= ( ^ [A3: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% log_def
thf(fact_694_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_695_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_696_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_697_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_698_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_699_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_700_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_701_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_702_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_703_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_704_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_705_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_706_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_707_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_708_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_709_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_710_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_711_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_712_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_713_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_714_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_715_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_716_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_717_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_718_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_719_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_720_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_721_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_722_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_723_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_724_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_725_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_726_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_727_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_728_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_729_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_730_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ( times_times_rat @ A @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_731_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_732_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_733_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_734_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_735_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_736_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ( times_times_rat @ C @ A )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_737_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_738_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_739_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_740_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_741_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_742_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_743_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_744_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_745_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_746_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_747_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_748_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_749_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_750_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_751_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_752_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_753_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_754_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_755_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_756_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_757_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_758_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_759_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_760_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_761_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_762_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_763_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_764_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_765_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_766_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_767_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_768_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_769_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_770_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_771_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_772_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_773_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_774_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_775_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_776_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_777_distrib__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% distrib_right
thf(fact_778_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_779_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_780_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_781_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_782_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_783_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_784_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_785_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_786_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_787_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_788_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_789_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_790_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_791_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_792_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_793_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_794_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_795_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_796_lambda__zero,axiom,
( ( ^ [H3: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_797_lambda__zero,axiom,
( ( ^ [H3: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_798_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_799_lambda__zero,axiom,
( ( ^ [H3: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_800_lambda__one,axiom,
( ( ^ [X: numera2417102609627094330l_num1] : X )
= ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% lambda_one
thf(fact_801_lambda__one,axiom,
( ( ^ [X: numera4273646738625120315l_num1] : X )
= ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% lambda_one
thf(fact_802_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_803_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_804_lambda__one,axiom,
( ( ^ [X: int] : X )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_805_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_806_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_807_class__dense__linordered__field_Obetween__same,axiom,
! [X2: rat] :
( ( times_times_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( plus_plus_rat @ X2 @ X2 ) )
= X2 ) ).
% class_dense_linordered_field.between_same
thf(fact_808_class__dense__linordered__field_Obetween__same,axiom,
! [X2: real] :
( ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ X2 @ X2 ) )
= X2 ) ).
% class_dense_linordered_field.between_same
thf(fact_809_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_810_div__mult__twopow__eq,axiom,
! [A: nat,N: nat,B: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ B )
= ( divide_divide_nat @ A @ ( times_times_nat @ B @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% div_mult_twopow_eq
thf(fact_811_four__x__squared,axiom,
! [X2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_812_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_813_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_814_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_815_nz__prod__eq,axiom,
! [C: rat,X2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ X2 )
= zero_zero_rat )
= ( X2 = zero_zero_rat ) ) ) ).
% nz_prod_eq
thf(fact_816_nz__prod__eq,axiom,
! [C: real,X2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_real ) ) ) ).
% nz_prod_eq
thf(fact_817_set__bit__0,axiom,
! [A: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_818_set__bit__0,axiom,
! [A: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_819_int_Onat__pow__0,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
thf(fact_820_nat__add__1__add__1,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_821_field__sum__of__halves,axiom,
! [X2: rat] :
( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= X2 ) ).
% field_sum_of_halves
thf(fact_822_field__sum__of__halves,axiom,
! [X2: real] :
( ( plus_plus_real @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X2 ) ).
% field_sum_of_halves
thf(fact_823_mult__delta__right,axiom,
! [B: $o,X2: rat,Y: rat] :
( ( B
=> ( ( times_times_rat @ X2 @ ( if_rat @ B @ Y @ zero_zero_rat ) )
= ( times_times_rat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_rat @ X2 @ ( if_rat @ B @ Y @ zero_zero_rat ) )
= zero_zero_rat ) ) ) ).
% mult_delta_right
thf(fact_824_mult__delta__right,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_825_mult__delta__right,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_826_mult__delta__right,axiom,
! [B: $o,X2: int,Y: int] :
( ( B
=> ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_827_mult__delta__left,axiom,
! [B: $o,X2: rat,Y: rat] :
( ( B
=> ( ( times_times_rat @ ( if_rat @ B @ X2 @ zero_zero_rat ) @ Y )
= ( times_times_rat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_rat @ ( if_rat @ B @ X2 @ zero_zero_rat ) @ Y )
= zero_zero_rat ) ) ) ).
% mult_delta_left
thf(fact_828_mult__delta__left,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_829_mult__delta__left,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_830_mult__delta__left,axiom,
! [B: $o,X2: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_831_int_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% int.nat_pow_zero
thf(fact_832_int_Oone__not__zero,axiom,
one_one_int != zero_zero_int ).
% int.one_not_zero
thf(fact_833_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_834_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_835_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_836_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_837_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_838_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_839_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_840_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_841_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_842_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: rat,X2: rat] :
( ( ( times_times_rat @ A @ X2 )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( X2 = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_843_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_844_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: rat] :
( ( times_times_rat @ zero_zero_rat @ X2 )
= zero_zero_rat ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_845_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_846_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_847_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_848_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: rat,X2: rat,Y: rat] :
( ( ( times_times_rat @ A @ X2 )
= ( times_times_rat @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_849_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_850_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: rat,X2: rat,B: rat] :
( ( ( times_times_rat @ A @ X2 )
= ( times_times_rat @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_851_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_852_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_853_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_854_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_855_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_856_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_857_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_858_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_859_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_860_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_861_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_862_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_863_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_864_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_865_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_866_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_867_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_868_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_869_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_870_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_871_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_872_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_873_zero__reorient,axiom,
! [X2: rat] :
( ( zero_zero_rat = X2 )
= ( X2 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_874_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_875_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_876_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_877_Groups_Omult__ac_I3_J,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_878_Groups_Omult__ac_I3_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_879_Groups_Omult__ac_I3_J,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_880_Groups_Omult__ac_I2_J,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_881_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_882_Groups_Omult__ac_I2_J,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_883_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_884_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_885_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_886_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_887_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_888_one__reorient,axiom,
! [X2: numera2417102609627094330l_num1] :
( ( one_on3868389512446148991l_num1 = X2 )
= ( X2 = one_on3868389512446148991l_num1 ) ) ).
% one_reorient
thf(fact_889_one__reorient,axiom,
! [X2: numera4273646738625120315l_num1] :
( ( one_on7795324986448017462l_num1 = X2 )
= ( X2 = one_on7795324986448017462l_num1 ) ) ).
% one_reorient
thf(fact_890_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_891_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_892_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_893_Groups_Oadd__ac_I3_J,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 ) ) ) ).
% Groups.add_ac(3)
thf(fact_894_Groups_Oadd__ac_I3_J,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 ) ) ) ).
% Groups.add_ac(3)
thf(fact_895_Groups_Oadd__ac_I3_J,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 ) ) ) ).
% Groups.add_ac(3)
thf(fact_896_Groups_Oadd__ac_I3_J,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% Groups.add_ac(3)
thf(fact_897_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_898_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_899_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_900_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_901_Groups_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 ) ) ) ).
% Groups.add_ac(1)
thf(fact_902_Groups_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 ) ) ) ).
% Groups.add_ac(1)
thf(fact_903_Groups_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 ) ) ) ).
% Groups.add_ac(1)
thf(fact_904_Groups_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% Groups.add_ac(1)
thf(fact_905_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_906_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_907_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_908_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_909_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_910_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_911_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_912_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_913_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X4: numera2417102609627094330l_num1] : ( P2 @ X4 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_914_exhaust__2,axiom,
! [X2: numera2417102609627094330l_num1] :
( ( X2 = one_on3868389512446148991l_num1 )
| ( X2
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_915_forall__3,axiom,
( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
! [X4: numera6367994245245682809l_num1] : ( P2 @ X4 ) )
= ( ^ [P3: numera6367994245245682809l_num1 > $o] :
( ( P3 @ one_on7819281148064737470l_num1 )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_916_exhaust__3,axiom,
! [X2: numera6367994245245682809l_num1] :
( ( X2 = one_on7819281148064737470l_num1 )
| ( X2
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X2
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_917_forall__4,axiom,
( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
! [X4: numera4273646738625120315l_num1] : ( P2 @ X4 ) )
= ( ^ [P3: numera4273646738625120315l_num1 > $o] :
( ( P3 @ one_on7795324986448017462l_num1 )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).
% forall_4
thf(fact_918_exhaust__4,axiom,
! [X2: numera4273646738625120315l_num1] :
( ( X2 = one_on7795324986448017462l_num1 )
| ( X2
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X2
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X2
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_919_div__twopow__def,axiom,
( div_twopow
= ( ^ [X: int,N4: nat] : ( divide_divide_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% div_twopow_def
thf(fact_920_iff__4k,axiom,
! [R: real,K: nat,M: nat,N: nat,M2: nat,N3: nat] :
( ( R
= ( semiri5074537144036343181t_real @ K ) )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ R ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ R ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) )
= ( ( M = M2 )
& ( N = N3 ) ) ) ) ) ).
% iff_4k
thf(fact_921_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_922_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_923_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_924_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_925_div2__even__ext__nat,axiom,
! [X2: nat,Y: nat] :
( ( ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X2 = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_926_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D2: nat,X3: nat,Y5: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_927_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_928_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_929_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_930_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_931_zdvd__period,axiom,
! [A: int,D: int,X2: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X2 @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X2 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_932_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_933_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_934_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_935_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_936_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_937_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_938_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X3: nat,Y5: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D2 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_939_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X3: nat,Y5: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y5 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_940_set__decode__0,axiom,
! [X2: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).
% set_decode_0
thf(fact_941_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_942_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_943_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_944_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_945_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_946_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_947_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_948_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_949_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_950_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_951_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_952_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_953_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_954_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_955_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_956_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_957_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_958_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_959_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_960_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_961_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_962_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_963_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_964_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_965_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_966_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_967_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_968_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_969_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_970_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_971_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_972_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_973_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_974_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_975_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_976_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_977_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_978_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_979_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_980_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_981_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_982_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_983_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_984_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_985_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_986_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_987_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_988_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_989_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_990_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_991_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_992_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_993_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_994_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_995_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_996_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_997_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_998_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_999_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1000_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1001_div2__less__self,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% div2_less_self
thf(fact_1002_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X: nat] :
( collect_nat
@ ^ [N4: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X2: rat,Y: rat] :
( ( if_rat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X2: rat,Y: rat] :
( ( if_rat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( t_of @ X ) ) @ one_one_real ) )
@ ( landau6322959426088225955t_real @ ( prod_f1623372399986984716at_rat @ at_top_nat @ ( prod_filter_rat_rat @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) @ ( topolo4023969691036296984in_rat @ zero_zero_rat @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ) ) )
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( ln_ln_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) @ ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------