TPTP Problem File: SLH0425^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Frequency_Moments/0085_Frequency_Moment_0/prob_01290_060507__19956254_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1090 ( 632 unt; 147 typ; 0 def)
% Number of atoms : 2099 (1133 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 7598 ( 158 ~; 41 |; 58 &;6832 @)
% ( 0 <=>; 509 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 528 ( 528 >; 0 *; 0 +; 0 <<)
% Number of symbols : 128 ( 125 usr; 18 con; 0-3 aty)
% Number of variables : 2171 ( 338 ^;1827 !; 6 ?;2171 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:17:30.484
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
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__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__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 (125)
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_Oeventually_001t__Nat__Onat,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
eventu6700955888398734894at_rat: ( produc5691113562410904374at_rat > $o ) > filter3199273883467263174at_rat > $o ).
thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
eventually_real: ( real > $o ) > filter_real > $o ).
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_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_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Num__Onum_J,type,
plus_plus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Rat__Orat_J,type,
plus_plus_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_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_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
semiri3786042916590614966l_num1: ( numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ) > nat > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
semiri3790374670636785663l_num1: ( numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ) > nat > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__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__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J_J,type,
ord_le4292863764789988646at_rat: filter3199273883467263174at_rat > filter3199273883467263174at_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le2908806416726583473t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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,
ord_le1994352800634783511t_real: set_Pr1128732697603872439t_real > set_Pr1128732697603872439t_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
ord_le4198349162570665613l_real: set_real_real > set_real_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $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__Real__Oreal,type,
power_power_real: real > nat > real ).
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_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__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
set_or1207661135979820486an_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Num__Onum,type,
set_or6990855429499425204an_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
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_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_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__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_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_r__of____,type,
r_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 (933)
thf(fact_0__C14_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] : ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) ).
% "14"
thf(fact_1__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ireal_A_In__of_Ax_J_A_L_A13_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_Ireal_A_In__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ 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] : ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (real (n_of x) + 13)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (real (n_of x)))\<close>
thf(fact_2__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ireal_A_In__of_Ax_J_A_L_A21_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_Ireal_A_In__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( 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] : ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (real (n_of x) + 21)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (real (n_of x)))\<close>
thf(fact_3__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_4__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ilog_A2_A_Ireal_A_In__of_Ax_J_A_L_A13_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_Iln_A_Ireal_A_In__of_Ax_J_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ 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] : ( ln_ln_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (log 2 (real (n_of x) + 13))) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (ln (real (n_of x))))\<close>
thf(fact_5__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_6_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_7_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_8_rel__simps_I19_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% rel_simps(19)
thf(fact_9_rel__simps_I17_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% rel_simps(17)
thf(fact_10_rel__simps_I22_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% rel_simps(22)
thf(fact_11_rel__simps_I21_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% rel_simps(21)
thf(fact_12_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_13_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_14_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_15_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_16_num_Oinject_I1_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% num.inject(1)
thf(fact_17_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_18_num_Oinject_I2_J,axiom,
! [X3: num,Y3: num] :
( ( ( bit1 @ X3 )
= ( bit1 @ Y3 ) )
= ( X3 = Y3 ) ) ).
% num.inject(2)
thf(fact_19_semiring__norm_I50_J,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% semiring_norm(50)
thf(fact_20_semiring__norm_I50_J,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% semiring_norm(50)
thf(fact_21_semiring__norm_I50_J,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% semiring_norm(50)
thf(fact_22_semiring__norm_I50_J,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% semiring_norm(50)
thf(fact_23_semiring__norm_I51_J,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% semiring_norm(51)
thf(fact_24_semiring__norm_I51_J,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% semiring_norm(51)
thf(fact_25_semiring__norm_I51_J,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% semiring_norm(51)
thf(fact_26_semiring__norm_I51_J,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% semiring_norm(51)
thf(fact_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% Num.of_nat_simps(1)
thf(fact_38_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% Num.of_nat_simps(1)
thf(fact_39_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% Num.of_nat_simps(1)
thf(fact_40_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% Num.of_nat_simps(1)
thf(fact_41_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_42_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_43_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_44_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_45_Collect__mem__eq,axiom,
! [A2: set_Pr1128732697603872439t_real] :
( ( collec8488528251386215510t_real
@ ^ [X: produc5691113562410904374at_rat > real] : ( member1610887461201275416t_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_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_47_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_48_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_49_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1795386414920522267l_num1 @ one_one_nat )
= one_on3868389512446148991l_num1 ) ).
% Num.of_nat_simps(2)
thf(fact_50_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri5667362542588693146l_num1 @ one_one_nat )
= one_on7795324986448017462l_num1 ) ).
% Num.of_nat_simps(2)
thf(fact_51_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% Num.of_nat_simps(2)
thf(fact_52_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% Num.of_nat_simps(2)
thf(fact_53_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% Num.of_nat_simps(2)
thf(fact_54_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_55_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_56_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_57_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1795386414920522267l_num1 @ ( numeral_numeral_nat @ N ) )
= ( numera2161328050825114965l_num1 @ N ) ) ).
% of_nat_numeral
thf(fact_58_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5667362542588693146l_num1 @ ( numeral_numeral_nat @ N ) )
= ( numera7754357348821619680l_num1 @ N ) ) ).
% of_nat_numeral
thf(fact_59_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_60_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_61_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_62_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_63_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_64_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_65_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_66_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_67_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_68_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_69_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_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_one__add__one,axiom,
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_80_one__add__one,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_81_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_82_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_83_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_84__C12_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( 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 ) ) ) ) @ g ) ) ).
% "12"
thf(fact_85__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_86_semiring__norm_I159_J,axiom,
zero_z5982384998485459395l_num1 != one_on3868389512446148991l_num1 ).
% semiring_norm(159)
thf(fact_87_semiring__norm_I159_J,axiom,
zero_z2241845390563828978l_num1 != one_on7795324986448017462l_num1 ).
% semiring_norm(159)
thf(fact_88_semiring__norm_I159_J,axiom,
zero_zero_rat != one_one_rat ).
% semiring_norm(159)
thf(fact_89_semiring__norm_I159_J,axiom,
zero_zero_nat != one_one_nat ).
% semiring_norm(159)
thf(fact_90_semiring__norm_I159_J,axiom,
zero_zero_real != one_one_real ).
% semiring_norm(159)
thf(fact_91_semiring__norm_I159_J,axiom,
zero_zero_int != one_one_int ).
% semiring_norm(159)
thf(fact_92_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_93_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_94_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_95_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_96_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_97_landau__o_Obig__trans,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_o.big_trans
thf(fact_98_landau__o_Obig__trans,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_o.big_trans
thf(fact_99_landau__o_Obig__trans,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_o.big_trans
thf(fact_100_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_101_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_102_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_103_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ X4 ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ X4 ) @ one_on3868389512446148991l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_104_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X4 ) @ one_on7795324986448017462l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_105_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X4 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_106_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_107_one__plus__numeral__commute,axiom,
! [X4: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_108_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_109_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_110_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_111_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_112_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_113_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_114_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_115_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_116_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_117_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_118_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_119_landau__o_Obig_Olift__trans_H,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_o.big.lift_trans'
thf(fact_120_landau__o_Obig_Olift__trans_H,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_o.big.lift_trans'
thf(fact_121_landau__o_Obig_Olift__trans_H,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_o.big.lift_trans'
thf(fact_122_landau__o_Obig_Olift__trans,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_o.big.lift_trans
thf(fact_123_landau__o_Obig_Olift__trans,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_o.big.lift_trans
thf(fact_124_landau__o_Obig_Olift__trans,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_o.big.lift_trans
thf(fact_125_numeral__Bit1,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit1 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) @ one_on3868389512446148991l_num1 ) ) ).
% numeral_Bit1
thf(fact_126_numeral__Bit1,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit1 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) @ one_on7795324986448017462l_num1 ) ) ).
% numeral_Bit1
thf(fact_127_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_128_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_129_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_130_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_131_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_132_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_133_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_134_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_135_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_136_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_137_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_138_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_139_num_Odistinct_I1_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% num.distinct(1)
thf(fact_140_num_Odistinct_I5_J,axiom,
! [X2: num,X3: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X3 ) ) ).
% num.distinct(5)
thf(fact_141_num_Odistinct_I3_J,axiom,
! [X3: num] :
( one
!= ( bit1 @ X3 ) ) ).
% num.distinct(3)
thf(fact_142_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_143_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_144_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_145_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_146_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_147_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_148_numeral__Bit0,axiom,
! [N: num] :
( ( numera2161328050825114965l_num1 @ ( bit0 @ N ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ ( numera2161328050825114965l_num1 @ N ) ) ) ).
% numeral_Bit0
thf(fact_149_numeral__Bit0,axiom,
! [N: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ).
% numeral_Bit0
thf(fact_150_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_151_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_152_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_153_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_154_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_155_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_156_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_157_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_158_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_159__C11_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( 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 ) ) ) ) @ g ) ) ).
% "11"
thf(fact_160__C13_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( r_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 ) ) ).
% "13"
thf(fact_161_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_162_Totient_Oof__nat__eq__1__iff,axiom,
! [X4: nat] :
( ( ( semiri1316708129612266289at_nat @ X4 )
= one_one_nat )
= ( X4 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_163_Totient_Oof__nat__eq__1__iff,axiom,
! [X4: nat] :
( ( ( semiri5074537144036343181t_real @ X4 )
= one_one_real )
= ( X4 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_164_Totient_Oof__nat__eq__1__iff,axiom,
! [X4: nat] :
( ( ( semiri1314217659103216013at_int @ X4 )
= one_one_int )
= ( X4 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_165_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_166_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_167_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_168_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_169_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_170_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_171_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_172_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_173_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_174_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_175_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_176_Transcendental_Olog__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
% Transcendental.log_one
thf(fact_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_arithmetic__simps_I1_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% arithmetic_simps(1)
thf(fact_186_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_187_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_188_arithmetic__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 ) ) ) ).
% arithmetic_simps(9)
thf(fact_189_arithmetic__simps_I7_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% arithmetic_simps(7)
thf(fact_190_arithmetic__simps_I4_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% arithmetic_simps(4)
thf(fact_191_arithmetic__simps_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% arithmetic_simps(3)
thf(fact_192_arithmetic__simps_I2_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% arithmetic_simps(2)
thf(fact_193_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_194_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_195_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_196_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_197_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_198_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_199_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_200_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_201_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_202_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_203_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_204_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_205_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_206_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_207_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_208_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_209_pth__7_I2_J,axiom,
! [X4: real] :
( ( plus_plus_real @ X4 @ zero_zero_real )
= X4 ) ).
% pth_7(2)
thf(fact_210_pth__7_I1_J,axiom,
! [X4: real] :
( ( plus_plus_real @ zero_zero_real @ X4 )
= X4 ) ).
% pth_7(1)
thf(fact_211_nat__arith_Orule0,axiom,
! [A: rat] :
( A
= ( plus_plus_rat @ A @ zero_zero_rat ) ) ).
% nat_arith.rule0
thf(fact_212_nat__arith_Orule0,axiom,
! [A: nat] :
( A
= ( plus_plus_nat @ A @ zero_zero_nat ) ) ).
% nat_arith.rule0
thf(fact_213_nat__arith_Orule0,axiom,
! [A: real] :
( A
= ( plus_plus_real @ A @ zero_zero_real ) ) ).
% nat_arith.rule0
thf(fact_214_nat__arith_Orule0,axiom,
! [A: int] :
( A
= ( plus_plus_int @ A @ zero_zero_int ) ) ).
% nat_arith.rule0
thf(fact_215_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_216_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_217_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_218_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_219_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_220_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_221_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_222_linordered__ab__group__add__class_Odouble__zero,axiom,
! [A: rat] :
( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% linordered_ab_group_add_class.double_zero
thf(fact_223_linordered__ab__group__add__class_Odouble__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% linordered_ab_group_add_class.double_zero
thf(fact_224_linordered__ab__group__add__class_Odouble__zero,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% linordered_ab_group_add_class.double_zero
thf(fact_225_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_226_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_227_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_228_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_229_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_230_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_231_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_232_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_233_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_234_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_235_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_254_zero__reorient,axiom,
! [X4: rat] :
( ( zero_zero_rat = X4 )
= ( X4 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_255_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_256_zero__reorient,axiom,
! [X4: real] :
( ( zero_zero_real = X4 )
= ( X4 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_257_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_258_one__reorient,axiom,
! [X4: numera2417102609627094330l_num1] :
( ( one_on3868389512446148991l_num1 = X4 )
= ( X4 = one_on3868389512446148991l_num1 ) ) ).
% one_reorient
thf(fact_259_one__reorient,axiom,
! [X4: numera4273646738625120315l_num1] :
( ( one_on7795324986448017462l_num1 = X4 )
= ( X4 = one_on7795324986448017462l_num1 ) ) ).
% one_reorient
thf(fact_260_one__reorient,axiom,
! [X4: real] :
( ( one_one_real = X4 )
= ( X4 = one_one_real ) ) ).
% one_reorient
thf(fact_261_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_262_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_263_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_264_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_265_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_266_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_267_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_268_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_269_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_270_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_271_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_272_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_273_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_274_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_275_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_276_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_277_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_278_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_279_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_280_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_281_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_282_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_283_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_284_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_285_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_286_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_287_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_288_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_289_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_290_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_291_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_292_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_293_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_294_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_295_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_296_lattice__ab__group__add__class_Odouble__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% lattice_ab_group_add_class.double_zero
thf(fact_297_lattice__ab__group__add__class_Odouble__zero,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% lattice_ab_group_add_class.double_zero
thf(fact_298_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_299_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_300_of__nat__code,axiom,
( semiri1795386414920522267l_num1
= ( ^ [N3: nat] :
( semiri3790374670636785663l_num1
@ ^ [I2: numera2417102609627094330l_num1] : ( plus_p2313304076027620419l_num1 @ I2 @ one_on3868389512446148991l_num1 )
@ N3
@ zero_z5982384998485459395l_num1 ) ) ) ).
% of_nat_code
thf(fact_301_of__nat__code,axiom,
( semiri5667362542588693146l_num1
= ( ^ [N3: nat] :
( semiri3786042916590614966l_num1
@ ^ [I2: numera4273646738625120315l_num1] : ( plus_p1441664204671982194l_num1 @ I2 @ one_on7795324986448017462l_num1 )
@ N3
@ zero_z2241845390563828978l_num1 ) ) ) ).
% of_nat_code
thf(fact_302_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N3: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I2: rat] : ( plus_plus_rat @ I2 @ one_one_rat )
@ N3
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_303_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N3: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
@ N3
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_304_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N3: nat] :
( semiri7260567687927622513x_real
@ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
@ N3
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_305_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N3: nat] :
( semiri8420488043553186161ux_int
@ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
@ N3
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_306_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_307_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_308_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_309_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_310_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_311_forall__4,axiom,
( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
! [X5: numera4273646738625120315l_num1] : ( P2 @ X5 ) )
= ( ^ [P3: numera4273646738625120315l_num1 > $o] :
( ( P3 @ one_on7795324986448017462l_num1 )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).
% forall_4
thf(fact_312_exhaust__4,axiom,
! [X4: numera4273646738625120315l_num1] :
( ( X4 = one_on7795324986448017462l_num1 )
| ( X4
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X4
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X4
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_313_forall__3,axiom,
( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
! [X5: numera6367994245245682809l_num1] : ( P2 @ X5 ) )
= ( ^ [P3: numera6367994245245682809l_num1 > $o] :
( ( P3 @ one_on7819281148064737470l_num1 )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_314_exhaust__3,axiom,
! [X4: numera6367994245245682809l_num1] :
( ( X4 = one_on7819281148064737470l_num1 )
| ( X4
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X4
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_315_exhaust__2,axiom,
! [X4: numera2417102609627094330l_num1] :
( ( X4 = one_on3868389512446148991l_num1 )
| ( X4
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_316_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X5: numera2417102609627094330l_num1] : ( P2 @ X5 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_317_greaterThan__eq__iff,axiom,
! [X4: rat,Y: rat] :
( ( ( set_or575021546402375026an_rat @ X4 )
= ( set_or575021546402375026an_rat @ Y ) )
= ( X4 = Y ) ) ).
% greaterThan_eq_iff
thf(fact_318_calculation_I1_J,axiom,
( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] : ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) )
@ ( 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 ) ) ) ) ) ).
% calculation(1)
thf(fact_319_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_320_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_321_set__plus__intro,axiom,
! [A: num,C2: set_num,B: num,D: set_num] :
( ( member_num @ A @ C2 )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( plus_plus_num @ A @ B ) @ ( plus_plus_set_num @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_322_set__plus__intro,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_set_int @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_323__092_060open_062_I_092_060lambda_062x_O_Alog_A2_A_Ilog_A2_A_Ireal_A_In__of_Ax_J_A_L_A13_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_Iln_A_Ireal_A_In__of_Ax_J_J_J_A_L_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_062__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ 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] : ( 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 ) ) ) ) ) ) ) ).
% \<open>(\<lambda>x. log 2 (log 2 (real (n_of x) + 13))) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (ln (real (n_of x))) + ln (1 / real_of_rat (\<delta>_of x)))\<close>
thf(fact_324_int_Oone__not__zero,axiom,
one_one_int != zero_zero_int ).
% int.one_not_zero
thf(fact_325_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_326_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_327_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_328_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_329_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_330_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_331_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_332_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_333_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_334_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_335_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_336_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_337_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_338_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_339_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_340_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_341_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_342_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_343_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_344_greaterThan__subset__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5849166863359141190n_real @ X4 ) @ ( set_or5849166863359141190n_real @ Y ) )
= ( ord_less_eq_real @ Y @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_345_greaterThan__subset__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X4 ) @ ( set_or1210151606488870762an_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_346_greaterThan__subset__iff,axiom,
! [X4: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_or6990855429499425204an_num @ X4 ) @ ( set_or6990855429499425204an_num @ Y ) )
= ( ord_less_eq_num @ Y @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_347_greaterThan__subset__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_or1207661135979820486an_int @ X4 ) @ ( set_or1207661135979820486an_int @ Y ) )
= ( ord_less_eq_int @ Y @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_348_greaterThan__subset__iff,axiom,
! [X4: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_or575021546402375026an_rat @ X4 ) @ ( set_or575021546402375026an_rat @ Y ) )
= ( ord_less_eq_rat @ Y @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_349_lattice__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% lattice_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_350_lattice__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% lattice_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_351_lattice__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% lattice_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_352_lattice__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% lattice_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_353_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_354_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_355_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_356_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_357_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_358_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_359_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_360_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_361_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_362_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_363_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_364_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_365_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_366_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_367_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_368_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_369_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_370_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_371_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_372_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_373_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_374_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_375_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_376_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_377_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_378_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_379_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_380_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_381_in__bigo__zero__iff,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat] :
( ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [Uu: produc5691113562410904374at_rat] : zero_zero_real ) )
= ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( F @ X )
= zero_zero_real )
@ F2 ) ) ).
% in_bigo_zero_iff
thf(fact_382_in__bigo__zero__iff,axiom,
! [F: real > real,F2: filter_real] :
( ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [Uu: real] : zero_zero_real ) )
= ( eventually_real
@ ^ [X: real] :
( ( F @ X )
= zero_zero_real )
@ F2 ) ) ).
% in_bigo_zero_iff
thf(fact_383_in__bigo__zero__iff,axiom,
! [F: nat > real,F2: filter_nat] :
( ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [Uu: nat] : zero_zero_real ) )
= ( eventually_nat
@ ^ [X: nat] :
( ( F @ X )
= zero_zero_real )
@ F2 ) ) ).
% in_bigo_zero_iff
thf(fact_384__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_385_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_386_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_387_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_388_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_389_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_390_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_391_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_392__092_060open_062_I_092_060lambda_062x_O_Areal_A_Ir__of_Ax_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_060delta_062__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( semiri5074537144036343181t_real @ ( r_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 @ ( delta_of @ X ) ) ) ) ) ) ).
% \<open>(\<lambda>x. real (r_of x)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (1 / real_of_rat (\<delta>_of x)))\<close>
thf(fact_393__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_394__092_060open_062_I_092_060lambda_062x_O_Areal_A_Ir__of_Ax_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_Iln_A_Ireal_A_In__of_Ax_J_J_J_A_L_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_062__of_Ax_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( semiri5074537144036343181t_real @ ( r_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] : ( 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 ) ) ) ) ) ) ) ).
% \<open>(\<lambda>x. real (r_of x)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (ln (real (n_of x))) + ln (1 / real_of_rat (\<delta>_of x)))\<close>
thf(fact_395_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_396_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_397_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_398_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_399_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_400_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_401_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_402_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_403_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_404_landau__o_OR__refl,axiom,
! [X4: real] : ( ord_less_eq_real @ X4 @ X4 ) ).
% landau_o.R_refl
thf(fact_405_landau__o_OR__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% landau_o.R_trans
thf(fact_406_landau__o_OR__linear,axiom,
! [X4: real,Y: real] :
( ~ ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ Y @ X4 ) ) ).
% landau_o.R_linear
thf(fact_407_landau__omega_OR__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% landau_omega.R_trans
thf(fact_408_landau__omega_OR__linear,axiom,
! [Y: real,X4: real] :
( ~ ( ord_less_eq_real @ Y @ X4 )
=> ( ord_less_eq_real @ X4 @ Y ) ) ).
% landau_omega.R_linear
thf(fact_409_landau__o_OR,axiom,
( ( ord_less_eq_real = ord_less_eq_real )
| ( ord_less_eq_real
= ( ^ [X: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X ) ) ) ) ).
% landau_o.R
thf(fact_410_landau__omega_OR,axiom,
( ( ( ^ [X: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X ) )
= ord_less_eq_real )
| ( ( ^ [X: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X ) )
= ( ^ [X: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X ) ) ) ) ).
% landau_omega.R
thf(fact_411_verit__eq__simplify_I6_J,axiom,
! [X4: real,Y: real] :
( ( X4 = Y )
=> ( ord_less_eq_real @ X4 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_412_verit__eq__simplify_I6_J,axiom,
! [X4: nat,Y: nat] :
( ( X4 = Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_413_verit__eq__simplify_I6_J,axiom,
! [X4: num,Y: num] :
( ( X4 = Y )
=> ( ord_less_eq_num @ X4 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_414_verit__eq__simplify_I6_J,axiom,
! [X4: int,Y: int] :
( ( X4 = Y )
=> ( ord_less_eq_int @ X4 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_415_verit__comp__simplify_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_416_verit__comp__simplify_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_417_verit__comp__simplify_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_418_verit__comp__simplify_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_419_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_420_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_421_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_422_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_423_landau__o_Obig_Odivide,axiom,
! [G1: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G22: produc5691113562410904374at_rat > real,F1: produc5691113562410904374at_rat > real,F22: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( G1 @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( G22 @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member1610887461201275416t_real @ F1 @ ( landau6322959426088225955t_real @ F2 @ F22 ) )
=> ( ( member1610887461201275416t_real @ G22 @ ( landau6322959426088225955t_real @ F2 @ G1 ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F1 @ X ) @ ( G1 @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F22 @ X ) @ ( G22 @ X ) ) ) ) ) ) ) ) ).
% landau_o.big.divide
thf(fact_424_landau__o_Obig_Odivide,axiom,
! [G1: real > real,F2: filter_real,G22: real > real,F1: real > real,F22: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( G1 @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G22 @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_real_real @ F1 @ ( landau308303187242894617l_real @ F2 @ F22 ) )
=> ( ( member_real_real @ G22 @ ( landau308303187242894617l_real @ F2 @ G1 ) )
=> ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F1 @ X ) @ ( G1 @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( divide_divide_real @ ( F22 @ X ) @ ( G22 @ X ) ) ) ) ) ) ) ) ).
% landau_o.big.divide
thf(fact_425_landau__o_Obig_Odivide,axiom,
! [G1: nat > real,F2: filter_nat,G22: nat > real,F1: nat > real,F22: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( G1 @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G22 @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real @ F1 @ ( landau_bigo_nat_real @ F2 @ F22 ) )
=> ( ( member_nat_real @ G22 @ ( landau_bigo_nat_real @ F2 @ G1 ) )
=> ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F1 @ X ) @ ( G1 @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( F22 @ X ) @ ( G22 @ X ) ) ) ) ) ) ) ) ).
% landau_o.big.divide
thf(fact_426_landau__o_Obig_Odivide__left,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( F @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( G @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member1610887461201275416t_real @ G @ ( landau6322959426088225955t_real @ F2 @ F ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( H @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% landau_o.big.divide_left
thf(fact_427_landau__o_Obig_Odivide__left,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_real_real @ G @ ( landau308303187242894617l_real @ F2 @ F ) )
=> ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( divide_divide_real @ ( H @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% landau_o.big.divide_left
thf(fact_428_landau__o_Obig_Odivide__left,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real @ G @ ( landau_bigo_nat_real @ F2 @ F ) )
=> ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( H @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% landau_o.big.divide_left
thf(fact_429_landau__o_Obig_Odivide__right,axiom,
! [H: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_o.big.divide_right
thf(fact_430_landau__o_Obig_Odivide__right,axiom,
! [H: real > real,F2: filter_real,F: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_o.big.divide_right
thf(fact_431_landau__o_Obig_Odivide__right,axiom,
! [H: nat > real,F2: filter_nat,F: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_o.big.divide_right
thf(fact_432_landau__o_Obig_Odivide__left__iff,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real,H: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( F @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( G @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( H @ X ) @ ( G @ X ) ) ) )
= ( member1610887461201275416t_real @ G @ ( landau6322959426088225955t_real @ F2 @ F ) ) ) ) ) ) ).
% landau_o.big.divide_left_iff
thf(fact_433_landau__o_Obig_Odivide__left__iff,axiom,
! [F: real > real,F2: filter_real,G: real > real,H: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( divide_divide_real @ ( H @ X ) @ ( G @ X ) ) ) )
= ( member_real_real @ G @ ( landau308303187242894617l_real @ F2 @ F ) ) ) ) ) ) ).
% landau_o.big.divide_left_iff
thf(fact_434_landau__o_Obig_Odivide__left__iff,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real,H: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G @ X )
!= zero_zero_real )
@ F2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( H @ X ) @ ( F @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( H @ X ) @ ( G @ X ) ) ) )
= ( member_nat_real @ G @ ( landau_bigo_nat_real @ F2 @ F ) ) ) ) ) ) ).
% landau_o.big.divide_left_iff
thf(fact_435_landau__o_Obig_Odivide__right__iff,axiom,
! [H: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) )
= ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ) ).
% landau_o.big.divide_right_iff
thf(fact_436_landau__o_Obig_Odivide__right__iff,axiom,
! [H: real > real,F2: filter_real,F: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) )
= ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ) ).
% landau_o.big.divide_right_iff
thf(fact_437_landau__o_Obig_Odivide__right__iff,axiom,
! [H: nat > real,F2: filter_nat,F: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) )
= ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_o.big.divide_right_iff
thf(fact_438_rel__simps_I46_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% rel_simps(46)
thf(fact_439_rel__simps_I46_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% rel_simps(46)
thf(fact_440_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_441_rel__simps_I46_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% rel_simps(46)
thf(fact_442_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_443_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_444_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_445_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_446_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_447_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_448_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_449_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_450_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_451_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_452_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_453_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_454_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_455_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_456_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_457_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_458_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_459_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_460_add__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_461_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_462_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_463_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_464_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_465_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_466_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_467_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_468_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_469_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_470_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_471_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_472_landau__o_Obig_Ocong,axiom,
! [F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( F @ X )
= ( G @ X ) )
@ F2 )
=> ( ( landau6322959426088225955t_real @ F2 @ F )
= ( landau6322959426088225955t_real @ F2 @ G ) ) ) ).
% landau_o.big.cong
thf(fact_473_landau__o_Obig_Ocong,axiom,
! [F: real > real,G: real > real,F2: filter_real] :
( ( eventually_real
@ ^ [X: real] :
( ( F @ X )
= ( G @ X ) )
@ F2 )
=> ( ( landau308303187242894617l_real @ F2 @ F )
= ( landau308303187242894617l_real @ F2 @ G ) ) ) ).
% landau_o.big.cong
thf(fact_474_landau__o_Obig_Ocong,axiom,
! [F: nat > real,G: nat > real,F2: filter_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F @ X )
= ( G @ X ) )
@ F2 )
=> ( ( landau_bigo_nat_real @ F2 @ F )
= ( landau_bigo_nat_real @ F2 @ G ) ) ) ).
% landau_o.big.cong
thf(fact_475_landau__o_Obig_Ocong__ex,axiom,
! [F1: produc5691113562410904374at_rat > real,F22: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G1: produc5691113562410904374at_rat > real,G22: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( F1 @ X )
= ( F22 @ X ) )
@ F2 )
=> ( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( G1 @ X )
= ( G22 @ X ) )
@ F2 )
=> ( ( member1610887461201275416t_real @ F1 @ ( landau6322959426088225955t_real @ F2 @ G1 ) )
= ( member1610887461201275416t_real @ F22 @ ( landau6322959426088225955t_real @ F2 @ G22 ) ) ) ) ) ).
% landau_o.big.cong_ex
thf(fact_476_landau__o_Obig_Ocong__ex,axiom,
! [F1: real > real,F22: real > real,F2: filter_real,G1: real > real,G22: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F1 @ X )
= ( F22 @ X ) )
@ F2 )
=> ( ( eventually_real
@ ^ [X: real] :
( ( G1 @ X )
= ( G22 @ X ) )
@ F2 )
=> ( ( member_real_real @ F1 @ ( landau308303187242894617l_real @ F2 @ G1 ) )
= ( member_real_real @ F22 @ ( landau308303187242894617l_real @ F2 @ G22 ) ) ) ) ) ).
% landau_o.big.cong_ex
thf(fact_477_landau__o_Obig_Ocong__ex,axiom,
! [F1: nat > real,F22: nat > real,F2: filter_nat,G1: nat > real,G22: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F1 @ X )
= ( F22 @ X ) )
@ F2 )
=> ( ( eventually_nat
@ ^ [X: nat] :
( ( G1 @ X )
= ( G22 @ X ) )
@ F2 )
=> ( ( member_nat_real @ F1 @ ( landau_bigo_nat_real @ F2 @ G1 ) )
= ( member_nat_real @ F22 @ ( landau_bigo_nat_real @ F2 @ G22 ) ) ) ) ) ).
% landau_o.big.cong_ex
thf(fact_478_landau__o_Obig_Oin__cong,axiom,
! [F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,H: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( F @ X )
= ( G @ X ) )
@ F2 )
=> ( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ H ) )
= ( member1610887461201275416t_real @ G @ ( landau6322959426088225955t_real @ F2 @ H ) ) ) ) ).
% landau_o.big.in_cong
thf(fact_479_landau__o_Obig_Oin__cong,axiom,
! [F: real > real,G: real > real,F2: filter_real,H: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( F @ X )
= ( G @ X ) )
@ F2 )
=> ( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ H ) )
= ( member_real_real @ G @ ( landau308303187242894617l_real @ F2 @ H ) ) ) ) ).
% landau_o.big.in_cong
thf(fact_480_landau__o_Obig_Oin__cong,axiom,
! [F: nat > real,G: nat > real,F2: filter_nat,H: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( F @ X )
= ( G @ X ) )
@ F2 )
=> ( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ H ) )
= ( member_nat_real @ G @ ( landau_bigo_nat_real @ F2 @ H ) ) ) ) ).
% landau_o.big.in_cong
thf(fact_481_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_482_verit__comp__simplify_I29_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% verit_comp_simplify(29)
thf(fact_483_verit__comp__simplify_I29_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(29)
thf(fact_484_verit__comp__simplify_I29_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(29)
thf(fact_485_verit__comp__simplify_I29_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% verit_comp_simplify(29)
thf(fact_486_rel__simps_I45_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% rel_simps(45)
thf(fact_487_rel__simps_I45_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% rel_simps(45)
thf(fact_488_rel__simps_I45_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% rel_simps(45)
thf(fact_489_rel__simps_I45_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% rel_simps(45)
thf(fact_490_add__sign__intros_I8_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_sign_intros(8)
thf(fact_491_add__sign__intros_I8_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(8)
thf(fact_492_add__sign__intros_I8_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(8)
thf(fact_493_add__sign__intros_I8_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_sign_intros(8)
thf(fact_494_add__sign__intros_I4_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_495_add__sign__intros_I4_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_496_add__sign__intros_I4_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_497_add__sign__intros_I4_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_498_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_499_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_500_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_501_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_502_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_503_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_504_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_505_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_506_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_507_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_508_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_509_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_510_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_511_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_512_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_513_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_514_add__nonneg__eq__0__iff,axiom,
! [X4: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X4 @ Y )
= zero_zero_rat )
= ( ( X4 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_515_add__nonneg__eq__0__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_516_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_517_add__nonneg__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_518_add__nonpos__eq__0__iff,axiom,
! [X4: rat,Y: rat] :
( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X4 @ Y )
= zero_zero_rat )
= ( ( X4 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_519_add__nonpos__eq__0__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_520_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_521_add__nonpos__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_522_verit__comp__simplify_I8_J,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% verit_comp_simplify(8)
thf(fact_523_verit__comp__simplify_I8_J,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% verit_comp_simplify(8)
thf(fact_524_verit__comp__simplify_I8_J,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% verit_comp_simplify(8)
thf(fact_525_verit__comp__simplify_I8_J,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% verit_comp_simplify(8)
thf(fact_526_verit__comp__simplify_I7_J,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% verit_comp_simplify(7)
thf(fact_527_verit__comp__simplify_I7_J,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% verit_comp_simplify(7)
thf(fact_528_verit__comp__simplify_I7_J,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% verit_comp_simplify(7)
thf(fact_529_verit__comp__simplify_I7_J,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% verit_comp_simplify(7)
thf(fact_530_verit__comp__simplify_I5_J,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% verit_comp_simplify(5)
thf(fact_531_verit__comp__simplify_I5_J,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% verit_comp_simplify(5)
thf(fact_532_verit__comp__simplify_I5_J,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% verit_comp_simplify(5)
thf(fact_533_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_534_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_535_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_536_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_537_log__def,axiom,
( log
= ( ^ [A3: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% log_def
thf(fact_538_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_539_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_540_set__plus__elim,axiom,
! [X4: real,A2: set_real,B2: set_real] :
( ( member_real @ X4 @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A4: real,B4: real] :
( ( X4
= ( plus_plus_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_541_set__plus__elim,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X4 @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X4
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_542_set__plus__elim,axiom,
! [X4: num,A2: set_num,B2: set_num] :
( ( member_num @ X4 @ ( plus_plus_set_num @ A2 @ B2 ) )
=> ~ ! [A4: num,B4: num] :
( ( X4
= ( plus_plus_num @ A4 @ B4 ) )
=> ( ( member_num @ A4 @ A2 )
=> ~ ( member_num @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_543_set__plus__elim,axiom,
! [X4: int,A2: set_int,B2: set_int] :
( ( member_int @ X4 @ ( plus_plus_set_int @ A2 @ B2 ) )
=> ~ ! [A4: int,B4: int] :
( ( X4
= ( plus_plus_int @ A4 @ B4 ) )
=> ( ( member_int @ A4 @ A2 )
=> ~ ( member_int @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_544_ln__ge__zero,axiom,
! [X4: real] :
( ( ord_less_eq_real @ one_one_real @ X4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% ln_ge_zero
thf(fact_545_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_546_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_547_ln__add__one__self__le__self,axiom,
! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% ln_add_one_self_le_self
thf(fact_548__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_549__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_550__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ireal_A_Ir__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_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] : ( ln_ln_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( r_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] : ( divide_divide_real @ one_one_real @ ( power_power_real @ ( field_7254667332652039916t_real @ ( delta_of @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (real (r_of x) + 1)) \<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_551__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_552__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_553__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_554_landau__o_Obig__subsetI,axiom,
! [F: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,G: produc5691113562410904374at_rat > real] :
( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F2 @ G ) )
=> ( ord_le1994352800634783511t_real @ ( landau6322959426088225955t_real @ F2 @ F ) @ ( landau6322959426088225955t_real @ F2 @ G ) ) ) ).
% landau_o.big_subsetI
thf(fact_555_landau__o_Obig__subsetI,axiom,
! [F: real > real,F2: filter_real,G: real > real] :
( ( member_real_real @ F @ ( landau308303187242894617l_real @ F2 @ G ) )
=> ( ord_le4198349162570665613l_real @ ( landau308303187242894617l_real @ F2 @ F ) @ ( landau308303187242894617l_real @ F2 @ G ) ) ) ).
% landau_o.big_subsetI
thf(fact_556_landau__o_Obig__subsetI,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real] :
( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F2 @ G ) )
=> ( ord_le2908806416726583473t_real @ ( landau_bigo_nat_real @ F2 @ F ) @ ( landau_bigo_nat_real @ F2 @ G ) ) ) ).
% landau_o.big_subsetI
thf(fact_557_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_558_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_559_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_560_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_561_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_562_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_563_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_564_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_565_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_566__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_567__092_060open_062_I_092_060lambda_062x_O_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_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_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] : ( ln_ln_real @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_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 ) ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (1 / real_of_rat (\<delta>_of x))) \<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_568_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_569_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_570_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_571_landau__o_Obig_Ofilter__mono_H,axiom,
! [F12: filter3199273883467263174at_rat,F23: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real] :
( ( ord_le4292863764789988646at_rat @ F12 @ F23 )
=> ( ord_le1994352800634783511t_real @ ( landau6322959426088225955t_real @ F23 @ F ) @ ( landau6322959426088225955t_real @ F12 @ F ) ) ) ).
% landau_o.big.filter_mono'
thf(fact_572_landau__o_Obig_Ofilter__mono_H,axiom,
! [F12: filter_real,F23: filter_real,F: real > real] :
( ( ord_le4104064031414453916r_real @ F12 @ F23 )
=> ( ord_le4198349162570665613l_real @ ( landau308303187242894617l_real @ F23 @ F ) @ ( landau308303187242894617l_real @ F12 @ F ) ) ) ).
% landau_o.big.filter_mono'
thf(fact_573_landau__o_Obig_Ofilter__mono_H,axiom,
! [F12: filter_nat,F23: filter_nat,F: nat > real] :
( ( ord_le2510731241096832064er_nat @ F12 @ F23 )
=> ( ord_le2908806416726583473t_real @ ( landau_bigo_nat_real @ F23 @ F ) @ ( landau_bigo_nat_real @ F12 @ F ) ) ) ).
% landau_o.big.filter_mono'
thf(fact_574_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_575_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_576_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_577_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_578_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_579_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ B ) )
=> ? [X6: nat] :
( ( P @ X6 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X6 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_580_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_581_le__num__One__iff,axiom,
! [X4: num] :
( ( ord_less_eq_num @ X4 @ one )
= ( X4 = one ) ) ).
% le_num_One_iff
thf(fact_582_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_583_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_584_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_585_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_586_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_587_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_588_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_589_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_590_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_591_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_592_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_593_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_594_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_595_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_596_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_597_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_598_landau__o_Obig_Ofilter__mono,axiom,
! [F12: filter3199273883467263174at_rat,F23: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real] :
( ( ord_le4292863764789988646at_rat @ F12 @ F23 )
=> ( ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F23 @ G ) )
=> ( member1610887461201275416t_real @ F @ ( landau6322959426088225955t_real @ F12 @ G ) ) ) ) ).
% landau_o.big.filter_mono
thf(fact_599_landau__o_Obig_Ofilter__mono,axiom,
! [F12: filter_real,F23: filter_real,F: real > real,G: real > real] :
( ( ord_le4104064031414453916r_real @ F12 @ F23 )
=> ( ( member_real_real @ F @ ( landau308303187242894617l_real @ F23 @ G ) )
=> ( member_real_real @ F @ ( landau308303187242894617l_real @ F12 @ G ) ) ) ) ).
% landau_o.big.filter_mono
thf(fact_600_landau__o_Obig_Ofilter__mono,axiom,
! [F12: filter_nat,F23: filter_nat,F: nat > real,G: nat > real] :
( ( ord_le2510731241096832064er_nat @ F12 @ F23 )
=> ( ( member_nat_real @ F @ ( landau_bigo_nat_real @ F23 @ G ) )
=> ( member_nat_real @ F @ ( landau_bigo_nat_real @ F12 @ G ) ) ) ) ).
% landau_o.big.filter_mono
thf(fact_601_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_602_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_603_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_604_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N3: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_605_set__zero__plus2,axiom,
! [A2: set_rat,B2: set_rat] :
( ( member_rat @ zero_zero_rat @ A2 )
=> ( ord_less_eq_set_rat @ B2 @ ( plus_plus_set_rat @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_606_set__zero__plus2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( plus_plus_set_nat @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_607_set__zero__plus2,axiom,
! [A2: set_real,B2: set_real] :
( ( member_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_set_real @ B2 @ ( plus_plus_set_real @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_608_set__zero__plus2,axiom,
! [A2: set_int,B2: set_int] :
( ( member_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_set_int @ B2 @ ( plus_plus_set_int @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_609_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_610_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_611_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_612_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_613_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_614_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_615__C15_C,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( t_of @ X ) ) @ ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ ( r_of @ X ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ 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] : ( 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 ) ) ) ) ) ) ) ) ).
% "15"
thf(fact_616__092_060open_062_I_092_060lambda_062x_O_Alog_A2_A_Ireal_A_It__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_Ireal_A_In__of_Ax_J_J_A_L_A1_A_P_A_Ireal__of__rat_A_I_092_060delta_062__of_Ax_J_J_092_060_094sup_0622_A_K_A_Iln_A_Iln_A_Ireal_A_In__of_Ax_J_J_J_A_L_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_062__of_Ax_J_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( 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] : ( 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 ) ) ) ) ) ) ) ) ) ).
% \<open>(\<lambda>x. log 2 (real (t_of x) + 1)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (real (n_of x)) + 1 / (real_of_rat (\<delta>_of x))\<^sup>2 * (ln (ln (real (n_of x))) + ln (1 / real_of_rat (\<delta>_of x))))\<close>
thf(fact_617__092_060open_062_I_092_060lambda_062x_O_Alog_A2_A_Ireal_A_Ir__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_Ireal_A_In__of_Ax_J_J_A_L_A1_A_P_A_Ireal__of__rat_A_I_092_060delta_062__of_Ax_J_J_092_060_094sup_0622_A_K_A_Iln_A_Iln_A_Ireal_A_In__of_Ax_J_J_J_A_L_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_062__of_Ax_J_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( r_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] : ( 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 ) ) ) ) ) ) ) ) ) ).
% \<open>(\<lambda>x. log 2 (real (r_of x) + 1)) \<in> O[sequentially \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (real (n_of x)) + 1 / (real_of_rat (\<delta>_of x))\<^sup>2 * (ln (ln (real (n_of x))) + ln (1 / real_of_rat (\<delta>_of x))))\<close>
thf(fact_618__092_060open_062_I_092_060lambda_062x_O_Alog_A2_A_Ireal_A_Ir__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_A1_A_P_A_Ireal__of__rat_A_I_092_060delta_062__of_Ax_J_J_092_060_094sup_0622_A_K_A_Iln_A_Iln_A_Ireal_A_In__of_Ax_J_J_J_A_L_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_062__of_Ax_J_J_J_J_092_060close_062,axiom,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( r_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 ) ) ) ) ) ) ) ) ).
% \<open>(\<lambda>x. log 2 (real (r_of x) + 1)) \<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 * (ln (ln (real (n_of x))) + ln (1 / real_of_rat (\<delta>_of x))))\<close>
thf(fact_619_calculation_I2_J,axiom,
( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] : ( ord_less_eq_real @ zero_zero_real @ ( 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 ) ) ) ) ) ) )
@ ( 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 ) ) ) ) ) ).
% calculation(2)
thf(fact_620_int_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% int.nat_pow_one
thf(fact_621_arithmetic__simps_I63_J,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% arithmetic_simps(63)
thf(fact_622_arithmetic__simps_I63_J,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% arithmetic_simps(63)
thf(fact_623_arithmetic__simps_I63_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_624_arithmetic__simps_I63_J,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% arithmetic_simps(63)
thf(fact_625_arithmetic__simps_I62_J,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% arithmetic_simps(62)
thf(fact_626_arithmetic__simps_I62_J,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% arithmetic_simps(62)
thf(fact_627_arithmetic__simps_I62_J,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_628_arithmetic__simps_I62_J,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% arithmetic_simps(62)
thf(fact_629_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: rat,X4: rat,B: rat] :
( ( ( times_times_rat @ A @ X4 )
= ( times_times_rat @ B @ X4 ) )
= ( ( A = B )
| ( X4 = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_630_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X4: real,B: real] :
( ( ( times_times_real @ A @ X4 )
= ( times_times_real @ B @ X4 ) )
= ( ( A = B )
| ( X4 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_631_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: rat,X4: rat,Y: rat] :
( ( ( times_times_rat @ A @ X4 )
= ( times_times_rat @ A @ Y ) )
= ( ( X4 = Y )
| ( A = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_632_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X4: real,Y: real] :
( ( ( times_times_real @ A @ X4 )
= ( times_times_real @ A @ Y ) )
= ( ( X4 = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_633_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_634_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_635_vector__space__over__itself_Oscale__zero__left,axiom,
! [X4: rat] :
( ( times_times_rat @ zero_zero_rat @ X4 )
= zero_zero_rat ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_636_vector__space__over__itself_Oscale__zero__left,axiom,
! [X4: real] :
( ( times_times_real @ zero_zero_real @ X4 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_637_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: rat,X4: rat] :
( ( ( times_times_rat @ A @ X4 )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( X4 = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_638_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X4: real] :
( ( ( times_times_real @ A @ X4 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X4 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_639_arithmetic__simps_I78_J,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_640_arithmetic__simps_I78_J,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_641_arithmetic__simps_I78_J,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_642_arithmetic__simps_I78_J,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_643_arithmetic__simps_I78_J,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% arithmetic_simps(78)
thf(fact_644_arithmetic__simps_I79_J,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% arithmetic_simps(79)
thf(fact_645_arithmetic__simps_I79_J,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% arithmetic_simps(79)
thf(fact_646_arithmetic__simps_I79_J,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% arithmetic_simps(79)
thf(fact_647_arithmetic__simps_I79_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% arithmetic_simps(79)
thf(fact_648_arithmetic__simps_I79_J,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% arithmetic_simps(79)
thf(fact_649_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X4: real] :
( ( times_times_real @ one_one_real @ X4 )
= X4 ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_650_arithmetic__simps_I58_J,axiom,
! [M: num,N: num] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% arithmetic_simps(58)
thf(fact_651_arithmetic__simps_I58_J,axiom,
! [M: num,N: num] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% arithmetic_simps(58)
thf(fact_652_arithmetic__simps_I58_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% arithmetic_simps(58)
thf(fact_653_arithmetic__simps_I58_J,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% arithmetic_simps(58)
thf(fact_654_arithmetic__simps_I58_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% arithmetic_simps(58)
thf(fact_655_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_656_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_657_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_658_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_659_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_660_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_661_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_662_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_663_int_Onat__pow__0,axiom,
! [X4: int] :
( ( power_power_int @ X4 @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
thf(fact_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_693_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_694_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_695_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_696_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_697_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_698_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_699__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_700__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_701__092_060open_062_I_092_060lambda_062x_O_Aln_A_Ireal_A_It__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_A1_A_P_A_Ireal__of__rat_A_I_092_060delta_062__of_Ax_J_J_092_060_094sup_0622_A_K_A_Iln_A_Iln_A_Ireal_A_In__of_Ax_J_J_J_A_L_Aln_A_I1_A_P_Areal__of__rat_A_I_092_060delta_062__of_Ax_J_J_J_J_092_060close_062,axiom,
( 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 ) ) ) ) ) ) ) ) ).
% \<open>(\<lambda>x. ln (real (t_of x) + 1)) \<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 * (ln (ln (real (n_of x))) + ln (1 / real_of_rat (\<delta>_of x))))\<close>
thf(fact_702_verit__la__generic,axiom,
! [A: int,X4: int] :
( ( ord_less_eq_int @ A @ X4 )
| ( A = X4 )
| ( ord_less_eq_int @ X4 @ A ) ) ).
% verit_la_generic
thf(fact_703_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X4: rat,A: rat,B: rat] :
( ( X4 != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ X4 )
= ( times_times_rat @ B @ X4 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_704_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X4: real,A: real,B: real] :
( ( X4 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X4 )
= ( times_times_real @ B @ X4 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_705_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: rat,X4: rat,Y: rat] :
( ( A != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ X4 )
= ( times_times_rat @ A @ Y ) )
=> ( X4 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_706_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X4: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X4 )
= ( times_times_real @ A @ Y ) )
=> ( X4 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_707_mult__delta__left,axiom,
! [B: $o,X4: rat,Y: rat] :
( ( B
=> ( ( times_times_rat @ ( if_rat @ B @ X4 @ zero_zero_rat ) @ Y )
= ( times_times_rat @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_rat @ ( if_rat @ B @ X4 @ zero_zero_rat ) @ Y )
= zero_zero_rat ) ) ) ).
% mult_delta_left
thf(fact_708_mult__delta__left,axiom,
! [B: $o,X4: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X4 @ zero_zero_real ) @ Y )
= ( times_times_real @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X4 @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_709_mult__delta__left,axiom,
! [B: $o,X4: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X4 @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X4 @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_710_mult__delta__left,axiom,
! [B: $o,X4: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X4 @ zero_zero_int ) @ Y )
= ( times_times_int @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X4 @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_711_mult__delta__right,axiom,
! [B: $o,X4: rat,Y: rat] :
( ( B
=> ( ( times_times_rat @ X4 @ ( if_rat @ B @ Y @ zero_zero_rat ) )
= ( times_times_rat @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_rat @ X4 @ ( if_rat @ B @ Y @ zero_zero_rat ) )
= zero_zero_rat ) ) ) ).
% mult_delta_right
thf(fact_712_mult__delta__right,axiom,
! [B: $o,X4: real,Y: real] :
( ( B
=> ( ( times_times_real @ X4 @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X4 @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_713_mult__delta__right,axiom,
! [B: $o,X4: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X4 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X4 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_714_mult__delta__right,axiom,
! [B: $o,X4: int,Y: int] :
( ( B
=> ( ( times_times_int @ X4 @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X4 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X4 @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_715_mult_Ocomm__neutral,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% mult.comm_neutral
thf(fact_716_mult_Ocomm__neutral,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ one_on7795324986448017462l_num1 )
= A ) ).
% mult.comm_neutral
thf(fact_717_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_718_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_719_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_720_comm__monoid__mult__class_Omult__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_721_comm__monoid__mult__class_Omult__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_722_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_723_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_724_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_725_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X4: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X4 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_726_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X4 )
= ( plus_plus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ B @ X4 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_727_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_728_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_729_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_730_mult__of__nat__commute,axiom,
! [X4: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).
% mult_of_nat_commute
thf(fact_731_mult__of__nat__commute,axiom,
! [X4: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).
% mult_of_nat_commute
thf(fact_732_mult__of__nat__commute,axiom,
! [X4: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).
% mult_of_nat_commute
thf(fact_733_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X4 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X4 ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_734_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X4: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X4 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X4 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_735_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_736_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_737_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_738_Groups_Omult__ac_I2_J,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_739_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_740_Groups_Omult__ac_I2_J,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_741_Groups_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 ) ) ) ).
% Groups.mult_ac(1)
thf(fact_742_Groups_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 ) ) ) ).
% Groups.mult_ac(1)
thf(fact_743_Groups_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 ) ) ) ).
% Groups.mult_ac(1)
thf(fact_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_mult__1s_I2_J,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ one ) )
= A ) ).
% mult_1s(2)
thf(fact_758_mult__1s_I2_J,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ one ) )
= A ) ).
% mult_1s(2)
thf(fact_759_mult__1s_I2_J,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_1s(2)
thf(fact_760_mult__1s_I2_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_1s(2)
thf(fact_761_mult__1s_I2_J,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_1s(2)
thf(fact_762_mult__1s_I1_J,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ one ) @ A )
= A ) ).
% mult_1s(1)
thf(fact_763_mult__1s_I1_J,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ one ) @ A )
= A ) ).
% mult_1s(1)
thf(fact_764_mult__1s_I1_J,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_1s(1)
thf(fact_765_mult__1s_I1_J,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_1s(1)
thf(fact_766_mult__1s_I1_J,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_1s(1)
thf(fact_767_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_768_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_769_landau__o_OR__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% landau_o.R_mult_left_mono
thf(fact_770_landau__o_OR__mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% landau_o.R_mult_right_mono
thf(fact_771_landau__omega_OR__mult__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ C @ A ) ) ) ) ).
% landau_omega.R_mult_left_mono
thf(fact_772_landau__omega_OR__mult__right__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ B @ C ) @ ( times_times_real @ A @ C ) ) ) ) ).
% landau_omega.R_mult_right_mono
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_793_sum__sqs__eq,axiom,
! [X4: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ X4 @ ( times_times_real @ Y @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( Y = X4 ) ) ).
% sum_sqs_eq
thf(fact_794_sum__sqs__eq,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ X4 @ ( times_times_int @ Y @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( Y = X4 ) ) ).
% sum_sqs_eq
thf(fact_795_mult__2,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p2313304076027620419l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_796_mult__2,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_797_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_798_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_799_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_landau__o_Obig_Odivide__eq2,axiom,
! [H: produc5691113562410904374at_rat > real,F2: filter3199273883467263174at_rat,F: produc5691113562410904374at_rat > real,G: produc5691113562410904374at_rat > real] :
( ( eventu6700955888398734894at_rat
@ ^ [X: produc5691113562410904374at_rat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau6322959426088225955t_real @ F2 @ G ) )
= ( member1610887461201275416t_real @ F
@ ( landau6322959426088225955t_real @ F2
@ ^ [X: produc5691113562410904374at_rat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_o.big.divide_eq2
thf(fact_811_landau__o_Obig_Odivide__eq2,axiom,
! [H: real > real,F2: filter_real,F: real > real,G: real > real] :
( ( eventually_real
@ ^ [X: real] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau308303187242894617l_real @ F2 @ G ) )
= ( member_real_real @ F
@ ( landau308303187242894617l_real @ F2
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_o.big.divide_eq2
thf(fact_812_landau__o_Obig_Odivide__eq2,axiom,
! [H: nat > real,F2: filter_nat,F: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real
@ ^ [X: nat] : ( divide_divide_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2 @ G ) )
= ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( times_times_real @ ( G @ X ) @ ( H @ X ) ) ) ) ) ) ).
% landau_o.big.divide_eq2
thf(fact_813_landau__o_Obig_Odivide__eq1,axiom,
! [H: nat > real,F2: filter_nat,F: nat > real,G: nat > real] :
( ( eventually_nat
@ ^ [X: nat] :
( ( H @ X )
!= zero_zero_real )
@ F2 )
=> ( ( member_nat_real @ F
@ ( landau_bigo_nat_real @ F2
@ ^ [X: nat] : ( divide_divide_real @ ( G @ X ) @ ( H @ X ) ) ) )
= ( member_nat_real
@ ^ [X: nat] : ( times_times_real @ ( F @ X ) @ ( H @ X ) )
@ ( landau_bigo_nat_real @ F2 @ G ) ) ) ) ).
% landau_o.big.divide_eq1
thf(fact_814_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_815_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_816_evt,axiom,
! [Delta: real,Epsilon: real,N: real,P: produc5691113562410904374at_rat > $o] :
( ! [X6: produc5691113562410904374at_rat] :
( ( ( ord_less_real @ zero_zero_real @ ( field_7254667332652039916t_real @ ( delta_of @ X6 ) ) )
& ( ord_less_real @ zero_zero_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X6 ) ) )
& ( ord_less_eq_real @ Delta @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( delta_of @ X6 ) ) ) )
& ( ord_less_eq_real @ Epsilon @ ( divide_divide_real @ one_one_real @ ( field_7254667332652039916t_real @ ( epsilon_of @ X6 ) ) ) )
& ( ord_less_eq_real @ N @ ( semiri5074537144036343181t_real @ ( n_of @ X6 ) ) ) )
=> ( P @ X6 ) )
=> ( eventu6700955888398734894at_rat @ P @ ( 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 ) ) ) ) ) ) ).
% evt
thf(fact_817_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_818_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_819_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_820_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_821_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_822_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_823_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_824_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_825_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_826_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_827_ln__inj__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X4 )
= ( ln_ln_real @ Y ) )
= ( X4 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_828_ln__less__cancel__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X4 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_829_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_830_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_831_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_832_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_833_ln__le__cancel__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_834_ln__eq__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ( ln_ln_real @ X4 )
= zero_zero_real )
= ( X4 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_835_ln__gt__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
= ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% ln_gt_zero_iff
thf(fact_836_ln__less__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
= ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_837_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_838_log__less__cancel__iff,axiom,
! [A: real,X4: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) )
= ( ord_less_real @ X4 @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_839_log__less__one__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ ( log @ A @ X4 ) @ one_one_real )
= ( ord_less_real @ X4 @ A ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_840_one__less__log__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ one_one_real @ ( log @ A @ X4 ) )
= ( ord_less_real @ A @ X4 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_841_log__less__zero__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ ( log @ A @ X4 ) @ zero_zero_real )
= ( ord_less_real @ X4 @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_842_zero__less__log__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X4 ) )
= ( ord_less_real @ one_one_real @ X4 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_843_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_844_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_845_ln__ge__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
= ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% ln_ge_zero_iff
thf(fact_846_ln__le__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
= ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_847_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_848_log__le__cancel__iff,axiom,
! [A: real,X4: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) )
= ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_849_log__le__one__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ one_one_real )
= ( ord_less_eq_real @ X4 @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_850_one__le__log__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X4 ) )
= ( ord_less_eq_real @ A @ X4 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_851_log__le__zero__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ zero_zero_real )
= ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_852_zero__le__log__cancel__iff,axiom,
! [A: real,X4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X4 ) )
= ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_853_log__pow__cancel,axiom,
! [A: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( power_power_real @ A @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_854_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_855_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_856_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_857_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_858_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_859_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_860_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_861_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_862_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_863_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_864_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_865_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_866_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_867_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_868_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_869_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_870_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_871_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_872_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_873_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_874_ln__less__self,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% ln_less_self
thf(fact_875_ln__bound,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% ln_bound
thf(fact_876_ln__gt__zero,axiom,
! [X4: real] :
( ( ord_less_real @ one_one_real @ X4 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% ln_gt_zero
thf(fact_877_ln__less__zero,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ X4 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_878_ln__gt__zero__imp__gt__one,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_879_ln__ge__zero__imp__ge__one,axiom,
! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_880_ln__mult,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X4 @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_881_log__base__change,axiom,
! [A: real,B: real,X4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ B @ X4 )
= ( divide_divide_real @ ( log @ A @ X4 ) @ ( log @ A @ B ) ) ) ) ) ).
% log_base_change
thf(fact_882_log__of__power__eq,axiom,
! [M: nat,B: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_883_less__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% less_log_of_power
thf(fact_884_ln__realpow,axiom,
! [X4: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ln_ln_real @ ( power_power_real @ X4 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X4 ) ) ) ) ).
% ln_realpow
thf(fact_885_log__mult,axiom,
! [A: real,X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( times_times_real @ X4 @ Y ) )
= ( plus_plus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_886_le__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% le_log_of_power
thf(fact_887_log__nat__power,axiom,
! [X4: real,B: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( log @ B @ ( power_power_real @ X4 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X4 ) ) ) ) ).
% log_nat_power
thf(fact_888_log__base__pow,axiom,
! [A: real,N: nat,X4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ ( power_power_real @ A @ N ) @ X4 )
= ( divide_divide_real @ ( log @ A @ X4 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log_base_pow
thf(fact_889_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_890_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_891_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,X4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( log @ A @ X4 )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X4 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_892_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_893_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_894_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_895_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_896_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_897_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_898_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_899_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_900_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_901_verit__comp__simplify_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% verit_comp_simplify(24)
thf(fact_902_verit__comp__simplify_I21_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% verit_comp_simplify(21)
thf(fact_903_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_904_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_905_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_906_mult__less__cancel1,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 ) ) ) ).
% mult_less_cancel1
thf(fact_907_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_908_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_909_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_910_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_911_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_912_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_913_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_914_verit__comp__simplify_I22_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% verit_comp_simplify(22)
thf(fact_915_verit__comp__simplify_I27_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% verit_comp_simplify(27)
thf(fact_916_verit__comp__simplify_I23_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% verit_comp_simplify(23)
thf(fact_917_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel1
thf(fact_918_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_919_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_920_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_921_verit__comp__simplify_I25_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% verit_comp_simplify(25)
thf(fact_922_verit__comp__simplify_I15_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% verit_comp_simplify(15)
thf(fact_923_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_924_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_925_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_926_le__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(1)
thf(fact_927_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_928_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_929_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_930_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_931_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_932_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X4: rat,Y: rat] :
( ( if_rat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X4: rat,Y: rat] :
( ( if_rat @ $true @ X4 @ Y )
= X4 ) ).
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,
! [X4: real,Y: real] :
( ( if_real @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X4: real,Y: real] :
( ( if_real @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member1610887461201275416t_real
@ ^ [X: produc5691113562410904374at_rat] : ( ln_ln_real @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( semiri5074537144036343181t_real @ ( n_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 @ ( semiri5074537144036343181t_real @ ( n_of @ X ) ) ) ) ) ).
%------------------------------------------------------------------------------