TPTP Problem File: SLH0979^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Frequency_Moments/0086_Frequency_Moment_2/prob_00298_012768__19973476_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1287 ( 549 unt; 213 typ; 0 def)
% Number of atoms : 2906 (1245 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8196 ( 230 ~; 84 |; 139 &;6538 @)
% ( 0 <=>;1205 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 684 ( 684 >; 0 *; 0 +; 0 <<)
% Number of symbols : 190 ( 187 usr; 18 con; 0-3 aty)
% Number of variables : 2619 ( 202 ^;2382 !; 35 ?;2619 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:18:12.707
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
list_s1210847774152347623at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_se7855581050983116737at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Extended____Nonnegative____Real__Oennreal_J,type,
list_E5688521862016077384nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Rat__Orat_J,type,
formal_Power_fps_rat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Rat__Orat_J,type,
list_rat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (187)
thf(sy_c_Bertrand_Omangoldt__odd,type,
mangoldt_odd: nat > real ).
thf(sy_c_Bertrand_Opsi__even,type,
psi_even: nat > real ).
thf(sy_c_Bertrand_Opsi__odd,type,
psi_odd: nat > real ).
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oequiv__rels,type,
equiva8721718519204927301v_rels: nat > list_s1210847774152347623at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
equiva2048684438135499664of_nat: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
extend7643940197134561352nnreal: real > extend8495563244428889912nnreal ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Float_Oreal__divl,type,
real_divl: nat > real > real > real ).
thf(sy_c_Float_Oreal__divr,type,
real_divr: nat > real > real > real ).
thf(sy_c_Float_Oround__up,type,
round_up: int > real > real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Rat__Orat,type,
formal7459771717576857490an_rat: rat > formal_Power_fps_rat ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Frequency__Moment__2_Os_092_060_094sub_0621,type,
frequency_Moment_s_1: rat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__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_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
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__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_Otimes__class_Otimes_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
times_4022348038934646771nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Rat__Orat_J,type,
zero_z5023345140362154345ps_rat: formal_Power_fps_rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_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__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_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
zero_z7294763051868718104at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Extended____Nonnegative____Real__Oennreal,type,
groups5253920722037313236nnreal: ( list_nat > extend8495563244428889912nnreal ) > set_list_nat > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
groups4393565826250045896at_int: ( list_nat > int ) > set_list_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups4396056296759096172at_nat: ( list_nat > nat ) > set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Real__Oreal,type,
groups8399112307953289288t_real: ( list_nat > real ) > set_list_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
groups4868793261593263428nnreal: ( nat > extend8495563244428889912nnreal ) > set_nat > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
groups4232809223866053280nnreal: ( real > extend8495563244428889912nnreal ) > set_real > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
groups178575644746855891at_int: ( set_Pr1261947904930325089at_nat > int ) > set_se7855581050983116737at_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
groups181066115255906167at_nat: ( set_Pr1261947904930325089at_nat > nat ) > set_se7855581050983116737at_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Real__Oreal,type,
groups5381467072955264723t_real: ( set_Pr1261947904930325089at_nat > real ) > set_se7855581050983116737at_nat > real ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set_001t__Extended____Nonnegative____Real__Oennreal,type,
groups2146370403815882837nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > $o ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set_001t__Int__Oint,type,
groups3245588053606888265et_int: ( int > int > int ) > int > $o ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set_001t__Nat__Onat,type,
groups3248078524115938541et_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set_001t__Real__Oreal,type,
groups5042480322358513993t_real: ( real > real > real ) > real > $o ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Extended____Nonnegative____Real__Oennreal,type,
groups2217173247284669407nnreal: list_E5688521862016077384nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Int__Oint,type,
groups4559388385066561235st_int: list_int > int ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Rat__Orat,type,
groups3926748795489115775st_rat: list_rat > rat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Real__Oreal,type,
groups6723090944982001619t_real: list_real > real ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
groups4697960971123825914at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_If_001t__Extended____Nonnegative____Real__Oennreal,type,
if_Ext9135588136721118450nnreal: $o > extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__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_List_Ocount__list_001t__Extended____Nonnegative____Real__Oennreal,type,
count_2060941923878149820nnreal: list_E5688521862016077384nnreal > extend8495563244428889912nnreal > nat ).
thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
count_list_int: list_int > int > nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
count_list_list_nat: list_list_nat > list_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
count_list_real: list_real > real > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
count_6440129622255701469at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Extended____Nonnegative____Real__Oennreal,type,
map_li5637531228398476638nnreal: ( list_nat > extend8495563244428889912nnreal ) > list_list_nat > list_E5688521862016077384nnreal ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
map_list_nat_int: ( list_nat > int ) > list_list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Real__Oreal,type,
map_list_nat_real: ( list_nat > real ) > list_list_nat > list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_li6003994582982014139at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_list_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
map_na4420896737966758094nnreal: ( nat > extend8495563244428889912nnreal ) > list_nat > list_E5688521862016077384nnreal ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
map_nat_real: ( nat > real ) > list_nat > list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
map_re3918317694826015018nnreal: ( real > extend8495563244428889912nnreal ) > list_real > list_E5688521862016077384nnreal ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Int__Oint,type,
map_real_int: ( real > int ) > list_real > list_int ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
map_real_nat: ( real > nat ) > list_real > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Extended____Nonnegative____Real__Oennreal,type,
map_se5413732074677250773nnreal: ( set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal ) > list_s1210847774152347623at_nat > list_E5688521862016077384nnreal ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
map_se4261650905360539081at_int: ( set_Pr1261947904930325089at_nat > int ) > list_s1210847774152347623at_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
map_se4264141375869589357at_nat: ( set_Pr1261947904930325089at_nat > nat ) > list_s1210847774152347623at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Real__Oreal,type,
map_se8952767809526791113t_real: ( set_Pr1261947904930325089at_nat > real ) > list_s1210847774152347623at_nat > list_real ).
thf(sy_c_List_Olist_Oset_001t__Extended____Nonnegative____Real__Oennreal,type,
set_Ex7800660098987911779nnreal: list_E5688521862016077384nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Rat__Orat,type,
set_rat2: list_rat > set_rat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_se5049602875457034614at_nat: list_s1210847774152347623at_nat > set_se7855581050983116737at_nat ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__Real__Oreal,type,
map_ta7765510035024898844t_real: ( list_nat > real ) > list_list_nat > list_real ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_ta8671482330076047857at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_list_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Omap__tailrec_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Extended____Nonnegative____Real__Oennreal,type,
map_ta4847886106086785803nnreal: ( set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal ) > list_s1210847774152347623at_nat > list_E5688521862016077384nnreal ).
thf(sy_c_List_Omap__tailrec_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
map_ta922926972790138367at_int: ( set_Pr1261947904930325089at_nat > int ) > list_s1210847774152347623at_nat > list_int ).
thf(sy_c_List_Omap__tailrec_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
map_ta925417443299188643at_nat: ( set_Pr1261947904930325089at_nat > nat ) > list_s1210847774152347623at_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Real__Oreal,type,
map_ta2204379005155838207t_real: ( set_Pr1261947904930325089at_nat > real ) > list_s1210847774152347623at_nat > list_real ).
thf(sy_c_List_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Orotate1_001t__Extended____Nonnegative____Real__Oennreal,type,
rotate2246681853897044133nnreal: list_E5688521862016077384nnreal > list_E5688521862016077384nnreal ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
rotate1_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Real__Oreal,type,
rotate1_real: list_real > list_real ).
thf(sy_c_List_Orotate1_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
rotate4238613965387346100at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Orotate_001t__Extended____Nonnegative____Real__Oennreal,type,
rotate856342757803299966nnreal: nat > list_E5688521862016077384nnreal > list_E5688521862016077384nnreal ).
thf(sy_c_List_Orotate_001t__Int__Oint,type,
rotate_int: nat > list_int > list_int ).
thf(sy_c_List_Orotate_001t__List__Olist_It__Nat__Onat_J,type,
rotate_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Real__Oreal,type,
rotate_real: nat > list_real > list_real ).
thf(sy_c_List_Orotate_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
rotate1033626827900196251at_nat: nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nonnegative____Real__Oennreal,type,
semiri6283507881447550617nnreal: nat > extend8495563244428889912nnreal ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__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_Osize__class_Osize_001t__List__Olist_It__Extended____Nonnegative____Real__Oennreal_J,type,
size_s6804550004907508956nnreal: list_E5688521862016077384nnreal > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
size_s8736152011456118867at_nat: list_s1210847774152347623at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $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_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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_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__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $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__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_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_n4168557817388953207nnreal: $o > extend8495563244428889912nnreal ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
zero_n3304061248610475627l_real: $o > real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
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_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_v__092_060delta_062,type,
delta: rat ).
thf(sy_v_exp__h__prod____,type,
exp_h_prod: list_nat > real ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_xs____,type,
xs: list_nat ).
% Relevant facts (1062)
thf(fact_0__092_060open_062_I_092_060Sum_062x_092_060leftarrow_062equiv__rels_An_O_Aof__bool_A_Ikernel__of_Axs_A_061_Ax_J_J_A_061_A_I1_058_058_063_Ha_J_092_060close_062,axiom,
( ( groups4561878855575611511st_nat
@ ( map_se4264141375869589357at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n2687167440665602831ol_nat
@ ( ( equiva2048684438135499664of_nat @ xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ na ) ) )
= one_one_nat ) ).
% \<open>(\<Sum>x\<leftarrow>equiv_rels n. of_bool (kernel_of xs = x)) = (1::?'a)\<close>
thf(fact_1__092_060open_062_I_092_060Sum_062x_092_060leftarrow_062equiv__rels_An_O_Aof__bool_A_Ikernel__of_Axs_A_061_Ax_J_J_A_061_A_I1_058_058_063_Ha_J_092_060close_062,axiom,
( ( groups4559388385066561235st_int
@ ( map_se4261650905360539081at_int
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n2684676970156552555ol_int
@ ( ( equiva2048684438135499664of_nat @ xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ na ) ) )
= one_one_int ) ).
% \<open>(\<Sum>x\<leftarrow>equiv_rels n. of_bool (kernel_of xs = x)) = (1::?'a)\<close>
thf(fact_2__092_060open_062_I_092_060Sum_062x_092_060leftarrow_062equiv__rels_An_O_Aof__bool_A_Ikernel__of_Axs_A_061_Ax_J_J_A_061_A_I1_058_058_063_Ha_J_092_060close_062,axiom,
( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n4168557817388953207nnreal
@ ( ( equiva2048684438135499664of_nat @ xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ na ) ) )
= one_on2969667320475766781nnreal ) ).
% \<open>(\<Sum>x\<leftarrow>equiv_rels n. of_bool (kernel_of xs = x)) = (1::?'a)\<close>
thf(fact_3__092_060open_062_I_092_060Sum_062x_092_060leftarrow_062equiv__rels_An_O_Aof__bool_A_Ikernel__of_Axs_A_061_Ax_J_J_A_061_A_I1_058_058_063_Ha_J_092_060close_062,axiom,
( ( groups6723090944982001619t_real
@ ( map_se8952767809526791113t_real
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n3304061248610475627l_real
@ ( ( equiva2048684438135499664of_nat @ xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ na ) ) )
= one_one_real ) ).
% \<open>(\<Sum>x\<leftarrow>equiv_rels n. of_bool (kernel_of xs = x)) = (1::?'a)\<close>
thf(fact_4_a,axiom,
( ( size_size_list_nat @ xs )
= na ) ).
% a
thf(fact_5_exp__h__prod__cong,axiom,
! [X2: list_nat,P: list_nat] :
( ( times_times_real
@ ( zero_n3304061248610475627l_real
@ ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ P ) ) )
@ ( exp_h_prod @ P ) )
= ( times_times_real
@ ( zero_n3304061248610475627l_real
@ ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ P ) ) )
@ ( exp_h_prod @ X2 ) ) ) ).
% exp_h_prod_cong
thf(fact_6_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_7_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_8_of__bool__eq_I2_J,axiom,
( ( zero_n4168557817388953207nnreal @ $true )
= one_on2969667320475766781nnreal ) ).
% of_bool_eq(2)
thf(fact_9_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_10_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= one_one_nat )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_11_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= one_one_int )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_12_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n4168557817388953207nnreal @ P2 )
= one_on2969667320475766781nnreal )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_13_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n3304061248610475627l_real @ P2 )
= one_one_real )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_14_more__arith__simps_I6_J,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% more_arith_simps(6)
thf(fact_15_more__arith__simps_I6_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% more_arith_simps(6)
thf(fact_16_more__arith__simps_I6_J,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% more_arith_simps(6)
thf(fact_17_more__arith__simps_I6_J,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
= A ) ).
% more_arith_simps(6)
thf(fact_18_more__arith__simps_I5_J,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_19_more__arith__simps_I5_J,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_20_more__arith__simps_I5_J,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_21_more__arith__simps_I5_J,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_22_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_23__092_060open_062_092_060And_062y_Ax_O_Akernel__of_Ax_A_061_Akernel__of_Ay_A_092_060Longrightarrow_062_Aexp__h__prod_Ax_A_061_Aexp__h__prod_Ay_092_060close_062,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( exp_h_prod @ X2 )
= ( exp_h_prod @ Y ) ) ) ).
% \<open>\<And>y x. kernel_of x = kernel_of y \<Longrightarrow> exp_h_prod x = exp_h_prod y\<close>
thf(fact_24_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_25_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_26_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_27_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_28_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_29_mult_Oleft__commute,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ B @ ( times_1893300245718287421nnreal @ A @ C ) )
= ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_30_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_31_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_32_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_33_mult_Ocommute,axiom,
( times_1893300245718287421nnreal
= ( ^ [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_34_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_35_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_36_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_37_mult_Oassoc,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
= ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_38_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_39_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_40_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_41_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
= ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_42_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_43_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_44_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_45_one__reorient,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( one_on2969667320475766781nnreal = X2 )
= ( X2 = one_on2969667320475766781nnreal ) ) ).
% one_reorient
thf(fact_46_of__bool__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= ( zero_n3304061248610475627l_real @ Q ) )
= ( P = Q ) ) ).
% of_bool_eq_iff
thf(fact_47_of__bool__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ( zero_n4168557817388953207nnreal @ P )
= ( zero_n4168557817388953207nnreal @ Q ) )
= ( P = Q ) ) ).
% of_bool_eq_iff
thf(fact_48_of__bool__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= ( zero_n2684676970156552555ol_int @ Q ) )
= ( P = Q ) ) ).
% of_bool_eq_iff
thf(fact_49_of__bool__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P = Q ) ) ).
% of_bool_eq_iff
thf(fact_50_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_51_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_52_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_53_mult_Ocomm__neutral,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
= A ) ).
% mult.comm_neutral
thf(fact_54_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_55_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_56_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_57_comm__monoid__mult__class_Omult__1,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_58_of__bool__conj,axiom,
! [P2: $o,Q2: $o] :
( ( zero_n3304061248610475627l_real
@ ( P2
& Q2 ) )
= ( times_times_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_59_of__bool__conj,axiom,
! [P2: $o,Q2: $o] :
( ( zero_n4168557817388953207nnreal
@ ( P2
& Q2 ) )
= ( times_1893300245718287421nnreal @ ( zero_n4168557817388953207nnreal @ P2 ) @ ( zero_n4168557817388953207nnreal @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_60_of__bool__conj,axiom,
! [P2: $o,Q2: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P2
& Q2 ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_61_of__bool__conj,axiom,
! [P2: $o,Q2: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P2
& Q2 ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_62_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_63_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_64_lambda__one,axiom,
( ( ^ [X: int] : X )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_65_lambda__one,axiom,
( ( ^ [X: extend8495563244428889912nnreal] : X )
= ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal ) ) ).
% lambda_one
thf(fact_66_equiv__rels__2,axiom,
! [N: nat,Xs: list_nat] :
( ( N
= ( size_size_list_nat @ Xs ) )
=> ( ( groups6723090944982001619t_real
@ ( map_se8952767809526791113t_real
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n3304061248610475627l_real
@ ( ( equiva2048684438135499664of_nat @ Xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ N ) ) )
= one_one_real ) ) ).
% equiv_rels_2
thf(fact_67_equiv__rels__2,axiom,
! [N: nat,Xs: list_nat] :
( ( N
= ( size_size_list_nat @ Xs ) )
=> ( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n4168557817388953207nnreal
@ ( ( equiva2048684438135499664of_nat @ Xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ N ) ) )
= one_on2969667320475766781nnreal ) ) ).
% equiv_rels_2
thf(fact_68_equiv__rels__2,axiom,
! [N: nat,Xs: list_nat] :
( ( N
= ( size_size_list_nat @ Xs ) )
=> ( ( groups4559388385066561235st_int
@ ( map_se4261650905360539081at_int
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n2684676970156552555ol_int
@ ( ( equiva2048684438135499664of_nat @ Xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ N ) ) )
= one_one_int ) ) ).
% equiv_rels_2
thf(fact_69_equiv__rels__2,axiom,
! [N: nat,Xs: list_nat] :
( ( N
= ( size_size_list_nat @ Xs ) )
=> ( ( groups4561878855575611511st_nat
@ ( map_se4264141375869589357at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] :
( zero_n2687167440665602831ol_nat
@ ( ( equiva2048684438135499664of_nat @ Xs )
= X ) )
@ ( equiva8721718519204927301v_rels @ N ) ) )
= one_one_nat ) ) ).
% equiv_rels_2
thf(fact_70_length__map,axiom,
! [F: list_nat > real,Xs: list_list_nat] :
( ( size_size_list_real @ ( map_list_nat_real @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_71_length__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( size_s8736152011456118867at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_72_length__map,axiom,
! [F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat] :
( ( size_size_list_real @ ( map_se8952767809526791113t_real @ F @ Xs ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_map
thf(fact_73_length__map,axiom,
! [F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat] :
( ( size_s6804550004907508956nnreal @ ( map_se5413732074677250773nnreal @ F @ Xs ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_map
thf(fact_74_length__map,axiom,
! [F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat] :
( ( size_size_list_int @ ( map_se4261650905360539081at_int @ F @ Xs ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_map
thf(fact_75_length__map,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat] :
( ( size_size_list_nat @ ( map_se4264141375869589357at_nat @ F @ Xs ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_map
thf(fact_76_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_77_sum__list__mult__const,axiom,
! [F: list_nat > real,C: real,Xs: list_list_nat] :
( ( groups6723090944982001619t_real
@ ( map_list_nat_real
@ ^ [X: list_nat] : ( times_times_real @ ( F @ X ) @ C )
@ Xs ) )
= ( times_times_real @ ( groups6723090944982001619t_real @ ( map_list_nat_real @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_78_sum__list__mult__const,axiom,
! [F: set_Pr1261947904930325089at_nat > real,C: real,Xs: list_s1210847774152347623at_nat] :
( ( groups6723090944982001619t_real
@ ( map_se8952767809526791113t_real
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_real @ ( F @ X ) @ C )
@ Xs ) )
= ( times_times_real @ ( groups6723090944982001619t_real @ ( map_se8952767809526791113t_real @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_79_sum__list__mult__const,axiom,
! [F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat] :
( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_1893300245718287421nnreal @ ( F @ X ) @ C )
@ Xs ) )
= ( times_1893300245718287421nnreal @ ( groups2217173247284669407nnreal @ ( map_se5413732074677250773nnreal @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_80_sum__list__mult__const,axiom,
! [F: set_Pr1261947904930325089at_nat > int,C: int,Xs: list_s1210847774152347623at_nat] :
( ( groups4559388385066561235st_int
@ ( map_se4261650905360539081at_int
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_int @ ( F @ X ) @ C )
@ Xs ) )
= ( times_times_int @ ( groups4559388385066561235st_int @ ( map_se4261650905360539081at_int @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_81_sum__list__mult__const,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,C: nat,Xs: list_s1210847774152347623at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_se4264141375869589357at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_nat @ ( F @ X ) @ C )
@ Xs ) )
= ( times_times_nat @ ( groups4561878855575611511st_nat @ ( map_se4264141375869589357at_nat @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_82_sum__list__const__mult,axiom,
! [C: real,F: list_nat > real,Xs: list_list_nat] :
( ( groups6723090944982001619t_real
@ ( map_list_nat_real
@ ^ [X: list_nat] : ( times_times_real @ C @ ( F @ X ) )
@ Xs ) )
= ( times_times_real @ C @ ( groups6723090944982001619t_real @ ( map_list_nat_real @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_83_sum__list__const__mult,axiom,
! [C: real,F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat] :
( ( groups6723090944982001619t_real
@ ( map_se8952767809526791113t_real
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_real @ C @ ( F @ X ) )
@ Xs ) )
= ( times_times_real @ C @ ( groups6723090944982001619t_real @ ( map_se8952767809526791113t_real @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_84_sum__list__const__mult,axiom,
! [C: extend8495563244428889912nnreal,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat] :
( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_1893300245718287421nnreal @ C @ ( F @ X ) )
@ Xs ) )
= ( times_1893300245718287421nnreal @ C @ ( groups2217173247284669407nnreal @ ( map_se5413732074677250773nnreal @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_85_sum__list__const__mult,axiom,
! [C: int,F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat] :
( ( groups4559388385066561235st_int
@ ( map_se4261650905360539081at_int
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_int @ C @ ( F @ X ) )
@ Xs ) )
= ( times_times_int @ C @ ( groups4559388385066561235st_int @ ( map_se4261650905360539081at_int @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_86_sum__list__const__mult,axiom,
! [C: nat,F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_se4264141375869589357at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_nat @ C @ ( F @ X ) )
@ Xs ) )
= ( times_times_nat @ C @ ( groups4561878855575611511st_nat @ ( map_se4264141375869589357at_nat @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_87_mem__Collect__eq,axiom,
! [A: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_88_mem__Collect__eq,axiom,
! [A: real,P2: real > $o] :
( ( member_real @ A @ ( collect_real @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A3: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_95_Collect__cong,axiom,
! [P2: int > $o,Q2: int > $o] :
( ! [X3: int] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_int @ P2 )
= ( collect_int @ Q2 ) ) ) ).
% Collect_cong
thf(fact_96_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_97_set__times__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_98_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_99_set__times__intro,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_100_set__times__intro,axiom,
! [A: extend8495563244428889912nnreal,C2: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,D: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ C2 )
=> ( ( member7908768830364227535nnreal @ B @ D )
=> ( member7908768830364227535nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ ( times_4022348038934646771nnreal @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_101_kernel__of__eq__len,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_102_map__eq__imp__length__eq,axiom,
! [F: list_nat > real,Xs: list_list_nat,G: nat > real,Ys: list_nat] :
( ( ( map_list_nat_real @ F @ Xs )
= ( map_nat_real @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_103_map__eq__imp__length__eq,axiom,
! [F: nat > real,Xs: list_nat,G: list_nat > real,Ys: list_list_nat] :
( ( ( map_nat_real @ F @ Xs )
= ( map_list_nat_real @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_104_map__eq__imp__length__eq,axiom,
! [F: list_nat > real,Xs: list_list_nat,G: list_nat > real,Ys: list_list_nat] :
( ( ( map_list_nat_real @ F @ Xs )
= ( map_list_nat_real @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_105_map__eq__imp__length__eq,axiom,
! [F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat,G: nat > real,Ys: list_nat] :
( ( ( map_se8952767809526791113t_real @ F @ Xs )
= ( map_nat_real @ G @ Ys ) )
=> ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_106_map__eq__imp__length__eq,axiom,
! [F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat,G: nat > extend8495563244428889912nnreal,Ys: list_nat] :
( ( ( map_se5413732074677250773nnreal @ F @ Xs )
= ( map_na4420896737966758094nnreal @ G @ Ys ) )
=> ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_107_map__eq__imp__length__eq,axiom,
! [F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat,G: nat > int,Ys: list_nat] :
( ( ( map_se4261650905360539081at_int @ F @ Xs )
= ( map_nat_int @ G @ Ys ) )
=> ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_108_map__eq__imp__length__eq,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_se4264141375869589357at_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_109_map__eq__imp__length__eq,axiom,
! [F: nat > real,Xs: list_nat,G: set_Pr1261947904930325089at_nat > real,Ys: list_s1210847774152347623at_nat] :
( ( ( map_nat_real @ F @ Xs )
= ( map_se8952767809526791113t_real @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_110_map__eq__imp__length__eq,axiom,
! [F: nat > extend8495563244428889912nnreal,Xs: list_nat,G: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Ys: list_s1210847774152347623at_nat] :
( ( ( map_na4420896737966758094nnreal @ F @ Xs )
= ( map_se5413732074677250773nnreal @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_111_map__eq__imp__length__eq,axiom,
! [F: nat > int,Xs: list_nat,G: set_Pr1261947904930325089at_nat > int,Ys: list_s1210847774152347623at_nat] :
( ( ( map_nat_int @ F @ Xs )
= ( map_se4261650905360539081at_int @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_112_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X: real] : ( times_times_real @ X @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_113_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_114_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X: int] : ( times_times_int @ X @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_115_mult__commute__abs,axiom,
! [C: extend8495563244428889912nnreal] :
( ( ^ [X: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ X @ C ) )
= ( times_1893300245718287421nnreal @ C ) ) ).
% mult_commute_abs
thf(fact_116_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_117_of__bool__less__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q2 ) )
= ( P2
=> Q2 ) ) ).
% of_bool_less_eq_iff
thf(fact_118_of__bool__less__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_le3935885782089961368nnreal @ ( zero_n4168557817388953207nnreal @ P2 ) @ ( zero_n4168557817388953207nnreal @ Q2 ) )
= ( P2
=> Q2 ) ) ).
% of_bool_less_eq_iff
thf(fact_119_of__bool__less__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) )
= ( P2
=> Q2 ) ) ).
% of_bool_less_eq_iff
thf(fact_120_of__bool__less__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) )
= ( P2
=> Q2 ) ) ).
% of_bool_less_eq_iff
thf(fact_121_verit__eq__simplify_I6_J,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_122_verit__eq__simplify_I6_J,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_123_verit__eq__simplify_I6_J,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_124_verit__eq__simplify_I6_J,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X2 = Y )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_125_verit__comp__simplify_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_126_verit__comp__simplify_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_127_verit__comp__simplify_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_128_verit__comp__simplify_I2_J,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_129_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_130_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_131_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_132_verit__la__disequality,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A = B )
| ~ ( ord_le3935885782089961368nnreal @ A @ B )
| ~ ( ord_le3935885782089961368nnreal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_133_rel__simps_I47_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% rel_simps(47)
thf(fact_134_rel__simps_I47_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% rel_simps(47)
thf(fact_135_rel__simps_I47_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% rel_simps(47)
thf(fact_136_rel__simps_I47_J,axiom,
ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ).
% rel_simps(47)
thf(fact_137_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real ) ).
% of_bool_less_eq_one
thf(fact_138_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_139_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_140_equiv__rels__def,axiom,
( equiva8721718519204927301v_rels
= ( ^ [N2: nat] : ( map_li6003994582982014139at_nat @ equiva2048684438135499664of_nat @ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) ) ).
% equiv_rels_def
thf(fact_141_set__times__elim,axiom,
! [X2: real,A3: set_real,B3: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A3 @ B3 ) )
=> ~ ! [A4: real,B4: real] :
( ( X2
= ( times_times_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A3 )
=> ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_142_set__times__elim,axiom,
! [X2: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A3 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A3 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_143_set__times__elim,axiom,
! [X2: int,A3: set_int,B3: set_int] :
( ( member_int @ X2 @ ( times_times_set_int @ A3 @ B3 ) )
=> ~ ! [A4: int,B4: int] :
( ( X2
= ( times_times_int @ A4 @ B4 ) )
=> ( ( member_int @ A4 @ A3 )
=> ~ ( member_int @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_144_set__times__elim,axiom,
! [X2: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ X2 @ ( times_4022348038934646771nnreal @ A3 @ B3 ) )
=> ~ ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( X2
= ( times_1893300245718287421nnreal @ A4 @ B4 ) )
=> ( ( member7908768830364227535nnreal @ A4 @ A3 )
=> ~ ( member7908768830364227535nnreal @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_145_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_146_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_147_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_148_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_149_dual__order_Orefl,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).
% dual_order.refl
thf(fact_150_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_151_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_152_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_153_order__refl,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ X2 ) ).
% order_refl
thf(fact_154_map__eq__map__tailrec,axiom,
map_list_nat_real = map_ta7765510035024898844t_real ).
% map_eq_map_tailrec
thf(fact_155_map__eq__map__tailrec,axiom,
map_li6003994582982014139at_nat = map_ta8671482330076047857at_nat ).
% map_eq_map_tailrec
thf(fact_156_map__eq__map__tailrec,axiom,
map_se8952767809526791113t_real = map_ta2204379005155838207t_real ).
% map_eq_map_tailrec
thf(fact_157_map__eq__map__tailrec,axiom,
map_se5413732074677250773nnreal = map_ta4847886106086785803nnreal ).
% map_eq_map_tailrec
thf(fact_158_map__eq__map__tailrec,axiom,
map_se4261650905360539081at_int = map_ta922926972790138367at_int ).
% map_eq_map_tailrec
thf(fact_159_map__eq__map__tailrec,axiom,
map_se4264141375869589357at_nat = map_ta925417443299188643at_nat ).
% map_eq_map_tailrec
thf(fact_160_sum__list__triv,axiom,
! [R: real,Xs: list_list_nat] :
( ( groups6723090944982001619t_real
@ ( map_list_nat_real
@ ^ [X: list_nat] : R
@ Xs ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s3023201423986296836st_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_161_sum__list__triv,axiom,
! [R: real,Xs: list_s1210847774152347623at_nat] :
( ( groups6723090944982001619t_real
@ ( map_se8952767809526791113t_real
@ ^ [X: set_Pr1261947904930325089at_nat] : R
@ Xs ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s8736152011456118867at_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_162_sum__list__triv,axiom,
! [R: real,Xs: list_nat] :
( ( groups6723090944982001619t_real
@ ( map_nat_real
@ ^ [X: nat] : R
@ Xs ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_163_sum__list__triv,axiom,
! [R: extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat] :
( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] : R
@ Xs ) )
= ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ ( size_s8736152011456118867at_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_164_sum__list__triv,axiom,
! [R: extend8495563244428889912nnreal,Xs: list_nat] :
( ( groups2217173247284669407nnreal
@ ( map_na4420896737966758094nnreal
@ ^ [X: nat] : R
@ Xs ) )
= ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ ( size_size_list_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_165_sum__list__triv,axiom,
! [R: int,Xs: list_s1210847774152347623at_nat] :
( ( groups4559388385066561235st_int
@ ( map_se4261650905360539081at_int
@ ^ [X: set_Pr1261947904930325089at_nat] : R
@ Xs ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( size_s8736152011456118867at_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_166_sum__list__triv,axiom,
! [R: int,Xs: list_nat] :
( ( groups4559388385066561235st_int
@ ( map_nat_int
@ ^ [X: nat] : R
@ Xs ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( size_size_list_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_167_sum__list__triv,axiom,
! [R: nat,Xs: list_s1210847774152347623at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_se4264141375869589357at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] : R
@ Xs ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( size_s8736152011456118867at_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_168_sum__list__triv,axiom,
! [R: nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : R
@ Xs ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( size_size_list_nat @ Xs ) ) @ R ) ) ).
% sum_list_triv
thf(fact_169_kernel__of__inj__on__rgfs__aux,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y ) )
=> ( ( equiva3371634703666331078on_rgf @ X2 )
=> ( ( equiva3371634703666331078on_rgf @ Y )
=> ( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( X2 = Y ) ) ) ) ) ).
% kernel_of_inj_on_rgfs_aux
thf(fact_170_prod_Ocomm__monoid__list__set__axioms,axiom,
groups5042480322358513993t_real @ times_times_real @ one_one_real ).
% prod.comm_monoid_list_set_axioms
thf(fact_171_prod_Ocomm__monoid__list__set__axioms,axiom,
groups3248078524115938541et_nat @ times_times_nat @ one_one_nat ).
% prod.comm_monoid_list_set_axioms
thf(fact_172_prod_Ocomm__monoid__list__set__axioms,axiom,
groups3245588053606888265et_int @ times_times_int @ one_one_int ).
% prod.comm_monoid_list_set_axioms
thf(fact_173_prod_Ocomm__monoid__list__set__axioms,axiom,
groups2146370403815882837nnreal @ times_1893300245718287421nnreal @ one_on2969667320475766781nnreal ).
% prod.comm_monoid_list_set_axioms
thf(fact_174_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_175_sum__list__0,axiom,
! [Xs: list_list_nat] :
( ( groups4697960971123825914at_nat
@ ( map_li6003994582982014139at_nat
@ ^ [X: list_nat] : zero_z7294763051868718104at_nat
@ Xs ) )
= zero_z7294763051868718104at_nat ) ).
% sum_list_0
thf(fact_176_sum__list__0,axiom,
! [Xs: list_list_nat] :
( ( groups6723090944982001619t_real
@ ( map_list_nat_real
@ ^ [X: list_nat] : zero_zero_real
@ Xs ) )
= zero_zero_real ) ).
% sum_list_0
thf(fact_177_sum__list__0,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( groups6723090944982001619t_real
@ ( map_se8952767809526791113t_real
@ ^ [X: set_Pr1261947904930325089at_nat] : zero_zero_real
@ Xs ) )
= zero_zero_real ) ).
% sum_list_0
thf(fact_178_sum__list__0,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] : zero_z7100319975126383169nnreal
@ Xs ) )
= zero_z7100319975126383169nnreal ) ).
% sum_list_0
thf(fact_179_sum__list__0,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( groups4559388385066561235st_int
@ ( map_se4261650905360539081at_int
@ ^ [X: set_Pr1261947904930325089at_nat] : zero_zero_int
@ Xs ) )
= zero_zero_int ) ).
% sum_list_0
thf(fact_180_sum__list__0,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_se4264141375869589357at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] : zero_zero_nat
@ Xs ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_181_sum__list__mono,axiom,
! [Xs: list_real,F: real > real,G: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( map_real_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_real_real @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_182_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > real,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( map_nat_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_nat_real @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_183_sum__list__mono,axiom,
! [Xs: list_real,F: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2217173247284669407nnreal @ ( map_re3918317694826015018nnreal @ F @ Xs ) ) @ ( groups2217173247284669407nnreal @ ( map_re3918317694826015018nnreal @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_184_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2217173247284669407nnreal @ ( map_na4420896737966758094nnreal @ F @ Xs ) ) @ ( groups2217173247284669407nnreal @ ( map_na4420896737966758094nnreal @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_185_sum__list__mono,axiom,
! [Xs: list_real,F: real > int,G: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ ( map_real_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_real_int @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_186_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > int,G: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ ( map_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_nat_int @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_187_sum__list__mono,axiom,
! [Xs: list_real,F: real > nat,G: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_real_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_real_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_188_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_189_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > real,G: list_nat > real] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( map_list_nat_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_list_nat_real @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_190_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > extend8495563244428889912nnreal,G: list_nat > extend8495563244428889912nnreal] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2217173247284669407nnreal @ ( map_li5637531228398476638nnreal @ F @ Xs ) ) @ ( groups2217173247284669407nnreal @ ( map_li5637531228398476638nnreal @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_191_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_192_le__zero__eq,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N @ zero_z7100319975126383169nnreal )
= ( N = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_193_semiring__norm_I63_J,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% semiring_norm(63)
thf(fact_194_semiring__norm_I63_J,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% semiring_norm(63)
thf(fact_195_semiring__norm_I63_J,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% semiring_norm(63)
thf(fact_196_semiring__norm_I63_J,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% semiring_norm(63)
thf(fact_197_semiring__norm_I63_J,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% semiring_norm(63)
thf(fact_198_semiring__norm_I64_J,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% semiring_norm(64)
thf(fact_199_semiring__norm_I64_J,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% semiring_norm(64)
thf(fact_200_semiring__norm_I64_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% semiring_norm(64)
thf(fact_201_semiring__norm_I64_J,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% semiring_norm(64)
thf(fact_202_semiring__norm_I64_J,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ zero_z7100319975126383169nnreal )
= zero_z7100319975126383169nnreal ) ).
% semiring_norm(64)
thf(fact_203_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_204_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_205_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_206_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_207_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_208_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_209_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_210_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_211_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: rat,X2: rat,B: rat] :
( ( ( times_times_rat @ A @ X2 )
= ( times_times_rat @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_212_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_213_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: rat,X2: rat,Y: rat] :
( ( ( times_times_rat @ A @ X2 )
= ( times_times_rat @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_214_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_215_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_216_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_217_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: rat] :
( ( times_times_rat @ zero_zero_rat @ X2 )
= zero_zero_rat ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_218_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_219_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: rat,X2: rat] :
( ( ( times_times_rat @ A @ X2 )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( X2 = zero_zero_rat ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_220_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_221_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_222_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_223_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_224_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_225_mult__eq__0__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
= ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% mult_eq_0_iff
thf(fact_226_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% Num.of_nat_simps(1)
thf(fact_227_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% Num.of_nat_simps(1)
thf(fact_228_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% Num.of_nat_simps(1)
thf(fact_229_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% Num.of_nat_simps(1)
thf(fact_230_Num_Oof__nat__simps_I1_J,axiom,
( ( semiri6283507881447550617nnreal @ zero_zero_nat )
= zero_z7100319975126383169nnreal ) ).
% Num.of_nat_simps(1)
thf(fact_231_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_232_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_233_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_234_Num_Oof__nat__simps_I5_J,axiom,
! [M: nat,N: nat] :
( ( semiri6283507881447550617nnreal @ ( times_times_nat @ M @ N ) )
= ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) ) ) ).
% Num.of_nat_simps(5)
thf(fact_235_map__eq__conv,axiom,
! [F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat,G: set_Pr1261947904930325089at_nat > real] :
( ( ( map_se8952767809526791113t_real @ F @ Xs )
= ( map_se8952767809526791113t_real @ G @ Xs ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_236_map__eq__conv,axiom,
! [F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat,G: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ( ( map_se5413732074677250773nnreal @ F @ Xs )
= ( map_se5413732074677250773nnreal @ G @ Xs ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_237_map__eq__conv,axiom,
! [F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat,G: set_Pr1261947904930325089at_nat > int] :
( ( ( map_se4261650905360539081at_int @ F @ Xs )
= ( map_se4261650905360539081at_int @ G @ Xs ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_238_map__eq__conv,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat,G: set_Pr1261947904930325089at_nat > nat] :
( ( ( map_se4264141375869589357at_nat @ F @ Xs )
= ( map_se4264141375869589357at_nat @ G @ Xs ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_239_map__eq__conv,axiom,
! [F: list_nat > real,Xs: list_list_nat,G: list_nat > real] :
( ( ( map_list_nat_real @ F @ Xs )
= ( map_list_nat_real @ G @ Xs ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_240_map__eq__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_241_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P2 )
= zero_zero_rat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_242_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n3304061248610475627l_real @ P2 )
= zero_zero_real )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_243_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n4168557817388953207nnreal @ P2 )
= zero_z7100319975126383169nnreal )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_244_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= zero_zero_int )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_245_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= zero_zero_nat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_246_of__bool__eq_I1_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $false )
= zero_zero_rat ) ).
% of_bool_eq(1)
thf(fact_247_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_248_of__bool__eq_I1_J,axiom,
( ( zero_n4168557817388953207nnreal @ $false )
= zero_z7100319975126383169nnreal ) ).
% of_bool_eq(1)
thf(fact_249_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_250_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_251_set__rotate1,axiom,
! [Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate1_list_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_252_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_253_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ( times_times_rat @ A @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_254_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_255_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_256_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_257_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_258_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_259_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ( times_times_rat @ C @ A )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_260_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_261_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_262_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_263_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_264_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_265_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% Num.of_nat_simps(2)
thf(fact_266_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% Num.of_nat_simps(2)
thf(fact_267_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% Num.of_nat_simps(2)
thf(fact_268_Num_Oof__nat__simps_I2_J,axiom,
( ( semiri6283507881447550617nnreal @ one_one_nat )
= one_on2969667320475766781nnreal ) ).
% Num.of_nat_simps(2)
thf(fact_269_sum__list__eq__0__iff,axiom,
! [Ns: list_E5688521862016077384nnreal] :
( ( ( groups2217173247284669407nnreal @ Ns )
= zero_z7100319975126383169nnreal )
= ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( set_Ex7800660098987911779nnreal @ Ns ) )
=> ( X = zero_z7100319975126383169nnreal ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_270_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ns ) )
=> ( X = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_271_subset__code_I1_J,axiom,
! [Xs: list_real,B3: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
= ( ! [X: real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( member_real @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_272_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_273_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B3 )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_274_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_275_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_276_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_277_zero__reorient,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X2 )
= ( X2 = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_278_zero__reorient,axiom,
! [X2: rat] :
( ( zero_zero_rat = X2 )
= ( X2 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_279_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_280_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_281_semiring__norm_I113_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% semiring_norm(113)
thf(fact_282_semiring__norm_I113_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% semiring_norm(113)
thf(fact_283_semiring__norm_I113_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% semiring_norm(113)
thf(fact_284_semiring__norm_I113_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% semiring_norm(113)
thf(fact_285_semiring__norm_I113_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% semiring_norm(113)
thf(fact_286_semiring__norm_I159_J,axiom,
zero_zero_nat != one_one_nat ).
% semiring_norm(159)
thf(fact_287_semiring__norm_I159_J,axiom,
zero_zero_int != one_one_int ).
% semiring_norm(159)
thf(fact_288_semiring__norm_I159_J,axiom,
zero_zero_real != one_one_real ).
% semiring_norm(159)
thf(fact_289_semiring__norm_I159_J,axiom,
zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).
% semiring_norm(159)
thf(fact_290_semiring__norm_I159_J,axiom,
zero_zero_rat != one_one_rat ).
% semiring_norm(159)
thf(fact_291_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_292_enum__rgfs__returns__rgfs,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( equiva3371634703666331078on_rgf @ X2 ) ) ).
% enum_rgfs_returns_rgfs
thf(fact_293_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ ( set_rat2 @ Xs ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ X3 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3926748795489115775st_rat @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_294_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ X3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6723090944982001619t_real @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_295_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_E5688521862016077384nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( set_Ex7800660098987911779nnreal @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X3 ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups2217173247284669407nnreal @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_296_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_eq_int @ zero_zero_int @ X3 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups4559388385066561235st_int @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_297_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_298_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ ( set_rat2 @ Xs ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ X3 ) )
=> ( ( ( groups3926748795489115775st_rat @ Xs )
= zero_zero_rat )
= ( ! [X: rat] :
( ( member_rat @ X @ ( set_rat2 @ Xs ) )
=> ( X = zero_zero_rat ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_299_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ X3 ) )
=> ( ( ( groups6723090944982001619t_real @ Xs )
= zero_zero_real )
= ( ! [X: real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( X = zero_zero_real ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_300_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_E5688521862016077384nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( set_Ex7800660098987911779nnreal @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X3 ) )
=> ( ( ( groups2217173247284669407nnreal @ Xs )
= zero_z7100319975126383169nnreal )
= ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( set_Ex7800660098987911779nnreal @ Xs ) )
=> ( X = zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_301_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_eq_int @ zero_zero_int @ X3 ) )
=> ( ( ( groups4559388385066561235st_int @ Xs )
= zero_zero_int )
= ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( X = zero_zero_int ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_302_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
=> ( ( ( groups4561878855575611511st_nat @ Xs )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( X = zero_zero_nat ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_303_sum__list__nonpos,axiom,
! [Xs: list_rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ ( set_rat2 @ Xs ) )
=> ( ord_less_eq_rat @ X3 @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3926748795489115775st_rat @ Xs ) @ zero_zero_rat ) ) ).
% sum_list_nonpos
thf(fact_304_sum__list__nonpos,axiom,
! [Xs: list_real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ X3 @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ Xs ) @ zero_zero_real ) ) ).
% sum_list_nonpos
thf(fact_305_sum__list__nonpos,axiom,
! [Xs: list_E5688521862016077384nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( set_Ex7800660098987911779nnreal @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ X3 @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2217173247284669407nnreal @ Xs ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_list_nonpos
thf(fact_306_sum__list__nonpos,axiom,
! [Xs: list_int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_eq_int @ X3 @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ Xs ) @ zero_zero_int ) ) ).
% sum_list_nonpos
thf(fact_307_sum__list__nonpos,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X3 @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ zero_zero_nat ) ) ).
% sum_list_nonpos
thf(fact_308_ex__map__conv,axiom,
! [Ys: list_real,F: list_nat > real] :
( ( ? [Xs3: list_list_nat] :
( Ys
= ( map_list_nat_real @ F @ Xs3 ) ) )
= ( ! [X: real] :
( ( member_real @ X @ ( set_real2 @ Ys ) )
=> ? [Y2: list_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_309_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs3: list_list_nat] :
( Ys
= ( map_li6003994582982014139at_nat @ F @ Xs3 ) ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y2: list_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_310_ex__map__conv,axiom,
! [Ys: list_real,F: set_Pr1261947904930325089at_nat > real] :
( ( ? [Xs3: list_s1210847774152347623at_nat] :
( Ys
= ( map_se8952767809526791113t_real @ F @ Xs3 ) ) )
= ( ! [X: real] :
( ( member_real @ X @ ( set_real2 @ Ys ) )
=> ? [Y2: set_Pr1261947904930325089at_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_311_ex__map__conv,axiom,
! [Ys: list_E5688521862016077384nnreal,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ( ? [Xs3: list_s1210847774152347623at_nat] :
( Ys
= ( map_se5413732074677250773nnreal @ F @ Xs3 ) ) )
= ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( set_Ex7800660098987911779nnreal @ Ys ) )
=> ? [Y2: set_Pr1261947904930325089at_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_312_ex__map__conv,axiom,
! [Ys: list_int,F: set_Pr1261947904930325089at_nat > int] :
( ( ? [Xs3: list_s1210847774152347623at_nat] :
( Ys
= ( map_se4261650905360539081at_int @ F @ Xs3 ) ) )
= ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Ys ) )
=> ? [Y2: set_Pr1261947904930325089at_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_313_ex__map__conv,axiom,
! [Ys: list_nat,F: set_Pr1261947904930325089at_nat > nat] :
( ( ? [Xs3: list_s1210847774152347623at_nat] :
( Ys
= ( map_se4264141375869589357at_nat @ F @ Xs3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y2: set_Pr1261947904930325089at_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_314_map__cong,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > real,G: set_Pr1261947904930325089at_nat > real] :
( ( Xs = Ys )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se8952767809526791113t_real @ F @ Xs )
= ( map_se8952767809526791113t_real @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_315_map__cong,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,G: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ( Xs = Ys )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se5413732074677250773nnreal @ F @ Xs )
= ( map_se5413732074677250773nnreal @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_316_map__cong,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > int,G: set_Pr1261947904930325089at_nat > int] :
( ( Xs = Ys )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se4261650905360539081at_int @ F @ Xs )
= ( map_se4261650905360539081at_int @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_317_map__cong,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se4264141375869589357at_nat @ F @ Xs )
= ( map_se4264141375869589357at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_318_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > real,G: list_nat > real] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_list_nat_real @ F @ Xs )
= ( map_list_nat_real @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_319_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_320_map__idI,axiom,
! [Xs: list_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_real_real @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_321_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_322_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_323_map__ext,axiom,
! [Xs: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > real,G: set_Pr1261947904930325089at_nat > real] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se8952767809526791113t_real @ F @ Xs )
= ( map_se8952767809526791113t_real @ G @ Xs ) ) ) ).
% map_ext
thf(fact_324_map__ext,axiom,
! [Xs: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,G: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se5413732074677250773nnreal @ F @ Xs )
= ( map_se5413732074677250773nnreal @ G @ Xs ) ) ) ).
% map_ext
thf(fact_325_map__ext,axiom,
! [Xs: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > int,G: set_Pr1261947904930325089at_nat > int] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se4261650905360539081at_int @ F @ Xs )
= ( map_se4261650905360539081at_int @ G @ Xs ) ) ) ).
% map_ext
thf(fact_326_map__ext,axiom,
! [Xs: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se4264141375869589357at_nat @ F @ Xs )
= ( map_se4264141375869589357at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_327_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > real,G: list_nat > real] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_list_nat_real @ F @ Xs )
= ( map_list_nat_real @ G @ Xs ) ) ) ).
% map_ext
thf(fact_328_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_329_list_Omap__ident__strong,axiom,
! [T: list_real,F: real > real] :
( ! [Z: real] :
( ( member_real @ Z @ ( set_real2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_real_real @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_330_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_331_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_332_list_Oinj__map__strong,axiom,
! [X2: list_s1210847774152347623at_nat,Xa: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > real,Fa: set_Pr1261947904930325089at_nat > real] :
( ! [Z: set_Pr1261947904930325089at_nat,Za: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( member2643936169264416010at_nat @ Za @ ( set_se5049602875457034614at_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_se8952767809526791113t_real @ F @ X2 )
= ( map_se8952767809526791113t_real @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_333_list_Oinj__map__strong,axiom,
! [X2: list_s1210847774152347623at_nat,Xa: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Fa: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ! [Z: set_Pr1261947904930325089at_nat,Za: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( member2643936169264416010at_nat @ Za @ ( set_se5049602875457034614at_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_se5413732074677250773nnreal @ F @ X2 )
= ( map_se5413732074677250773nnreal @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_334_list_Oinj__map__strong,axiom,
! [X2: list_s1210847774152347623at_nat,Xa: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > int,Fa: set_Pr1261947904930325089at_nat > int] :
( ! [Z: set_Pr1261947904930325089at_nat,Za: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( member2643936169264416010at_nat @ Za @ ( set_se5049602875457034614at_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_se4261650905360539081at_int @ F @ X2 )
= ( map_se4261650905360539081at_int @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_335_list_Oinj__map__strong,axiom,
! [X2: list_s1210847774152347623at_nat,Xa: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > nat,Fa: set_Pr1261947904930325089at_nat > nat] :
( ! [Z: set_Pr1261947904930325089at_nat,Za: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( member2643936169264416010at_nat @ Za @ ( set_se5049602875457034614at_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_se4264141375869589357at_nat @ F @ X2 )
= ( map_se4264141375869589357at_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_336_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > real,Fa: list_nat > real] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_list_nat_real @ F @ X2 )
= ( map_list_nat_real @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_337_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,Fa: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_338_list_Omap__cong0,axiom,
! [X2: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > real,G: set_Pr1261947904930325089at_nat > real] :
( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se8952767809526791113t_real @ F @ X2 )
= ( map_se8952767809526791113t_real @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_339_list_Omap__cong0,axiom,
! [X2: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,G: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se5413732074677250773nnreal @ F @ X2 )
= ( map_se5413732074677250773nnreal @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_340_list_Omap__cong0,axiom,
! [X2: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > int,G: set_Pr1261947904930325089at_nat > int] :
( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se4261650905360539081at_int @ F @ X2 )
= ( map_se4261650905360539081at_int @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_341_list_Omap__cong0,axiom,
! [X2: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat] :
( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se4264141375869589357at_nat @ F @ X2 )
= ( map_se4264141375869589357at_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_342_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > real,G: list_nat > real] :
( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_list_nat_real @ F @ X2 )
= ( map_list_nat_real @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_343_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_344_list_Omap__cong,axiom,
! [X2: list_s1210847774152347623at_nat,Ya: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > real,G: set_Pr1261947904930325089at_nat > real] :
( ( X2 = Ya )
=> ( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se8952767809526791113t_real @ F @ X2 )
= ( map_se8952767809526791113t_real @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_345_list_Omap__cong,axiom,
! [X2: list_s1210847774152347623at_nat,Ya: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,G: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] :
( ( X2 = Ya )
=> ( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se5413732074677250773nnreal @ F @ X2 )
= ( map_se5413732074677250773nnreal @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_346_list_Omap__cong,axiom,
! [X2: list_s1210847774152347623at_nat,Ya: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > int,G: set_Pr1261947904930325089at_nat > int] :
( ( X2 = Ya )
=> ( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se4261650905360539081at_int @ F @ X2 )
= ( map_se4261650905360539081at_int @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_347_list_Omap__cong,axiom,
! [X2: list_s1210847774152347623at_nat,Ya: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat] :
( ( X2 = Ya )
=> ( ! [Z: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z @ ( set_se5049602875457034614at_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se4264141375869589357at_nat @ F @ X2 )
= ( map_se4264141375869589357at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_348_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > real,G: list_nat > real] :
( ( X2 = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_list_nat_real @ F @ X2 )
= ( map_list_nat_real @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_349_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( X2 = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_350_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_351_zero__le,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 ) ).
% zero_le
thf(fact_352_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_353_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_354_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_355_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_356_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_357_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_358_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_359_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_360_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: rat,A: rat,B: rat] :
( ( X2 != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ X2 )
= ( times_times_rat @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_361_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_362_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: rat,X2: rat,Y: rat] :
( ( A != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ X2 )
= ( times_times_rat @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_363_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X2: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_364_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_365_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_366_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_367_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_368_no__zero__divisors,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A != zero_z7100319975126383169nnreal )
=> ( ( B != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ A @ B )
!= zero_z7100319975126383169nnreal ) ) ) ).
% no_zero_divisors
thf(fact_369_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_370_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_371_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_372_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_373_divisors__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% divisors_zero
thf(fact_374_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_375_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_376_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_377_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_378_mult__not__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
!= zero_z7100319975126383169nnreal )
=> ( ( A != zero_z7100319975126383169nnreal )
& ( B != zero_z7100319975126383169nnreal ) ) ) ).
% mult_not_zero
thf(fact_379_semiring__norm_I111_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% semiring_norm(111)
thf(fact_380_semiring__norm_I111_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% semiring_norm(111)
thf(fact_381_semiring__norm_I111_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% semiring_norm(111)
thf(fact_382_semiring__norm_I111_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% semiring_norm(111)
thf(fact_383_semiring__norm_I111_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% semiring_norm(111)
thf(fact_384_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% semiring_norm(112)
thf(fact_385_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% semiring_norm(112)
thf(fact_386_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% semiring_norm(112)
thf(fact_387_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% semiring_norm(112)
thf(fact_388_semiring__norm_I112_J,axiom,
~ ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% semiring_norm(112)
thf(fact_389_rotate1__map,axiom,
! [F: list_nat > real,Xs: list_list_nat] :
( ( rotate1_real @ ( map_list_nat_real @ F @ Xs ) )
= ( map_list_nat_real @ F @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_390_rotate1__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( rotate4238613965387346100at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( map_li6003994582982014139at_nat @ F @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_391_rotate1__map,axiom,
! [F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat] :
( ( rotate1_real @ ( map_se8952767809526791113t_real @ F @ Xs ) )
= ( map_se8952767809526791113t_real @ F @ ( rotate4238613965387346100at_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_392_rotate1__map,axiom,
! [F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat] :
( ( rotate2246681853897044133nnreal @ ( map_se5413732074677250773nnreal @ F @ Xs ) )
= ( map_se5413732074677250773nnreal @ F @ ( rotate4238613965387346100at_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_393_rotate1__map,axiom,
! [F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat] :
( ( rotate1_int @ ( map_se4261650905360539081at_int @ F @ Xs ) )
= ( map_se4261650905360539081at_int @ F @ ( rotate4238613965387346100at_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_394_rotate1__map,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat] :
( ( rotate1_nat @ ( map_se4264141375869589357at_nat @ F @ Xs ) )
= ( map_se4264141375869589357at_nat @ F @ ( rotate4238613965387346100at_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_395_lambda__zero,axiom,
( ( ^ [H: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_396_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_397_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_398_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_399_lambda__zero,axiom,
( ( ^ [H: extend8495563244428889912nnreal] : zero_z7100319975126383169nnreal )
= ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal ) ) ).
% lambda_zero
thf(fact_400_member__le__sum__list,axiom,
! [X2: extend8495563244428889912nnreal,Xs: list_E5688521862016077384nnreal] :
( ( member7908768830364227535nnreal @ X2 @ ( set_Ex7800660098987911779nnreal @ Xs ) )
=> ( ord_le3935885782089961368nnreal @ X2 @ ( groups2217173247284669407nnreal @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_401_member__le__sum__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_402_mult__sign__intros_I4_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 @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_403_mult__sign__intros_I4_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 @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_404_mult__sign__intros_I4_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 @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_405_mult__sign__intros_I3_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_sign_intros(3)
thf(fact_406_mult__sign__intros_I3_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_407_mult__sign__intros_I3_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_sign_intros(3)
thf(fact_408_mult__sign__intros_I3_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(3)
thf(fact_409_mult__sign__intros_I2_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_sign_intros(2)
thf(fact_410_mult__sign__intros_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_411_mult__sign__intros_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_sign_intros(2)
thf(fact_412_mult__sign__intros_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(2)
thf(fact_413_mult__sign__intros_I1_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 @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_414_mult__sign__intros_I1_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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_415_mult__sign__intros_I1_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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_416_mult__sign__intros_I1_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 @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_417_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_418_mult__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 @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_419_mult__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 @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_420_mult__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 @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_421_mult__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ C @ D2 )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_422_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_423_mult__mono_H,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 @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_424_mult__mono_H,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 @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_425_mult__mono_H,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 @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_426_mult__mono_H,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ C @ D2 )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_427_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_428_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_429_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_430_split__mult__pos__le,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 @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_431_split__mult__pos__le,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 @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_432_split__mult__pos__le,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 @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_433_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_434_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_435_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_436_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_437_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_438_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_439_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 ) ) ) ) ).
% mult_left_mono
thf(fact_440_mult__left__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_441_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_442_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_443_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_444_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_445_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_446_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_447_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 ) ) ) ) ).
% mult_right_mono
thf(fact_448_mult__right__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_449_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_450_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_451_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_452_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_453_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_454_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_455_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_456_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_457_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_458_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_459_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_460_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_461_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_462_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_463_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_464_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_465_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_466_ordered__comm__semiring__class_Ocomm__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 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_467_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_468_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_469_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_470_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_471_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_472_zero__less__one__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% zero_less_one_class.zero_le_one
thf(fact_473_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_474_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_475_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_476_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_477_split__of__bool__asm,axiom,
! [P2: rat > $o,P: $o] :
( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_rat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_478_split__of__bool__asm,axiom,
! [P2: real > $o,P: $o] :
( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_real ) )
| ( ~ P
& ~ ( P2 @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_479_split__of__bool__asm,axiom,
! [P2: extend8495563244428889912nnreal > $o,P: $o] :
( ( P2 @ ( zero_n4168557817388953207nnreal @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_on2969667320475766781nnreal ) )
| ( ~ P
& ~ ( P2 @ zero_z7100319975126383169nnreal ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_480_split__of__bool__asm,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_int ) )
| ( ~ P
& ~ ( P2 @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_481_split__of__bool__asm,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_nat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_482_split__of__bool,axiom,
! [P2: rat > $o,P: $o] :
( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( ( P
=> ( P2 @ one_one_rat ) )
& ( ~ P
=> ( P2 @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_483_split__of__bool,axiom,
! [P2: real > $o,P: $o] :
( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
= ( ( P
=> ( P2 @ one_one_real ) )
& ( ~ P
=> ( P2 @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_484_split__of__bool,axiom,
! [P2: extend8495563244428889912nnreal > $o,P: $o] :
( ( P2 @ ( zero_n4168557817388953207nnreal @ P ) )
= ( ( P
=> ( P2 @ one_on2969667320475766781nnreal ) )
& ( ~ P
=> ( P2 @ zero_z7100319975126383169nnreal ) ) ) ) ).
% split_of_bool
thf(fact_485_split__of__bool,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ( P
=> ( P2 @ one_one_int ) )
& ( ~ P
=> ( P2 @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_486_split__of__bool,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ( P
=> ( P2 @ one_one_nat ) )
& ( ~ P
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_487_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_488_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_489_of__bool__def,axiom,
( zero_n4168557817388953207nnreal
= ( ^ [P3: $o] : ( if_Ext9135588136721118450nnreal @ P3 @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ) ) ).
% of_bool_def
thf(fact_490_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_491_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_492_enum__rgfs__len,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( ( size_size_list_nat @ X2 )
= N ) ) ).
% enum_rgfs_len
thf(fact_493_order__trans__rules_I26_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_494_order__trans__rules_I26_J,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_495_order__trans__rules_I26_J,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_496_order__trans__rules_I26_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A = B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_497_order__trans__rules_I25_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_498_order__trans__rules_I25_J,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_499_order__trans__rules_I25_J,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_500_order__trans__rules_I25_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( B = C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_501_order__trans__rules_I24_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_502_order__trans__rules_I24_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_503_order__trans__rules_I24_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_504_order__trans__rules_I24_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_505_order__trans__rules_I23_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans_rules(23)
thf(fact_506_order__trans__rules_I23_J,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans_rules(23)
thf(fact_507_order__trans__rules_I23_J,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans_rules(23)
thf(fact_508_order__trans__rules_I23_J,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) ) ) ).
% order_trans_rules(23)
thf(fact_509_order__trans__rules_I10_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_510_order__trans__rules_I10_J,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_511_order__trans__rules_I10_J,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_512_order__trans__rules_I10_J,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_513_order__trans__rules_I10_J,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_514_order__trans__rules_I10_J,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_515_order__trans__rules_I10_J,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_516_order__trans__rules_I10_J,axiom,
! [A: extend8495563244428889912nnreal,F: int > extend8495563244428889912nnreal,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_517_order__trans__rules_I10_J,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_518_order__trans__rules_I10_J,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_519_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_520_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_521_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_522_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_523_order__trans__rules_I9_J,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_524_order__trans__rules_I9_J,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_525_order__trans__rules_I9_J,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_526_order__trans__rules_I9_J,axiom,
! [A: int,B: int,F: int > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_527_order__trans__rules_I9_J,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_528_order__trans__rules_I9_J,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_529_order__trans__rules_I8_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_530_order__trans__rules_I8_J,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_531_order__trans__rules_I8_J,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_532_order__trans__rules_I8_J,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_533_order__trans__rules_I8_J,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_534_order__trans__rules_I8_J,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_535_order__trans__rules_I8_J,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_536_order__trans__rules_I8_J,axiom,
! [A: int,F: extend8495563244428889912nnreal > int,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_537_order__trans__rules_I8_J,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_538_order__trans__rules_I8_J,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_539_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_540_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_541_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_542_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_543_order__trans__rules_I7_J,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_544_order__trans__rules_I7_J,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_545_order__trans__rules_I7_J,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_546_order__trans__rules_I7_J,axiom,
! [A: int,B: int,F: int > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_547_order__trans__rules_I7_J,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_548_order__trans__rules_I7_J,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_549_linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linear
thf(fact_550_linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linear
thf(fact_551_linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linear
thf(fact_552_linear,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
| ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).
% linear
thf(fact_553_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_554_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_555_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_556_nle__le,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A @ B ) )
= ( ( ord_le3935885782089961368nnreal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_557_le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% le_cases
thf(fact_558_le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% le_cases
thf(fact_559_le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% le_cases
thf(fact_560_le__cases,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).
% le_cases
thf(fact_561_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_562_le__cases3,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_563_le__cases3,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_564_le__cases3,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ~ ( ord_le3935885782089961368nnreal @ Y @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X2 @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ Y ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
=> ~ ( ord_le3935885782089961368nnreal @ Y @ X2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ X2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z2 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_565_antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_566_antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_567_antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_568_antisym__conv,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_569_order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_570_order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_571_order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_572_order_Oeq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] : ( Y4 = Z3 ) )
= ( ^ [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
& ( ord_le3935885782089961368nnreal @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_573_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_574_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
& ( ord_less_eq_int @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_575_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
& ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_576_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] : ( Y4 = Z3 ) )
= ( ^ [X: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y2 )
& ( ord_le3935885782089961368nnreal @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_577_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_578_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_579_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_580_order__antisym,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_581_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_582_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_583_order_Otrans,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 ) ) ) ).
% order.trans
thf(fact_584_order_Otrans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% order.trans
thf(fact_585_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_586_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_587_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_588_linorder__wlog,axiom,
! [P2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_589_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_590_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_591_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_592_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] : ( Y4 = Z3 ) )
= ( ^ [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
& ( ord_le3935885782089961368nnreal @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_593_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_594_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_595_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_596_dual__order_Oantisym,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_597_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_598_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_599_dual__order_Otrans,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 ) ) ) ).
% dual_order.trans
thf(fact_600_dual__order_Otrans,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_601_mult__left__le,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_602_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_603_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_604_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_605_mult__left__le,axiom,
! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ one_on2969667320475766781nnreal )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_606_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_607_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_608_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_609_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_610_mult__le__one,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ one_on2969667320475766781nnreal )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ one_on2969667320475766781nnreal )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ one_on2969667320475766781nnreal ) ) ) ) ).
% mult_le_one
thf(fact_611_mult__right__le__one__le,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_612_mult__right__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_613_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_614_mult__left__le__one__le,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_615_mult__left__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_616_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_617_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri1316708129612266289at_nat @ X2 )
= one_one_nat )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_618_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri1314217659103216013at_int @ X2 )
= one_one_int )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_619_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri5074537144036343181t_real @ X2 )
= one_one_real )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_620_Totient_Oof__nat__eq__1__iff,axiom,
! [X2: nat] :
( ( ( semiri6283507881447550617nnreal @ X2 )
= one_on2969667320475766781nnreal )
= ( X2 = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_621_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_622_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_623_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_624_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri6283507881447550617nnreal @ N )
= one_on2969667320475766781nnreal )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_625_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_626_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_627_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_628_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_on2969667320475766781nnreal
= ( semiri6283507881447550617nnreal @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_629_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_630_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_631_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_632_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_633_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_634_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_635_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_636_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_637_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_638_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n3304061248610475627l_real @ P2 ) ) ).
% of_nat_of_bool
thf(fact_639_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri6283507881447550617nnreal @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n4168557817388953207nnreal @ P2 ) ) ).
% of_nat_of_bool
thf(fact_640_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% of_nat_of_bool
thf(fact_641_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% of_nat_of_bool
thf(fact_642_of__nat__ge__1__iff,axiom,
! [X2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ X2 ) )
= ( ord_less_eq_nat @ one_one_nat @ X2 ) ) ).
% of_nat_ge_1_iff
thf(fact_643_of__nat__ge__1__iff,axiom,
! [X2: nat] :
( ( ord_less_eq_int @ one_one_int @ ( semiri1314217659103216013at_int @ X2 ) )
= ( ord_less_eq_nat @ one_one_nat @ X2 ) ) ).
% of_nat_ge_1_iff
thf(fact_644_of__nat__ge__1__iff,axiom,
! [X2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ X2 ) )
= ( ord_less_eq_nat @ one_one_nat @ X2 ) ) ).
% of_nat_ge_1_iff
thf(fact_645_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_646_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_647_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= ( semiri6283507881447550617nnreal @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_648_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_649_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_650_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_651_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_661_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_662_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_663_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_664_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_665_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z7100319975126383169nnreal
= ( semiri6283507881447550617nnreal @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_666_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_667_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_668_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_669_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_670_int__if,axiom,
! [P2: $o,A: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_671_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_672_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_673_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_674_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_675_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_676_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_677_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_678_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_679_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_680_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_681_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_682_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_683_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_684_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_685_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_686_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_687_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_688_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_689_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_690_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_691_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_692_size__neq__size__imp__neq,axiom,
! [X2: char,Y: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_693_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_694_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_695_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_696_mult__of__nat__commute,axiom,
! [X2: nat,Y: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ X2 ) @ Y )
= ( times_1893300245718287421nnreal @ Y @ ( semiri6283507881447550617nnreal @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_697_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_698_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_699_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_700_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_701_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N ) ) ).
% of_nat_0_le_iff
thf(fact_702_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_703_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_704_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_705_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ I ) @ ( semiri6283507881447550617nnreal @ J ) ) ) ).
% of_nat_mono
thf(fact_706_kuhn__labelling__lemma_H,axiom,
! [P2: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X3: nat > real] :
( ( P2 @ X3 )
=> ( P2 @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P2 @ X3 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X4 @ I3 ) @ one_one_nat )
& ! [X4: nat > real,I3: nat] :
( ( ( P2 @ X4 )
& ( Q2 @ I3 )
& ( ( X4 @ I3 )
= zero_zero_real ) )
=> ( ( L2 @ X4 @ I3 )
= zero_zero_nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P2 @ X4 )
& ( Q2 @ I3 )
& ( ( X4 @ I3 )
= one_one_real ) )
=> ( ( L2 @ X4 @ I3 )
= one_one_nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P2 @ X4 )
& ( Q2 @ I3 )
& ( ( L2 @ X4 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I3 ) @ ( F @ X4 @ I3 ) ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P2 @ X4 )
& ( Q2 @ I3 )
& ( ( L2 @ X4 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_707_mult__eq__1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ( ( times_times_rat @ A @ B )
= one_one_rat )
= ( ( A = one_one_rat )
& ( B = one_one_rat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_708_mult__eq__1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= one_one_nat )
= ( ( A = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_709_mult__eq__1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= one_one_int )
= ( ( A = one_one_int )
& ( B = one_one_int ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_710_mult__eq__1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= one_one_real )
= ( ( A = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_711_mult__eq__1,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
=> ( ( ord_le3935885782089961368nnreal @ A @ one_on2969667320475766781nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B @ one_on2969667320475766781nnreal )
=> ( ( ( times_1893300245718287421nnreal @ A @ B )
= one_on2969667320475766781nnreal )
= ( ( A = one_on2969667320475766781nnreal )
& ( B = one_on2969667320475766781nnreal ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_712_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_713_real__of__nat__ge__one__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ one_one_nat @ N ) ) ).
% real_of_nat_ge_one_iff
thf(fact_714_mult__if__delta,axiom,
! [P2: $o,Q: rat] :
( ( P2
=> ( ( times_times_rat @ ( if_rat @ P2 @ one_one_rat @ zero_zero_rat ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_rat @ ( if_rat @ P2 @ one_one_rat @ zero_zero_rat ) @ Q )
= zero_zero_rat ) ) ) ).
% mult_if_delta
thf(fact_715_mult__if__delta,axiom,
! [P2: $o,Q: real] :
( ( P2
=> ( ( times_times_real @ ( if_real @ P2 @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_real @ ( if_real @ P2 @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_716_mult__if__delta,axiom,
! [P2: $o,Q: nat] :
( ( P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_717_mult__if__delta,axiom,
! [P2: $o,Q: int] :
( ( P2
=> ( ( times_times_int @ ( if_int @ P2 @ one_one_int @ zero_zero_int ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_int @ ( if_int @ P2 @ one_one_int @ zero_zero_int ) @ Q )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_718_mult__if__delta,axiom,
! [P2: $o,Q: extend8495563244428889912nnreal] :
( ( P2
=> ( ( times_1893300245718287421nnreal @ ( if_Ext9135588136721118450nnreal @ P2 @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_1893300245718287421nnreal @ ( if_Ext9135588136721118450nnreal @ P2 @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) @ Q )
= zero_z7100319975126383169nnreal ) ) ) ).
% mult_if_delta
thf(fact_719_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_720_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_721_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_722_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_723_nonneg__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_eq_int
thf(fact_724_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_725_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_726_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_727_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_728_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_729_mult__mono__nonpos__nonpos,axiom,
! [C: real,A: real,D2: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ( ord_less_eq_real @ D2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ C @ D2 ) ) ) ) ) ) ).
% mult_mono_nonpos_nonpos
thf(fact_730_mult__mono__nonpos__nonneg,axiom,
! [A: rat,C: rat,D2: rat,B: rat] :
( ( ord_less_eq_rat @ A @ C )
=> ( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ D2 )
=> ( ( ord_less_eq_rat @ D2 @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ C @ D2 ) ) ) ) ) ) ).
% mult_mono_nonpos_nonneg
thf(fact_731_mult__mono__nonpos__nonneg,axiom,
! [A: int,C: int,D2: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ D2 )
=> ( ( ord_less_eq_int @ D2 @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ C @ D2 ) ) ) ) ) ) ).
% mult_mono_nonpos_nonneg
thf(fact_732_mult__mono__nonpos__nonneg,axiom,
! [A: real,C: real,D2: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ D2 )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ C @ D2 ) ) ) ) ) ) ).
% mult_mono_nonpos_nonneg
thf(fact_733_landau__o_OR__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% landau_o.R_refl
thf(fact_734_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_735_landau__o_OR__linear,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% landau_o.R_linear
thf(fact_736_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_737_landau__omega_OR__linear,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% landau_omega.R_linear
thf(fact_738_landau__o_OR,axiom,
( ( ord_less_eq_real = ord_less_eq_real )
| ( ord_less_eq_real
= ( ^ [X: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).
% landau_o.R
thf(fact_739_landau__omega_OR,axiom,
( ( ( ^ [X: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X ) )
= ord_less_eq_real )
| ( ( ^ [X: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X ) )
= ( ^ [X: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).
% landau_omega.R
thf(fact_740_mult__mono__nonneg__nonpos,axiom,
! [A: rat,C: rat,D2: rat,B: rat] :
( ( ord_less_eq_rat @ A @ C )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ D2 )
=> ( ( ord_less_eq_rat @ D2 @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ D2 @ C ) ) ) ) ) ) ).
% mult_mono_nonneg_nonpos
thf(fact_741_mult__mono__nonneg__nonpos,axiom,
! [A: int,C: int,D2: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ D2 )
=> ( ( ord_less_eq_int @ D2 @ B )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ D2 @ C ) ) ) ) ) ) ).
% mult_mono_nonneg_nonpos
thf(fact_742_mult__mono__nonneg__nonpos,axiom,
! [A: real,C: real,D2: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ D2 )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ D2 @ C ) ) ) ) ) ) ).
% mult_mono_nonneg_nonpos
thf(fact_743_set__n__lists,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys2 )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys2 ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_744_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_745_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_746_subsetI,axiom,
! [A3: set_list_nat,B3: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ( member_list_nat @ X3 @ B3 ) )
=> ( ord_le6045566169113846134st_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_747_subsetI,axiom,
! [A3: set_real,B3: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( member_real @ X3 @ B3 ) )
=> ( ord_less_eq_set_real @ A3 @ B3 ) ) ).
% subsetI
thf(fact_748_subsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_749_Set_Obasic__monos_I7_J,axiom,
! [A3: set_list_nat,B3: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
=> ( ( member_list_nat @ X2 @ A3 )
=> ( member_list_nat @ X2 @ B3 ) ) ) ).
% Set.basic_monos(7)
thf(fact_750_Set_Obasic__monos_I7_J,axiom,
! [A3: set_real,B3: set_real,X2: real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
=> ( ( member_real @ X2 @ A3 )
=> ( member_real @ X2 @ B3 ) ) ) ).
% Set.basic_monos(7)
thf(fact_751_Set_Obasic__monos_I7_J,axiom,
! [A3: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% Set.basic_monos(7)
thf(fact_752_Set_Obasic__monos_I6_J,axiom,
! [P2: int > $o,Q2: int > $o] :
( ! [X3: int] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) ) ) ).
% Set.basic_monos(6)
thf(fact_753_Set_Obasic__monos_I6_J,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Set.basic_monos(6)
thf(fact_754_basic__trans__rules_I31_J,axiom,
! [A3: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
=> ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% basic_trans_rules(31)
thf(fact_755_basic__trans__rules_I31_J,axiom,
! [A3: set_real,B3: set_real,C: real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
=> ( ( member_real @ C @ A3 )
=> ( member_real @ C @ B3 ) ) ) ).
% basic_trans_rules(31)
thf(fact_756_basic__trans__rules_I31_J,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% basic_trans_rules(31)
thf(fact_757_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A5 )
=> ( member_list_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_758_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X: real] :
( ( member_real @ X @ A5 )
=> ( member_real @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_759_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_760_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A5 )
=> ( member_list_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_761_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A5 )
=> ( member_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_762_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_763_Collect__mono__iff,axiom,
! [P2: int > $o,Q2: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) )
= ( ! [X: int] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_764_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_765_Collect__subset,axiom,
! [A3: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A3 )
& ( P2 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_766_Collect__subset,axiom,
! [A3: set_real,P2: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A3 )
& ( P2 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_767_Collect__subset,axiom,
! [A3: set_int,P2: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A3 )
& ( P2 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_768_Collect__subset,axiom,
! [A3: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P2 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_769_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_770_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ A5 )
@ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_771_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_772_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_773_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_774_sum__list__ennreal,axiom,
! [Xs: list_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( groups2217173247284669407nnreal
@ ( map_re3918317694826015018nnreal
@ ^ [X: real] : ( extend7643940197134561352nnreal @ ( F @ X ) )
@ Xs ) )
= ( extend7643940197134561352nnreal @ ( groups6723090944982001619t_real @ ( map_real_real @ F @ Xs ) ) ) ) ) ).
% sum_list_ennreal
thf(fact_775_sum__list__ennreal,axiom,
! [Xs: list_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( groups2217173247284669407nnreal
@ ( map_na4420896737966758094nnreal
@ ^ [X: nat] : ( extend7643940197134561352nnreal @ ( F @ X ) )
@ Xs ) )
= ( extend7643940197134561352nnreal @ ( groups6723090944982001619t_real @ ( map_nat_real @ F @ Xs ) ) ) ) ) ).
% sum_list_ennreal
thf(fact_776_sum__list__ennreal,axiom,
! [Xs: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > real] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( groups2217173247284669407nnreal
@ ( map_se5413732074677250773nnreal
@ ^ [X: set_Pr1261947904930325089at_nat] : ( extend7643940197134561352nnreal @ ( F @ X ) )
@ Xs ) )
= ( extend7643940197134561352nnreal @ ( groups6723090944982001619t_real @ ( map_se8952767809526791113t_real @ F @ Xs ) ) ) ) ) ).
% sum_list_ennreal
thf(fact_777_sum__list__ennreal,axiom,
! [Xs: list_list_nat,F: list_nat > real] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( groups2217173247284669407nnreal
@ ( map_li5637531228398476638nnreal
@ ^ [X: list_nat] : ( extend7643940197134561352nnreal @ ( F @ X ) )
@ Xs ) )
= ( extend7643940197134561352nnreal @ ( groups6723090944982001619t_real @ ( map_list_nat_real @ F @ Xs ) ) ) ) ) ).
% sum_list_ennreal
thf(fact_778_imp__le__cong,axiom,
! [X2: int,X5: int,P2: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_779_conj__le__cong,axiom,
! [X2: int,X5: int,P2: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_780_int_Oone__not__zero,axiom,
one_one_int != zero_zero_int ).
% int.one_not_zero
thf(fact_781_ennreal__0,axiom,
( ( extend7643940197134561352nnreal @ zero_zero_real )
= zero_z7100319975126383169nnreal ) ).
% ennreal_0
thf(fact_782_ennreal__1,axiom,
( ( extend7643940197134561352nnreal @ one_one_real )
= one_on2969667320475766781nnreal ) ).
% ennreal_1
thf(fact_783_ennreal__eq__1,axiom,
! [X2: real] :
( ( ( extend7643940197134561352nnreal @ X2 )
= one_on2969667320475766781nnreal )
= ( X2 = one_one_real ) ) ).
% ennreal_eq_1
thf(fact_784_ennreal__inj,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( extend7643940197134561352nnreal @ A )
= ( extend7643940197134561352nnreal @ B ) )
= ( A = B ) ) ) ) ).
% ennreal_inj
thf(fact_785_ennreal__le__iff,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ).
% ennreal_le_iff
thf(fact_786_ennreal__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal )
= ( X2 = zero_zero_real ) ) ) ).
% ennreal_eq_zero_iff
thf(fact_787_ennreal__ge__1,axiom,
! [X2: real] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% ennreal_ge_1
thf(fact_788_ennreal__le__1,axiom,
! [X2: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% ennreal_le_1
thf(fact_789_ennreal__le__of__nat__iff,axiom,
! [R: real,I: nat] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( semiri6283507881447550617nnreal @ I ) )
= ( ord_less_eq_real @ R @ ( semiri5074537144036343181t_real @ I ) ) ) ).
% ennreal_le_of_nat_iff
thf(fact_790_of__nat__le__ennreal__iff,axiom,
! [R: real,I: nat] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ I ) @ ( extend7643940197134561352nnreal @ R ) )
= ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ R ) ) ) ).
% of_nat_le_ennreal_iff
thf(fact_791_ennreal__cong,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ( extend7643940197134561352nnreal @ X2 )
= ( extend7643940197134561352nnreal @ Y ) ) ) ).
% ennreal_cong
thf(fact_792_ennreal__leI,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y ) ) ) ).
% ennreal_leI
thf(fact_793_ennreal__of__nat__eq__real__of__nat,axiom,
( semiri6283507881447550617nnreal
= ( ^ [I4: nat] : ( extend7643940197134561352nnreal @ ( semiri5074537144036343181t_real @ I4 ) ) ) ) ).
% ennreal_of_nat_eq_real_of_nat
thf(fact_794_ennreal__neg,axiom,
! [X2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_neg
thf(fact_795_le__ennreal__iff,axiom,
! [R: real,X2: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ ( extend7643940197134561352nnreal @ R ) )
= ( ? [Q3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q3 )
& ( X2
= ( extend7643940197134561352nnreal @ Q3 ) )
& ( ord_less_eq_real @ Q3 @ R ) ) ) ) ) ).
% le_ennreal_iff
thf(fact_796_ennreal__le__iff2,axiom,
! [X2: real,Y: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ Y )
& ( ord_less_eq_real @ X2 @ Y ) )
| ( ( ord_less_eq_real @ X2 @ zero_zero_real )
& ( ord_less_eq_real @ Y @ zero_zero_real ) ) ) ) ).
% ennreal_le_iff2
thf(fact_797_ennreal__eq__0__iff,axiom,
! [X2: real] :
( ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal )
= ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% ennreal_eq_0_iff
thf(fact_798_mult__right__ennreal__cancel,axiom,
! [A: extend8495563244428889912nnreal,C: real,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ ( extend7643940197134561352nnreal @ C ) )
= ( times_1893300245718287421nnreal @ B @ ( extend7643940197134561352nnreal @ C ) ) )
= ( ( A = B )
| ( ord_less_eq_real @ C @ zero_zero_real ) ) ) ).
% mult_right_ennreal_cancel
thf(fact_799_ennreal__mult,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ) ).
% ennreal_mult
thf(fact_800_ennreal__mult_H,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).
% ennreal_mult'
thf(fact_801_ennreal__mult_H_H,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).
% ennreal_mult''
thf(fact_802_pred__subset__eq,axiom,
! [R2: set_list_nat,S: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ R2 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ S ) )
= ( ord_le6045566169113846134st_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_803_pred__subset__eq,axiom,
! [R2: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ R2 )
@ ^ [X: real] : ( member_real @ X @ S ) )
= ( ord_less_eq_set_real @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_804_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_805_real__arch__simple,axiom,
! [X2: real] :
? [N3: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_806_nz__prod__eq,axiom,
! [C: rat,X2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ X2 )
= zero_zero_rat )
= ( X2 = zero_zero_rat ) ) ) ).
% nz_prod_eq
thf(fact_807_nz__prod__eq,axiom,
! [C: real,X2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_real ) ) ) ).
% nz_prod_eq
thf(fact_808_mult__delta__left,axiom,
! [B: $o,X2: rat,Y: rat] :
( ( B
=> ( ( times_times_rat @ ( if_rat @ B @ X2 @ zero_zero_rat ) @ Y )
= ( times_times_rat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_rat @ ( if_rat @ B @ X2 @ zero_zero_rat ) @ Y )
= zero_zero_rat ) ) ) ).
% mult_delta_left
thf(fact_809_mult__delta__left,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_810_mult__delta__left,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_811_mult__delta__left,axiom,
! [B: $o,X2: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_812_mult__delta__left,axiom,
! [B: $o,X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( B
=> ( ( times_1893300245718287421nnreal @ ( if_Ext9135588136721118450nnreal @ B @ X2 @ zero_z7100319975126383169nnreal ) @ Y )
= ( times_1893300245718287421nnreal @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_1893300245718287421nnreal @ ( if_Ext9135588136721118450nnreal @ B @ X2 @ zero_z7100319975126383169nnreal ) @ Y )
= zero_z7100319975126383169nnreal ) ) ) ).
% mult_delta_left
thf(fact_813_mult__delta__right,axiom,
! [B: $o,X2: rat,Y: rat] :
( ( B
=> ( ( times_times_rat @ X2 @ ( if_rat @ B @ Y @ zero_zero_rat ) )
= ( times_times_rat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_rat @ X2 @ ( if_rat @ B @ Y @ zero_zero_rat ) )
= zero_zero_rat ) ) ) ).
% mult_delta_right
thf(fact_814_mult__delta__right,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_815_mult__delta__right,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_816_mult__delta__right,axiom,
! [B: $o,X2: int,Y: int] :
( ( B
=> ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_817_mult__delta__right,axiom,
! [B: $o,X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( B
=> ( ( times_1893300245718287421nnreal @ X2 @ ( if_Ext9135588136721118450nnreal @ B @ Y @ zero_z7100319975126383169nnreal ) )
= ( times_1893300245718287421nnreal @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_1893300245718287421nnreal @ X2 @ ( if_Ext9135588136721118450nnreal @ B @ Y @ zero_z7100319975126383169nnreal ) )
= zero_z7100319975126383169nnreal ) ) ) ).
% mult_delta_right
thf(fact_818_ennreal__mult__left__cong,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( A != zero_z7100319975126383169nnreal )
=> ( B = C ) )
=> ( ( times_1893300245718287421nnreal @ A @ B )
= ( times_1893300245718287421nnreal @ A @ C ) ) ) ).
% ennreal_mult_left_cong
thf(fact_819_ennreal__mult__right__cong,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( A != zero_z7100319975126383169nnreal )
=> ( B = C ) )
=> ( ( times_1893300245718287421nnreal @ B @ A )
= ( times_1893300245718287421nnreal @ C @ A ) ) ) ).
% ennreal_mult_right_cong
thf(fact_820_equiv__rels__enum,axiom,
! [X2: list_nat] :
( ( equiva3371634703666331078on_rgf @ X2 )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ X2 ) ) @ X2 )
= one_one_nat ) ) ).
% equiv_rels_enum
thf(fact_821_length__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_822_round__up__le1,axiom,
! [X2: real,Prec: int] :
( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ( ord_less_eq_int @ zero_zero_int @ Prec )
=> ( ord_less_eq_real @ ( round_up @ Prec @ X2 ) @ one_one_real ) ) ) ).
% round_up_le1
thf(fact_823_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_824_mangoldt__odd__pos,axiom,
! [D2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( mangoldt_odd @ D2 ) ) ).
% mangoldt_odd_pos
thf(fact_825_round__up__0,axiom,
! [P: int] :
( ( round_up @ P @ zero_zero_real )
= zero_zero_real ) ).
% round_up_0
thf(fact_826_count__notin,axiom,
! [X2: real,Xs: list_real] :
( ~ ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_827_count__notin,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_828_count__notin,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_829_round__up__mono,axiom,
! [X2: real,Y: real,P: int] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( round_up @ P @ X2 ) @ ( round_up @ P @ Y ) ) ) ).
% round_up_mono
thf(fact_830_round__up,axiom,
! [X2: real,Prec: int] : ( ord_less_eq_real @ X2 @ ( round_up @ Prec @ X2 ) ) ).
% round_up
thf(fact_831_count__list__0__iff,axiom,
! [Xs: list_real,X2: real] :
( ( ( count_list_real @ Xs @ X2 )
= zero_zero_nat )
= ( ~ ( member_real @ X2 @ ( set_real2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_832_count__list__0__iff,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( count_list_nat @ Xs @ X2 )
= zero_zero_nat )
= ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_833_count__list__0__iff,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( count_list_list_nat @ Xs @ X2 )
= zero_zero_nat )
= ( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_834_count__le__length,axiom,
! [Xs: list_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% count_le_length
thf(fact_835_count__le__length,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X2 ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_836_count__list__map__ge,axiom,
! [Xs: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > real] : ( ord_less_eq_nat @ ( count_6440129622255701469at_nat @ Xs @ X2 ) @ ( count_list_real @ ( map_se8952767809526791113t_real @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_837_count__list__map__ge,axiom,
! [Xs: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal] : ( ord_less_eq_nat @ ( count_6440129622255701469at_nat @ Xs @ X2 ) @ ( count_2060941923878149820nnreal @ ( map_se5413732074677250773nnreal @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_838_count__list__map__ge,axiom,
! [Xs: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > int] : ( ord_less_eq_nat @ ( count_6440129622255701469at_nat @ Xs @ X2 ) @ ( count_list_int @ ( map_se4261650905360539081at_int @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_839_count__list__map__ge,axiom,
! [Xs: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] : ( ord_less_eq_nat @ ( count_6440129622255701469at_nat @ Xs @ X2 ) @ ( count_list_nat @ ( map_se4264141375869589357at_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_840_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > real] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_list_real @ ( map_list_nat_real @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_841_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > set_Pr1261947904930325089at_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_842_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_843_round__up__le0,axiom,
! [X2: real,P: int] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( round_up @ P @ X2 ) @ zero_zero_real ) ) ).
% round_up_le0
thf(fact_844_count__list__gr__1,axiom,
! [X2: real,Xs: list_real] :
( ( member_real @ X2 @ ( set_real2 @ Xs ) )
= ( ord_less_eq_nat @ one_one_nat @ ( count_list_real @ Xs @ X2 ) ) ) ).
% count_list_gr_1
thf(fact_845_count__list__gr__1,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ord_less_eq_nat @ one_one_nat @ ( count_list_nat @ Xs @ X2 ) ) ) ).
% count_list_gr_1
thf(fact_846_count__list__gr__1,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ord_less_eq_nat @ one_one_nat @ ( count_list_list_nat @ Xs @ X2 ) ) ) ).
% count_list_gr_1
thf(fact_847_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_848_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_849_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_850_fps__tan__0,axiom,
( ( formal7459771717576857490an_rat @ zero_zero_rat )
= zero_z5023345140362154345ps_rat ) ).
% fps_tan_0
thf(fact_851_psi__odd__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_real @ ( psi_odd @ M ) @ ( psi_odd @ N ) ) ) ).
% psi_odd_mono
thf(fact_852_set__rotate,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate_list_nat @ N @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_853_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_854_rotate__map,axiom,
! [N: nat,F: list_nat > real,Xs: list_list_nat] :
( ( rotate_real @ N @ ( map_list_nat_real @ F @ Xs ) )
= ( map_list_nat_real @ F @ ( rotate_list_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_855_rotate__map,axiom,
! [N: nat,F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( rotate1033626827900196251at_nat @ N @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( map_li6003994582982014139at_nat @ F @ ( rotate_list_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_856_rotate__map,axiom,
! [N: nat,F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat] :
( ( rotate_real @ N @ ( map_se8952767809526791113t_real @ F @ Xs ) )
= ( map_se8952767809526791113t_real @ F @ ( rotate1033626827900196251at_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_857_rotate__map,axiom,
! [N: nat,F: set_Pr1261947904930325089at_nat > extend8495563244428889912nnreal,Xs: list_s1210847774152347623at_nat] :
( ( rotate856342757803299966nnreal @ N @ ( map_se5413732074677250773nnreal @ F @ Xs ) )
= ( map_se5413732074677250773nnreal @ F @ ( rotate1033626827900196251at_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_858_rotate__map,axiom,
! [N: nat,F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat] :
( ( rotate_int @ N @ ( map_se4261650905360539081at_int @ F @ Xs ) )
= ( map_se4261650905360539081at_int @ F @ ( rotate1033626827900196251at_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_859_rotate__map,axiom,
! [N: nat,F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat] :
( ( rotate_nat @ N @ ( map_se4264141375869589357at_nat @ F @ Xs ) )
= ( map_se4264141375869589357at_nat @ F @ ( rotate1033626827900196251at_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_860_psi__odd__pos,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( psi_odd @ N ) ) ).
% psi_odd_pos
thf(fact_861_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_862_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_863_psi__residues__compare,axiom,
! [N: nat] : ( ord_less_eq_real @ ( psi_odd @ N ) @ ( psi_even @ N ) ) ).
% psi_residues_compare
thf(fact_864_sum__list__eval,axiom,
! [F: set_Pr1261947904930325089at_nat > real,Xs: list_s1210847774152347623at_nat] :
( ( groups6723090944982001619t_real @ ( map_se8952767809526791113t_real @ F @ Xs ) )
= ( groups5381467072955264723t_real
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( count_6440129622255701469at_nat @ Xs @ X ) ) @ ( F @ X ) )
@ ( set_se5049602875457034614at_nat @ Xs ) ) ) ).
% sum_list_eval
thf(fact_865_sum__list__eval,axiom,
! [F: list_nat > real,Xs: list_list_nat] :
( ( groups6723090944982001619t_real @ ( map_list_nat_real @ F @ Xs ) )
= ( groups8399112307953289288t_real
@ ^ [X: list_nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( count_list_list_nat @ Xs @ X ) ) @ ( F @ X ) )
@ ( set_list_nat2 @ Xs ) ) ) ).
% sum_list_eval
thf(fact_866_sum__list__eval,axiom,
! [F: set_Pr1261947904930325089at_nat > int,Xs: list_s1210847774152347623at_nat] :
( ( groups4559388385066561235st_int @ ( map_se4261650905360539081at_int @ F @ Xs ) )
= ( groups178575644746855891at_int
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( count_6440129622255701469at_nat @ Xs @ X ) ) @ ( F @ X ) )
@ ( set_se5049602875457034614at_nat @ Xs ) ) ) ).
% sum_list_eval
thf(fact_867_sum__list__eval,axiom,
! [F: list_nat > int,Xs: list_list_nat] :
( ( groups4559388385066561235st_int @ ( map_list_nat_int @ F @ Xs ) )
= ( groups4393565826250045896at_int
@ ^ [X: list_nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( count_list_list_nat @ Xs @ X ) ) @ ( F @ X ) )
@ ( set_list_nat2 @ Xs ) ) ) ).
% sum_list_eval
thf(fact_868_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_869_sum__ennreal,axiom,
! [I5: set_list_nat,F: list_nat > real] :
( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( groups5253920722037313236nnreal
@ ^ [I4: list_nat] : ( extend7643940197134561352nnreal @ ( F @ I4 ) )
@ I5 )
= ( extend7643940197134561352nnreal @ ( groups8399112307953289288t_real @ F @ I5 ) ) ) ) ).
% sum_ennreal
thf(fact_870_sum__ennreal,axiom,
! [I5: set_real,F: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( groups4232809223866053280nnreal
@ ^ [I4: real] : ( extend7643940197134561352nnreal @ ( F @ I4 ) )
@ I5 )
= ( extend7643940197134561352nnreal @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ).
% sum_ennreal
thf(fact_871_sum__ennreal,axiom,
! [I5: set_nat,F: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( groups4868793261593263428nnreal
@ ^ [I4: nat] : ( extend7643940197134561352nnreal @ ( F @ I4 ) )
@ I5 )
= ( extend7643940197134561352nnreal @ ( groups6591440286371151544t_real @ F @ I5 ) ) ) ) ).
% sum_ennreal
thf(fact_872_int_Oadd_Ofinprod__one__eqI,axiom,
! [A3: set_list_nat,F: list_nat > int] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ( ( F @ X3 )
= zero_zero_int ) )
=> ( ( groups4393565826250045896at_int @ F @ A3 )
= zero_zero_int ) ) ).
% int.add.finprod_one_eqI
thf(fact_873_int_Oadd_Ofinprod__one__eqI,axiom,
! [A3: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ( F @ X3 )
= zero_zero_int ) )
=> ( ( groups1932886352136224148al_int @ F @ A3 )
= zero_zero_int ) ) ).
% int.add.finprod_one_eqI
thf(fact_874_int_Oadd_Ofinprod__one__eqI,axiom,
! [A3: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( F @ X3 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ F @ A3 )
= zero_zero_int ) ) ).
% int.add.finprod_one_eqI
thf(fact_875_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_876_sum__list__map__eq__sum__count,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,Xs: list_s1210847774152347623at_nat] :
( ( groups4561878855575611511st_nat @ ( map_se4264141375869589357at_nat @ F @ Xs ) )
= ( groups181066115255906167at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] : ( times_times_nat @ ( count_6440129622255701469at_nat @ Xs @ X ) @ ( F @ X ) )
@ ( set_se5049602875457034614at_nat @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_877_sum__list__map__eq__sum__count,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( groups4396056296759096172at_nat
@ ^ [X: list_nat] : ( times_times_nat @ ( count_list_list_nat @ Xs @ X ) @ ( F @ X ) )
@ ( set_list_nat2 @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_878_sum__nonpos,axiom,
! [A3: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_879_sum__nonpos,axiom,
! [A3: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_880_sum__nonpos,axiom,
! [A3: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_881_sum__nonpos,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_882_sum__nonpos,axiom,
! [A3: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A3 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_883_sum__nonpos,axiom,
! [A3: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_884_sum__nonpos,axiom,
! [A3: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_885_sum__nonpos,axiom,
! [A3: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A3 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_886_sum__nonpos,axiom,
! [A3: set_real,F: real > extend8495563244428889912nnreal] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4232809223866053280nnreal @ F @ A3 ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_nonpos
thf(fact_887_sum__nonpos,axiom,
! [A3: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ A3 ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_nonpos
thf(fact_888_sum__nonneg,axiom,
! [A3: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_889_sum__nonneg,axiom,
! [A3: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_890_sum__nonneg,axiom,
! [A3: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_891_sum__nonneg,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_892_sum__nonneg,axiom,
! [A3: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_893_sum__nonneg,axiom,
! [A3: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_894_sum__nonneg,axiom,
! [A3: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_895_sum__nonneg,axiom,
! [A3: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_896_sum__nonneg,axiom,
! [A3: set_real,F: real > extend8495563244428889912nnreal] :
( ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups4232809223866053280nnreal @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_897_sum__nonneg,axiom,
! [A3: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups4868793261593263428nnreal @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_898_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A3: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A3 )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_899_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A3: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A3 )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_900_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A3: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A3 )
!= zero_zero_int )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_901_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A3: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A3 )
!= zero_zero_int )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_902_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A3: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A3 )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_903_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A3: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A3 )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_904_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > extend8495563244428889912nnreal,A3: set_real] :
( ( ( groups4232809223866053280nnreal @ G @ A3 )
!= zero_z7100319975126383169nnreal )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_905_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > extend8495563244428889912nnreal,A3: set_nat] :
( ( ( groups4868793261593263428nnreal @ G @ A3 )
!= zero_z7100319975126383169nnreal )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_906_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A3: set_real] :
( ( ( groups1300246762558778688al_rat @ G @ A3 )
!= zero_zero_rat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_907_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A3: set_nat] :
( ( ( groups2906978787729119204at_rat @ G @ A3 )
!= zero_zero_rat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_908_sum__mono,axiom,
! [K2: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K2 ) @ ( groups1932886352136224148al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_909_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_910_sum__mono,axiom,
! [K2: set_list_nat,F: list_nat > real,G: list_nat > real] :
( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups8399112307953289288t_real @ F @ K2 ) @ ( groups8399112307953289288t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_911_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_912_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_913_sum__mono,axiom,
! [K2: set_list_nat,F: list_nat > extend8495563244428889912nnreal,G: list_nat > extend8495563244428889912nnreal] :
( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ K2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups5253920722037313236nnreal @ F @ K2 ) @ ( groups5253920722037313236nnreal @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_914_sum__mono,axiom,
! [K2: set_real,F: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4232809223866053280nnreal @ F @ K2 ) @ ( groups4232809223866053280nnreal @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_915_sum__mono,axiom,
! [K2: set_nat,F: nat > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ K2 ) @ ( groups4868793261593263428nnreal @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_916_ln__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_917_ln__ge__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_iff
thf(fact_918_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_eq_int @ A @ I4 )
& ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_919_ln__less__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X2 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_920_ln__inj__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X2 )
= ( ln_ln_real @ Y ) )
= ( X2 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_921_ln__le__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_922_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= zero_zero_real )
= ( X2 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_923_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_924_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_925_ln__less__self,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_less_self
thf(fact_926_ln__bound,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_bound
thf(fact_927_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_928_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_929_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_930_ln__ge__zero__imp__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_931_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_932_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_933__092_060delta_062__range,axiom,
ord_less_rat @ zero_zero_rat @ delta ).
% \<delta>_range
thf(fact_934_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_935_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_936_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_int @ A @ I4 )
& ( ord_less_int @ I4 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_937_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_938_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_939_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_940_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_eq_int @ A @ I4 )
& ( ord_less_int @ I4 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_941_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_int @ A @ I4 )
& ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_942_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_943_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_944_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_945_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_946_nat__mult__le__cancel__disj,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 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_947_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_948_ennreal__less__zero__iff,axiom,
! [X2: real] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% ennreal_less_zero_iff
thf(fact_949_one__less__ennreal,axiom,
! [X2: real] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ).
% one_less_ennreal
thf(fact_950_ennreal__less__one__iff,axiom,
! [X2: real] :
( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_real @ X2 @ one_one_real ) ) ).
% ennreal_less_one_iff
thf(fact_951_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_952_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_953_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_954_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_955_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_956_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_957_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_958_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_959_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_960_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_961_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_962_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_963_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_964_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_965_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_966_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P2 @ M2 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_967_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_968_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_969_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% int.lless_eq
thf(fact_970_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_971_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_972_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_973_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_974_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_975_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_976_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_977_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_978_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_979_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_980_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
thf(fact_981_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_982_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_983_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_984_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_985_ennreal__zero__less__mult__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( times_1893300245718287421nnreal @ A @ B ) )
= ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A )
& ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B ) ) ) ).
% ennreal_zero_less_mult_iff
thf(fact_986_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_987_ennreal__lessI,axiom,
! [Q: real,R: real] :
( ( ord_less_real @ zero_zero_real @ Q )
=> ( ( ord_less_real @ R @ Q )
=> ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q ) ) ) ) ).
% ennreal_lessI
thf(fact_988_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_989_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_990_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_991_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_992_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_993_ennreal__less__iff,axiom,
! [R: real,Q: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q ) )
= ( ord_less_real @ R @ Q ) ) ) ).
% ennreal_less_iff
thf(fact_994_ennreal__approx__unit,axiom,
! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A4 )
=> ( ( ord_le7381754540660121996nnreal @ A4 @ one_on2969667320475766781nnreal )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A4 @ Z2 ) @ Y ) ) )
=> ( ord_le3935885782089961368nnreal @ Z2 @ Y ) ) ).
% ennreal_approx_unit
thf(fact_995_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_996_seq__mono__lemma,axiom,
! [M: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_real @ ( D2 @ N4 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_997_real__divl__pos__less1__bound,axiom,
! [X2: real,Prec: nat] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( real_divl @ Prec @ one_one_real @ X2 ) ) ) ) ).
% real_divl_pos_less1_bound
thf(fact_998_s1__gt__0,axiom,
ord_less_nat @ zero_zero_nat @ ( frequency_Moment_s_1 @ delta ) ).
% s1_gt_0
thf(fact_999_real__divl__lower__bound,axiom,
! [X2: real,Y: real,Prec: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_divl @ Prec @ X2 @ Y ) ) ) ) ).
% real_divl_lower_bound
thf(fact_1000_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1001_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( P2 @ K4 )
& ( ord_less_nat @ K4 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1002_bounded__Max__nat,axiom,
! [P2: nat > $o,X2: nat,M5: nat] :
( ( P2 @ X2 )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1003_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1004_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1005_real__divr__pos__less1__lower__bound,axiom,
! [X2: real,Prec: nat] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( real_divr @ Prec @ one_one_real @ X2 ) ) ) ) ).
% real_divr_pos_less1_lower_bound
thf(fact_1006_real__divr__nonneg__neg__upper__bound,axiom,
! [X2: real,Y: real,Prec: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_divr @ Prec @ X2 @ Y ) @ zero_zero_real ) ) ) ).
% real_divr_nonneg_neg_upper_bound
thf(fact_1007_real__divr__nonpos__pos__upper__bound,axiom,
! [X2: real,Y: real,Prec: nat] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( real_divr @ Prec @ X2 @ Y ) @ zero_zero_real ) ) ) ).
% real_divr_nonpos_pos_upper_bound
thf(fact_1008_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P2 @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P2 @ I3 ) )
=> ( P2 @ K3 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_1009_ennreal__minus__zero,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% ennreal_minus_zero
thf(fact_1010_zero__minus__ennreal,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% zero_minus_ennreal
thf(fact_1011_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1012_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1013_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1014_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1015_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1016_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1017_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1018_ennreal__minus__if,axiom,
! [A: real,B: real] :
( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
= ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( if_real @ ( ord_less_eq_real @ B @ A ) @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) @ A ) ) ) ).
% ennreal_minus_if
thf(fact_1019_ennreal__minus,axiom,
! [Q: real,R: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q )
=> ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q ) )
= ( extend7643940197134561352nnreal @ ( minus_minus_real @ R @ Q ) ) ) ) ).
% ennreal_minus
thf(fact_1020_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1021_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1022_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1023_nat__distrib_I4_J,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% nat_distrib(4)
thf(fact_1024_nat__distrib_I3_J,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% nat_distrib(3)
thf(fact_1025_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1026_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1027_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_1028_ennreal__minus__eq__0,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A @ B ) ) ).
% ennreal_minus_eq_0
thf(fact_1029_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1030_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1031_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1032_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1033_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1034_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1035_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1036_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N3: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1037_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1038_ennreal__minus__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ D2 @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ C @ D2 ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1039_ennreal__mono__minus,axiom,
! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).
% ennreal_mono_minus
thf(fact_1040_diff__le__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ A ) ).
% diff_le_self_ennreal
thf(fact_1041_ennreal__diff__le__mono__left,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1042_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1043_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1044_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1045_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1046_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1047_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_1048_zdiff__int__split,axiom,
! [P2: int > $o,X2: nat,Y: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1049_diff__gr0__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ).
% diff_gr0_ennreal
thf(fact_1050_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1051_minusinfinity,axiom,
! [D2: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1052_plusinfinity,axiom,
! [D2: int,P4: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_1: int] : ( P4 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1053_ln__eq__minus__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= ( minus_minus_real @ X2 @ one_one_real ) )
=> ( X2 = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1054_decr__mult__lemma,axiom,
! [D2: int,P2: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( minus_minus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1055_ln__le__minus__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1056_Bolzano,axiom,
! [A: real,B: real,P2: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B4: real,C4: real] :
( ( P2 @ A4 @ B4 )
=> ( ( P2 @ B4 @ C4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C4 )
=> ( P2 @ A4 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
=> ( P2 @ A4 @ B4 ) ) ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1057_diff__diff__commute__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% diff_diff_commute_ennreal
thf(fact_1058_ln__one__minus__pos__upper__bound,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1059_int_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% int.nat_pow_one
thf(fact_1060_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1061_int_Onat__pow__0,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
% Helper facts (11)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X2: rat,Y: rat] :
( ( if_rat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X2: rat,Y: rat] :
( ( if_rat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups6723090944982001619t_real
@ ( map_list_nat_real
@ ^ [P3: list_nat] :
( times_times_real
@ ( zero_n3304061248610475627l_real
@ ( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ P3 ) ) )
@ one_one_real )
@ ( equiva7426478223624825838m_rgfs @ na ) ) )
= one_one_real ) ).
%------------------------------------------------------------------------------