TPTP Problem File: SLH0109^1.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 : Fishers_Inequality/0034_Incidence_Matrices/prob_00376_016513__27983326_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1487 ( 589 unt; 210 typ; 0 def)
% Number of atoms : 3657 (1370 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9586 ( 346 ~; 109 |; 134 &;7470 @)
% ( 0 <=>;1527 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 490 ( 490 >; 0 *; 0 +; 0 <<)
% Number of symbols : 184 ( 181 usr; 17 con; 0-3 aty)
% Number of variables : 3118 ( 175 ^;2819 !; 124 ?;3118 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:47:57.446
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Real__Oreal_J_J,type,
list_vec_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Matrix__Ovec_Itf__b_J_J,type,
multiset_vec_b: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Real__Oreal_J_J,type,
set_vec_real: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
list_vec_nat: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Int__Oint_J_J,type,
list_vec_int: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
set_vec_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
set_vec_int: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_Itf__b_J_J,type,
list_vec_b: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__b_J_J,type,
set_vec_b: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Real__Oreal_J,type,
multiset_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
multiset_int: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Real__Oreal_J,type,
vec_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Int__Oint_J,type,
vec_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $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__Matrix__Ovec_Itf__b_J,type,
vec_b: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__b_J,type,
mat_b: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (181)
thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
archim2898591450579166408c_real: real > real ).
thf(sy_c_General_Ofilter__min__append_001t__Matrix__Ovec_Itf__b_J,type,
filter4530267383450833409_vec_b: ( vec_b > vec_b > $o ) > list_vec_b > list_vec_b > list_vec_b ).
thf(sy_c_General_Ofilter__min__append_001t__Nat__Onat,type,
filter1442860272890367977nd_nat: ( nat > nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_General_Ofilter__min__append_001t__Real__Oreal,type,
filter2283823222820419141d_real: ( real > real > $o ) > list_real > list_real > list_real ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001tf__b,type,
one_one_b: b ).
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__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_It__Int__Oint_J,type,
uminus8720015189474472720ec_int: vec_int > vec_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_It__Real__Oreal_J,type,
uminus8989278663012614928c_real: vec_real > vec_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__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Int__Oint_J,type,
zero_z3170743180189231877et_int: multiset_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Matrix__Ovec_Itf__b_J_J,type,
zero_z1737153457778485697_vec_b: multiset_vec_b ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
zero_z8811559133707751557t_real: multiset_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__b,type,
zero_zero_b: b ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001tf__b,type,
if_b: $o > b > b > b ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Int__Oint,type,
incide4964164200581851450ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Nat__Onat,type,
incide4966654671090901726ix_nat: mat_nat > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Real__Oreal,type,
incide4475037519619858106x_real: mat_real > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001tf__b,type,
incide7367983062745021297trix_b: mat_b > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Int__Oint,type,
incide3973235006681262014ck_int: vec_int > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Nat__Onat,type,
incide3975725477190312290ck_nat: vec_nat > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Real__Oreal,type,
incide970706021007448894k_real: vec_real > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001tf__b,type,
incide5355957740755015149lock_b: vec_b > set_nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_List_OListMem_001t__Matrix__Ovec_Itf__b_J,type,
listMem_vec_b: vec_b > list_vec_b > $o ).
thf(sy_c_List_OListMem_001t__Nat__Onat,type,
listMem_nat: nat > list_nat > $o ).
thf(sy_c_List_OListMem_001t__Real__Oreal,type,
listMem_real: real > list_real > $o ).
thf(sy_c_List_Ocan__select_001t__Matrix__Ovec_Itf__b_J,type,
can_select_vec_b: ( vec_b > $o ) > set_vec_b > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Real__Oreal,type,
can_select_real: ( real > $o ) > set_real > $o ).
thf(sy_c_List_Oinsert_001t__Matrix__Ovec_Itf__b_J,type,
insert_vec_b: vec_b > list_vec_b > list_vec_b ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
insert_real: real > list_real > list_real ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Int__Oint_J,type,
set_vec_int2: list_vec_int > set_vec_int ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Nat__Onat_J,type,
set_vec_nat2: list_vec_nat > set_vec_nat ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Real__Oreal_J,type,
set_vec_real2: list_vec_real > set_vec_real ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_Itf__b_J,type,
set_vec_b2: list_vec_b > set_vec_b ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist__ex1_001t__Matrix__Ovec_Itf__b_J,type,
list_ex1_vec_b: ( vec_b > $o ) > list_vec_b > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Omember_001t__Matrix__Ovec_Itf__b_J,type,
member_vec_b: list_vec_b > vec_b > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Real__Oreal,type,
member_real: list_real > real > $o ).
thf(sy_c_List__Index_Olast__index_001t__Matrix__Ovec_Itf__b_J,type,
list_l8470360163612209885_vec_b: list_vec_b > vec_b > nat ).
thf(sy_c_List__Index_Olast__index_001t__Nat__Onat,type,
list_last_index_nat: list_nat > nat > nat ).
thf(sy_c_List__Index_Olast__index_001t__Real__Oreal,type,
list_last_index_real: list_real > real > nat ).
thf(sy_c_Matrix_Ocols_001t__Int__Oint,type,
cols_int: mat_int > list_vec_int ).
thf(sy_c_Matrix_Ocols_001t__Nat__Onat,type,
cols_nat: mat_nat > list_vec_nat ).
thf(sy_c_Matrix_Ocols_001t__Real__Oreal,type,
cols_real: mat_real > list_vec_real ).
thf(sy_c_Matrix_Ocols_001tf__b,type,
cols_b: mat_b > list_vec_b ).
thf(sy_c_Matrix_Odim__vec_001t__Int__Oint,type,
dim_vec_int: vec_int > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Nat__Onat,type,
dim_vec_nat: vec_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Real__Oreal,type,
dim_vec_real: vec_real > nat ).
thf(sy_c_Matrix_Odim__vec_001tf__b,type,
dim_vec_b: vec_b > nat ).
thf(sy_c_Matrix_Osmult__vec_001t__Int__Oint,type,
smult_vec_int: int > vec_int > vec_int ).
thf(sy_c_Matrix_Osmult__vec_001t__Nat__Onat,type,
smult_vec_nat: nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Osmult__vec_001t__Real__Oreal,type,
smult_vec_real: real > vec_real > vec_real ).
thf(sy_c_Matrix_Ounit__vec_001t__Int__Oint,type,
unit_vec_int: nat > nat > vec_int ).
thf(sy_c_Matrix_Ounit__vec_001t__Nat__Onat,type,
unit_vec_nat: nat > nat > vec_nat ).
thf(sy_c_Matrix_Ounit__vec_001t__Real__Oreal,type,
unit_vec_real: nat > nat > vec_real ).
thf(sy_c_Matrix_Ounit__vec_001tf__b,type,
unit_vec_b: nat > nat > vec_b ).
thf(sy_c_Matrix_Ounit__vecs__first_001tf__b,type,
unit_vecs_first_b: nat > nat > list_vec_b ).
thf(sy_c_Matrix_Oupdate__vec_001t__Int__Oint,type,
update_vec_int: vec_int > nat > int > vec_int ).
thf(sy_c_Matrix_Oupdate__vec_001tf__b,type,
update_vec_b: vec_b > nat > b > vec_b ).
thf(sy_c_Matrix_Ovec__first_001t__Int__Oint,type,
vec_first_int: vec_int > nat > vec_int ).
thf(sy_c_Matrix_Ovec__first_001tf__b,type,
vec_first_b: vec_b > nat > vec_b ).
thf(sy_c_Matrix_Ovec__index_001t__Int__Oint,type,
vec_index_int: vec_int > nat > int ).
thf(sy_c_Matrix_Ovec__index_001t__Nat__Onat,type,
vec_index_nat: vec_nat > nat > nat ).
thf(sy_c_Matrix_Ovec__index_001t__Real__Oreal,type,
vec_index_real: vec_real > nat > real ).
thf(sy_c_Matrix_Ovec__index_001tf__b,type,
vec_index_b: vec_b > nat > b ).
thf(sy_c_Matrix_Ovec__last_001t__Int__Oint,type,
vec_last_int: vec_int > nat > vec_int ).
thf(sy_c_Matrix_Ovec__last_001tf__b,type,
vec_last_b: vec_b > nat > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Int__Oint,type,
matrix2748772424961467270ec_int: nat > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Nat__Onat,type,
matrix2751262895470517546ec_nat: nat > vec_nat ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Real__Oreal,type,
matrix5166576126360777478c_real: nat > vec_real ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001tf__b,type,
matrix8789069900454870053_vec_b: nat > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Int__Oint,type,
matrix3634415343793898042ec_int: vec_int > int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001t__Int__Oint_001t__Int__Oint,type,
matrix8301520909418075407nt_int: vec_int > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001t__Int__Oint_001tf__b,type,
matrix1865072738833226460_int_b: vec_int > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001tf__b_001t__Int__Oint,type,
matrix1311240063772730166_b_int: vec_b > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001tf__b_001tf__b,type,
matrix7059812428859951221ec_b_b: vec_b > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Int__Oint,type,
matrix1697308990001484774nt_int: int > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Nat__Onat,type,
matrix1699799460510535050nt_nat: int > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Real__Oreal,type,
matrix1706393078865277798t_real: int > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001tf__b,type,
matrix6038540757728371653_int_b: int > b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Int__Oint,type,
matrix697955278100430218at_int: nat > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Nat__Onat,type,
matrix700445748609480494at_nat: nat > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Real__Oreal,type,
matrix8742843541027031818t_real: nat > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001tf__b,type,
matrix8283685725398817569_nat_b: nat > b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Int__Oint,type,
matrix4084289606792104422al_int: real > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Nat__Onat,type,
matrix4086780077301154698al_nat: real > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Real__Oreal,type,
matrix3070681271257819494l_real: real > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001tf__b,type,
matrix6537263852557659589real_b: real > b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001t__Int__Oint,type,
matrix5484708082667875359_b_int: b > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001t__Nat__Onat,type,
matrix5487198553176925635_b_nat: b > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001t__Real__Oreal,type,
matrix2280091663418064671b_real: b > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001tf__b,type,
matrix4781043112069605324ne_b_b: b > b ).
thf(sy_c_Missing__List_Oremdups__sort_001t__Nat__Onat,type,
missin6101193410121742181rt_nat: list_nat > list_nat ).
thf(sy_c_Missing__List_Oremdups__sort_001t__Real__Oreal,type,
missin5757521373528793025t_real: list_real > list_real ).
thf(sy_c_Multiset_Oadd__mset_001t__Int__Oint,type,
add_mset_int: int > multiset_int > multiset_int ).
thf(sy_c_Multiset_Oadd__mset_001t__Matrix__Ovec_Itf__b_J,type,
add_mset_vec_b: vec_b > multiset_vec_b > multiset_vec_b ).
thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
add_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Real__Oreal,type,
add_mset_real: real > multiset_real > multiset_real ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Real__Oreal,type,
linord36121425647212990t_real: multiset_real > list_real ).
thf(sy_c_Multiset_Omset_001t__Matrix__Ovec_Itf__b_J,type,
mset_vec_b: list_vec_b > multiset_vec_b ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Real__Oreal,type,
mset_real: list_real > multiset_real ).
thf(sy_c_Multiset_Oset__mset_001t__Int__Oint,type,
set_mset_int: multiset_int > set_int ).
thf(sy_c_Multiset_Oset__mset_001t__Matrix__Ovec_Itf__b_J,type,
set_mset_vec_b: multiset_vec_b > set_vec_b ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Real__Oreal,type,
set_mset_real: multiset_real > set_real ).
thf(sy_c_Multiset__More_Olist__of__mset_001t__Matrix__Ovec_Itf__b_J,type,
multis2398011375292950429_vec_b: multiset_vec_b > list_vec_b ).
thf(sy_c_Multiset__More_Olist__of__mset_001t__Nat__Onat,type,
multis105632648212199813et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset__More_Olist__of__mset_001t__Real__Oreal,type,
multis3585236986679824865t_real: multiset_real > list_real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Int__Oint_J,type,
ord_le1599922481286804176et_int: multiset_int > multiset_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
ord_le7573655249420395216t_real: multiset_real > multiset_real > $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__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Int__Oint_J,type,
ord_less_eq_vec_int: vec_int > vec_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Nat__Onat_J,type,
ord_less_eq_vec_nat: vec_nat > vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Real__Oreal_J,type,
ord_less_eq_vec_real: vec_real > vec_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Int__Oint_J,type,
ord_le2424384866860593884et_int: multiset_int > multiset_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
ord_le2426415917361421532t_real: multiset_real > multiset_real > $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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__b_J_J,type,
ord_le4862985661309304830_vec_b: set_vec_b > set_vec_b > $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_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__b_J,type,
collect_vec_b: ( vec_b > $o ) > set_vec_b ).
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_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Int__Oint_J,type,
member_vec_int: vec_int > set_vec_int > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Nat__Onat_J,type,
member_vec_nat: vec_nat > set_vec_nat > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Real__Oreal_J,type,
member_vec_real: vec_real > set_vec_real > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__b_J,type,
member_vec_b2: vec_b > set_vec_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real2: real > set_real > $o ).
thf(sy_v_bl,type,
bl: set_nat ).
thf(sy_v_c____,type,
c: vec_b ).
thf(sy_v_matrix,type,
matrix: mat_b ).
% Relevant facts (1267)
thf(fact_0_bleq,axiom,
( bl
= ( incide5355957740755015149lock_b @ c ) ) ).
% bleq
thf(fact_1_zero__one__matrix__axioms,axiom,
incide7367983062745021297trix_b @ matrix ).
% zero_one_matrix_axioms
thf(fact_2__092_060open_062c_A_092_060in_062_D_Amset_A_Icols_AM_J_092_060close_062,axiom,
member_vec_b2 @ c @ ( set_mset_vec_b @ ( mset_vec_b @ ( cols_b @ matrix ) ) ) ).
% \<open>c \<in># mset (cols M)\<close>
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060lbrakk_062bl_A_061_Amap__col__to__block_Ac_059_Ac_A_092_060in_062_D_Amset_A_Icols_AM_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [C: vec_b] :
( ( bl
= ( incide5355957740755015149lock_b @ C ) )
=> ~ ( member_vec_b2 @ C @ ( set_mset_vec_b @ ( mset_vec_b @ ( cols_b @ matrix ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>c. \<lbrakk>bl = map_col_to_block c; c \<in># mset (cols M)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_remdups__sort_I2_J,axiom,
! [Xs: list_real] :
( ( set_real2 @ ( missin5757521373528793025t_real @ Xs ) )
= ( set_real2 @ Xs ) ) ).
% remdups_sort(2)
thf(fact_5_remdups__sort_I2_J,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( missin6101193410121742181rt_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% remdups_sort(2)
thf(fact_6_map__col__to__block__elem__not,axiom,
! [C2: vec_b,I: nat] :
( ( member_vec_b2 @ C2 @ ( set_vec_b2 @ ( cols_b @ matrix ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_b @ C2 ) )
=> ( ( ~ ( member_nat2 @ I @ ( incide5355957740755015149lock_b @ C2 ) ) )
= ( ( vec_index_b @ C2 @ I )
= zero_zero_b ) ) ) ) ).
% map_col_to_block_elem_not
thf(fact_7_mset__eq__setD,axiom,
! [Xs: list_real,Ys: list_real] :
( ( ( mset_real @ Xs )
= ( mset_real @ Ys ) )
=> ( ( set_real2 @ Xs )
= ( set_real2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_8_mset__eq__setD,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_9_mset__eq__setD,axiom,
! [Xs: list_vec_b,Ys: list_vec_b] :
( ( ( mset_vec_b @ Xs )
= ( mset_vec_b @ Ys ) )
=> ( ( set_vec_b2 @ Xs )
= ( set_vec_b2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_10_set__mset__mset,axiom,
! [Xs: list_real] :
( ( set_mset_real @ ( mset_real @ Xs ) )
= ( set_real2 @ Xs ) ) ).
% set_mset_mset
thf(fact_11_set__mset__mset,axiom,
! [Xs: list_nat] :
( ( set_mset_nat @ ( mset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_mset_mset
thf(fact_12_set__mset__mset,axiom,
! [Xs: list_vec_b] :
( ( set_mset_vec_b @ ( mset_vec_b @ Xs ) )
= ( set_vec_b2 @ Xs ) ) ).
% set_mset_mset
thf(fact_13_in__multiset__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_14_in__multiset__in__set,axiom,
! [X: real,Xs: list_real] :
( ( member_real2 @ X @ ( set_mset_real @ ( mset_real @ Xs ) ) )
= ( member_real2 @ X @ ( set_real2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_15_in__multiset__in__set,axiom,
! [X: vec_b,Xs: list_vec_b] :
( ( member_vec_b2 @ X @ ( set_mset_vec_b @ ( mset_vec_b @ Xs ) ) )
= ( member_vec_b2 @ X @ ( set_vec_b2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_16_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_17_in__set__member,axiom,
! [X: real,Xs: list_real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
= ( member_real @ Xs @ X ) ) ).
% in_set_member
thf(fact_18_in__set__member,axiom,
! [X: vec_b,Xs: list_vec_b] :
( ( member_vec_b2 @ X @ ( set_vec_b2 @ Xs ) )
= ( member_vec_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_19_list__ex1__iff,axiom,
( list_ex1_vec_b
= ( ^ [P: vec_b > $o,Xs2: list_vec_b] :
? [X2: vec_b] :
( ( member_vec_b2 @ X2 @ ( set_vec_b2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: vec_b] :
( ( ( member_vec_b2 @ Y @ ( set_vec_b2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_20_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P: real > $o,Xs2: list_real] :
? [X2: real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: real] :
( ( ( member_real2 @ Y @ ( set_real2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_21_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ ( set_nat2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_22_in__set__insert,axiom,
! [X: vec_b,Xs: list_vec_b] :
( ( member_vec_b2 @ X @ ( set_vec_b2 @ Xs ) )
=> ( ( insert_vec_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_23_in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_24_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_25_last__index__eq__index__conv,axiom,
! [X: vec_b,Xs: list_vec_b,Y2: vec_b] :
( ( ( member_vec_b2 @ X @ ( set_vec_b2 @ Xs ) )
| ( member_vec_b2 @ Y2 @ ( set_vec_b2 @ Xs ) ) )
=> ( ( ( list_l8470360163612209885_vec_b @ Xs @ X )
= ( list_l8470360163612209885_vec_b @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_26_last__index__eq__index__conv,axiom,
! [X: real,Xs: list_real,Y2: real] :
( ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
| ( member_real2 @ Y2 @ ( set_real2 @ Xs ) ) )
=> ( ( ( list_last_index_real @ Xs @ X )
= ( list_last_index_real @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_27_last__index__eq__index__conv,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
| ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) ) )
=> ( ( ( list_last_index_nat @ Xs @ X )
= ( list_last_index_nat @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_28_ListMem__iff,axiom,
( listMem_vec_b
= ( ^ [X2: vec_b,Xs2: list_vec_b] : ( member_vec_b2 @ X2 @ ( set_vec_b2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_29_ListMem__iff,axiom,
( listMem_real
= ( ^ [X2: real,Xs2: list_real] : ( member_real2 @ X2 @ ( set_real2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_30_ListMem__iff,axiom,
( listMem_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_31_filter__min__append__minimal,axiom,
! [Xs: list_vec_b,Rel: vec_b > vec_b > $o,Ys: list_vec_b,X: vec_b,Y2: vec_b] :
( ! [X3: vec_b,Y3: vec_b] :
( ( member_vec_b2 @ X3 @ ( set_vec_b2 @ Xs ) )
=> ( ( member_vec_b2 @ Y3 @ ( set_vec_b2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: vec_b,Y3: vec_b] :
( ( member_vec_b2 @ X3 @ ( set_vec_b2 @ Ys ) )
=> ( ( member_vec_b2 @ Y3 @ ( set_vec_b2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_vec_b2 @ X @ ( set_vec_b2 @ ( filter4530267383450833409_vec_b @ Rel @ Xs @ Ys ) ) )
=> ( ( member_vec_b2 @ Y2 @ ( set_vec_b2 @ ( filter4530267383450833409_vec_b @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_32_filter__min__append__minimal,axiom,
! [Xs: list_real,Rel: real > real > $o,Ys: list_real,X: real,Y2: real] :
( ! [X3: real,Y3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ( ( member_real2 @ Y3 @ ( set_real2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: real,Y3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Ys ) )
=> ( ( member_real2 @ Y3 @ ( set_real2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_real2 @ X @ ( set_real2 @ ( filter2283823222820419141d_real @ Rel @ Xs @ Ys ) ) )
=> ( ( member_real2 @ Y2 @ ( set_real2 @ ( filter2283823222820419141d_real @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_33_filter__min__append__minimal,axiom,
! [Xs: list_nat,Rel: nat > nat > $o,Ys: list_nat,X: nat,Y2: nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( member_nat2 @ Y3 @ ( set_nat2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ ( filter1442860272890367977nd_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ ( filter1442860272890367977nd_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_34_in__map__col__valid__index,axiom,
! [I: nat,C2: vec_int] :
( ( member_nat2 @ I @ ( incide3973235006681262014ck_int @ C2 ) )
=> ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) ) ) ).
% in_map_col_valid_index
thf(fact_35_in__map__col__valid__index,axiom,
! [I: nat,C2: vec_b] :
( ( member_nat2 @ I @ ( incide5355957740755015149lock_b @ C2 ) )
=> ( ord_less_nat @ I @ ( dim_vec_b @ C2 ) ) ) ).
% in_map_col_valid_index
thf(fact_36_zero__one__matrix_Oin__map__col__valid__index,axiom,
! [Matrix: mat_b,I: nat,C2: vec_int] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( member_nat2 @ I @ ( incide3973235006681262014ck_int @ C2 ) )
=> ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index
thf(fact_37_zero__one__matrix_Oin__map__col__valid__index,axiom,
! [Matrix: mat_b,I: nat,C2: vec_b] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( member_nat2 @ I @ ( incide5355957740755015149lock_b @ C2 ) )
=> ( ord_less_nat @ I @ ( dim_vec_b @ C2 ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index
thf(fact_38_ex__mset,axiom,
! [X4: multiset_vec_b] :
? [Xs3: list_vec_b] :
( ( mset_vec_b @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_39_ex__mset,axiom,
! [X4: multiset_real] :
? [Xs3: list_real] :
( ( mset_real @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_40_ex__mset,axiom,
! [X4: multiset_nat] :
? [Xs3: list_nat] :
( ( mset_nat @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_41_zero__one__matrix_Omap__col__to__block__elem__not,axiom,
! [Matrix: mat_nat,C2: vec_nat,I: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( member_vec_nat @ C2 @ ( set_vec_nat2 @ ( cols_nat @ Matrix ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_nat @ C2 ) )
=> ( ( ~ ( member_nat2 @ I @ ( incide3975725477190312290ck_nat @ C2 ) ) )
= ( ( vec_index_nat @ C2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem_not
thf(fact_42_zero__one__matrix_Omap__col__to__block__elem__not,axiom,
! [Matrix: mat_int,C2: vec_int,I: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( member_vec_int @ C2 @ ( set_vec_int2 @ ( cols_int @ Matrix ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) )
=> ( ( ~ ( member_nat2 @ I @ ( incide3973235006681262014ck_int @ C2 ) ) )
= ( ( vec_index_int @ C2 @ I )
= zero_zero_int ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem_not
thf(fact_43_zero__one__matrix_Omap__col__to__block__elem__not,axiom,
! [Matrix: mat_real,C2: vec_real,I: nat] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ( member_vec_real @ C2 @ ( set_vec_real2 @ ( cols_real @ Matrix ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_real @ C2 ) )
=> ( ( ~ ( member_nat2 @ I @ ( incide970706021007448894k_real @ C2 ) ) )
= ( ( vec_index_real @ C2 @ I )
= zero_zero_real ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem_not
thf(fact_44_zero__one__matrix_Omap__col__to__block__elem__not,axiom,
! [Matrix: mat_b,C2: vec_b,I: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( member_vec_b2 @ C2 @ ( set_vec_b2 @ ( cols_b @ Matrix ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_b @ C2 ) )
=> ( ( ~ ( member_nat2 @ I @ ( incide5355957740755015149lock_b @ C2 ) ) )
= ( ( vec_index_b @ C2 @ I )
= zero_zero_b ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem_not
thf(fact_45_eq__vecI,axiom,
! [W: vec_b,V: vec_b] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_vec_b @ W ) )
=> ( ( vec_index_b @ V @ I2 )
= ( vec_index_b @ W @ I2 ) ) )
=> ( ( ( dim_vec_b @ V )
= ( dim_vec_b @ W ) )
=> ( V = W ) ) ) ).
% eq_vecI
thf(fact_46_eq__vecI,axiom,
! [W: vec_int,V: vec_int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_vec_int @ W ) )
=> ( ( vec_index_int @ V @ I2 )
= ( vec_index_int @ W @ I2 ) ) )
=> ( ( ( dim_vec_int @ V )
= ( dim_vec_int @ W ) )
=> ( V = W ) ) ) ).
% eq_vecI
thf(fact_47_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_48_map__col__to__block__elem,axiom,
! [I: nat,C2: vec_nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide3975725477190312290ck_nat @ C2 ) )
= ( ( vec_index_nat @ C2 @ I )
= one_one_nat ) ) ) ).
% map_col_to_block_elem
thf(fact_49_map__col__to__block__elem,axiom,
! [I: nat,C2: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide3973235006681262014ck_int @ C2 ) )
= ( ( vec_index_int @ C2 @ I )
= one_one_int ) ) ) ).
% map_col_to_block_elem
thf(fact_50_map__col__to__block__elem,axiom,
! [I: nat,C2: vec_real] :
( ( ord_less_nat @ I @ ( dim_vec_real @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide970706021007448894k_real @ C2 ) )
= ( ( vec_index_real @ C2 @ I )
= one_one_real ) ) ) ).
% map_col_to_block_elem
thf(fact_51_map__col__to__block__elem,axiom,
! [I: nat,C2: vec_b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide5355957740755015149lock_b @ C2 ) )
= ( ( vec_index_b @ C2 @ I )
= one_one_b ) ) ) ).
% map_col_to_block_elem
thf(fact_52_vec__eq__iff,axiom,
( ( ^ [Y4: vec_b,Z: vec_b] : ( Y4 = Z ) )
= ( ^ [X2: vec_b,Y: vec_b] :
( ( ( dim_vec_b @ X2 )
= ( dim_vec_b @ Y ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_b @ Y ) )
=> ( ( vec_index_b @ X2 @ I3 )
= ( vec_index_b @ Y @ I3 ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_53_vec__eq__iff,axiom,
( ( ^ [Y4: vec_int,Z: vec_int] : ( Y4 = Z ) )
= ( ^ [X2: vec_int,Y: vec_int] :
( ( ( dim_vec_int @ X2 )
= ( dim_vec_int @ Y ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_int @ Y ) )
=> ( ( vec_index_int @ X2 @ I3 )
= ( vec_index_int @ Y @ I3 ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_54_can__select__set__list__ex1,axiom,
! [P2: vec_b > $o,A: list_vec_b] :
( ( can_select_vec_b @ P2 @ ( set_vec_b2 @ A ) )
= ( list_ex1_vec_b @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_55_can__select__set__list__ex1,axiom,
! [P2: real > $o,A: list_real] :
( ( can_select_real @ P2 @ ( set_real2 @ A ) )
= ( list_ex1_real @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_56_can__select__set__list__ex1,axiom,
! [P2: nat > $o,A: list_nat] :
( ( can_select_nat @ P2 @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_57_ex__gt__imp__less__multiset,axiom,
! [N2: multiset_nat,M: multiset_nat] :
( ? [Y5: nat] :
( ( member_nat2 @ Y5 @ ( set_mset_nat @ N2 ) )
& ! [X5: nat] :
( ( member_nat2 @ X5 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ X5 @ Y5 ) ) )
=> ( ord_le5777773500796000884et_nat @ M @ N2 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_58_ex__gt__imp__less__multiset,axiom,
! [N2: multiset_int,M: multiset_int] :
( ? [Y5: int] :
( ( member_int @ Y5 @ ( set_mset_int @ N2 ) )
& ! [X5: int] :
( ( member_int @ X5 @ ( set_mset_int @ M ) )
=> ( ord_less_int @ X5 @ Y5 ) ) )
=> ( ord_le1599922481286804176et_int @ M @ N2 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_59_ex__gt__imp__less__multiset,axiom,
! [N2: multiset_real,M: multiset_real] :
( ? [Y5: real] :
( ( member_real2 @ Y5 @ ( set_mset_real @ N2 ) )
& ! [X5: real] :
( ( member_real2 @ X5 @ ( set_mset_real @ M ) )
=> ( ord_less_real @ X5 @ Y5 ) ) )
=> ( ord_le7573655249420395216t_real @ M @ N2 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_60_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_61_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_62_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_63_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_64_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_65_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_66_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_67_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_68_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_69_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_70_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_71_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_72_can__select__def,axiom,
( can_select_vec_b
= ( ^ [P: vec_b > $o,A2: set_vec_b] :
? [X2: vec_b] :
( ( member_vec_b2 @ X2 @ A2 )
& ( P @ X2 )
& ! [Y: vec_b] :
( ( ( member_vec_b2 @ Y @ A2 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_73_can__select__def,axiom,
( can_select_nat
= ( ^ [P: nat > $o,A2: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( P @ X2 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ A2 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_74_can__select__def,axiom,
( can_select_real
= ( ^ [P: real > $o,A2: set_real] :
? [X2: real] :
( ( member_real2 @ X2 @ A2 )
& ( P @ X2 )
& ! [Y: real] :
( ( ( member_real2 @ Y @ A2 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_75_multiset__nonemptyE,axiom,
! [A: multiset_vec_b] :
( ( A != zero_z1737153457778485697_vec_b )
=> ~ ! [X5: vec_b] :
~ ( member_vec_b2 @ X5 @ ( set_mset_vec_b @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_76_multiset__nonemptyE,axiom,
! [A: multiset_real] :
( ( A != zero_z8811559133707751557t_real )
=> ~ ! [X5: real] :
~ ( member_real2 @ X5 @ ( set_mset_real @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_77_multiset__nonemptyE,axiom,
! [A: multiset_nat] :
( ( A != zero_z7348594199698428585et_nat )
=> ~ ! [X5: nat] :
~ ( member_nat2 @ X5 @ ( set_mset_nat @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_78_mem__Collect__eq,axiom,
! [A3: vec_b,P2: vec_b > $o] :
( ( member_vec_b2 @ A3 @ ( collect_vec_b @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat2 @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A3: real,P2: real > $o] :
( ( member_real2 @ A3 @ ( collect_real @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A: set_vec_b] :
( ( collect_vec_b
@ ^ [X2: vec_b] : ( member_vec_b2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_84_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_85_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_86_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_87_zero__reorient,axiom,
! [X: b] :
( ( zero_zero_b = X )
= ( X = zero_zero_b ) ) ).
% zero_reorient
thf(fact_88_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_89_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_90_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_91_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C2: vec_nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_nat @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide3975725477190312290ck_nat @ C2 ) )
= ( ( vec_index_nat @ C2 @ I )
= one_one_nat ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_92_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C2: vec_int] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide3973235006681262014ck_int @ C2 ) )
= ( ( vec_index_int @ C2 @ I )
= one_one_int ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_93_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C2: vec_real] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_real @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide970706021007448894k_real @ C2 ) )
= ( ( vec_index_real @ C2 @ I )
= one_one_real ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_94_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C2: vec_b] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_b @ C2 ) )
=> ( ( member_nat2 @ I @ ( incide5355957740755015149lock_b @ C2 ) )
= ( ( vec_index_b @ C2 @ I )
= one_one_b ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_95_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_96_index__update__vec1,axiom,
! [I: nat,V: vec_b,A3: b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( ( vec_index_b @ ( update_vec_b @ V @ I @ A3 ) @ I )
= A3 ) ) ).
% index_update_vec1
thf(fact_97_index__update__vec1,axiom,
! [I: nat,V: vec_int,A3: int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ ( update_vec_int @ V @ I @ A3 ) @ I )
= A3 ) ) ).
% index_update_vec1
thf(fact_98_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_99_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_100_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_101_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_102_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_103_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_104_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_105_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_106_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_107_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_108_zero__neq__one,axiom,
zero_zero_b != one_one_b ).
% zero_neq_one
thf(fact_109_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_110_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_111_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_112_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_113_index__update__vec2,axiom,
! [I4: nat,I: nat,V: vec_b,A3: b] :
( ( I4 != I )
=> ( ( vec_index_b @ ( update_vec_b @ V @ I @ A3 ) @ I4 )
= ( vec_index_b @ V @ I4 ) ) ) ).
% index_update_vec2
thf(fact_114_index__update__vec2,axiom,
! [I4: nat,I: nat,V: vec_int,A3: int] :
( ( I4 != I )
=> ( ( vec_index_int @ ( update_vec_int @ V @ I @ A3 ) @ I4 )
= ( vec_index_int @ V @ I4 ) ) ) ).
% index_update_vec2
thf(fact_115_dim__update__vec,axiom,
! [V: vec_b,I: nat,A3: b] :
( ( dim_vec_b @ ( update_vec_b @ V @ I @ A3 ) )
= ( dim_vec_b @ V ) ) ).
% dim_update_vec
thf(fact_116_dim__update__vec,axiom,
! [V: vec_int,I: nat,A3: int] :
( ( dim_vec_int @ ( update_vec_int @ V @ I @ A3 ) )
= ( dim_vec_int @ V ) ) ).
% dim_update_vec
thf(fact_117_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_118_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_119_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_120_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_121_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_122_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_123_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_124_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_125_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_126_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_127_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_128_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_129_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_130_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_131_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_132_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_133_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_134_dim__vec__last,axiom,
! [V: vec_b,N: nat] :
( ( dim_vec_b @ ( vec_last_b @ V @ N ) )
= N ) ).
% dim_vec_last
thf(fact_135_dim__vec__last,axiom,
! [V: vec_int,N: nat] :
( ( dim_vec_int @ ( vec_last_int @ V @ N ) )
= N ) ).
% dim_vec_last
thf(fact_136_dim__vec__first,axiom,
! [V: vec_b,N: nat] :
( ( dim_vec_b @ ( vec_first_b @ V @ N ) )
= N ) ).
% dim_vec_first
thf(fact_137_dim__vec__first,axiom,
! [V: vec_int,N: nat] :
( ( dim_vec_int @ ( vec_first_int @ V @ N ) )
= N ) ).
% dim_vec_first
thf(fact_138_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( J = I )
=> ( ( vec_index_b @ ( unit_vec_b @ N @ I ) @ J )
= one_one_b ) )
& ( ( J != I )
=> ( ( vec_index_b @ ( unit_vec_b @ N @ I ) @ J )
= zero_zero_b ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_139_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( J = I )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I ) @ J )
= one_one_nat ) )
& ( ( J != I )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I ) @ J )
= zero_zero_nat ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_140_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( J = I )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I ) @ J )
= one_one_int ) )
& ( ( J != I )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I ) @ J )
= zero_zero_int ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_141_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( J = I )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I ) @ J )
= one_one_real ) )
& ( ( J != I )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I ) @ J )
= zero_zero_real ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_142_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_143_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_144_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_b @ ( matrix8789069900454870053_vec_b @ N ) @ I )
= one_one_b ) ) ).
% all_ones_index
thf(fact_145_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_nat @ ( matrix2751262895470517546ec_nat @ N ) @ I )
= one_one_nat ) ) ).
% all_ones_index
thf(fact_146_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_int @ ( matrix2748772424961467270ec_int @ N ) @ I )
= one_one_int ) ) ).
% all_ones_index
thf(fact_147_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_real @ ( matrix5166576126360777478c_real @ N ) @ I )
= one_one_real ) ) ).
% all_ones_index
thf(fact_148_set__sorted__list__of__multiset,axiom,
! [M: multiset_real] :
( ( set_real2 @ ( linord36121425647212990t_real @ M ) )
= ( set_mset_real @ M ) ) ).
% set_sorted_list_of_multiset
thf(fact_149_set__sorted__list__of__multiset,axiom,
! [M: multiset_nat] :
( ( set_nat2 @ ( linord3047872887403683810et_nat @ M ) )
= ( set_mset_nat @ M ) ) ).
% set_sorted_list_of_multiset
thf(fact_150_mset__lt__single__right__iff,axiom,
! [M: multiset_nat,Y2: nat] :
( ( ord_le5777773500796000884et_nat @ M @ ( add_mset_nat @ Y2 @ zero_z7348594199698428585et_nat ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ X2 @ Y2 ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_151_mset__lt__single__right__iff,axiom,
! [M: multiset_int,Y2: int] :
( ( ord_le1599922481286804176et_int @ M @ ( add_mset_int @ Y2 @ zero_z3170743180189231877et_int ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_mset_int @ M ) )
=> ( ord_less_int @ X2 @ Y2 ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_152_mset__lt__single__right__iff,axiom,
! [M: multiset_real,Y2: real] :
( ( ord_le7573655249420395216t_real @ M @ ( add_mset_real @ Y2 @ zero_z8811559133707751557t_real ) )
= ( ! [X2: real] :
( ( member_real2 @ X2 @ ( set_mset_real @ M ) )
=> ( ord_less_real @ X2 @ Y2 ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_153_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_b @ ( unit_vec_b @ N @ I ) @ I )
= one_one_b ) ) ).
% index_unit_vec(2)
thf(fact_154_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I ) @ I )
= one_one_nat ) ) ).
% index_unit_vec(2)
thf(fact_155_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I ) @ I )
= one_one_int ) ) ).
% index_unit_vec(2)
thf(fact_156_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I ) @ I )
= one_one_real ) ) ).
% index_unit_vec(2)
thf(fact_157_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_158_index__unit__vec_I3_J,axiom,
! [N: nat,I: nat] :
( ( dim_vec_b @ ( unit_vec_b @ N @ I ) )
= N ) ).
% index_unit_vec(3)
thf(fact_159_index__unit__vec_I3_J,axiom,
! [N: nat,I: nat] :
( ( dim_vec_int @ ( unit_vec_int @ N @ I ) )
= N ) ).
% index_unit_vec(3)
thf(fact_160_mset__sorted__list__of__multiset,axiom,
! [M: multiset_real] :
( ( mset_real @ ( linord36121425647212990t_real @ M ) )
= M ) ).
% mset_sorted_list_of_multiset
thf(fact_161_mset__sorted__list__of__multiset,axiom,
! [M: multiset_nat] :
( ( mset_nat @ ( linord3047872887403683810et_nat @ M ) )
= M ) ).
% mset_sorted_list_of_multiset
thf(fact_162_dim__vec__all__ones,axiom,
! [N: nat] :
( ( dim_vec_b @ ( matrix8789069900454870053_vec_b @ N ) )
= N ) ).
% dim_vec_all_ones
thf(fact_163_dim__vec__all__ones,axiom,
! [N: nat] :
( ( dim_vec_int @ ( matrix2748772424961467270ec_int @ N ) )
= N ) ).
% dim_vec_all_ones
thf(fact_164_mset__lt__single__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y2 @ zero_z7348594199698428585et_nat ) )
= ( ord_less_nat @ X @ Y2 ) ) ).
% mset_lt_single_iff
thf(fact_165_mset__lt__single__iff,axiom,
! [X: int,Y2: int] :
( ( ord_le1599922481286804176et_int @ ( add_mset_int @ X @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ Y2 @ zero_z3170743180189231877et_int ) )
= ( ord_less_int @ X @ Y2 ) ) ).
% mset_lt_single_iff
thf(fact_166_mset__lt__single__iff,axiom,
! [X: real,Y2: real] :
( ( ord_le7573655249420395216t_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) @ ( add_mset_real @ Y2 @ zero_z8811559133707751557t_real ) )
= ( ord_less_real @ X @ Y2 ) ) ).
% mset_lt_single_iff
thf(fact_167_union__single__eq__member,axiom,
! [X: vec_b,M: multiset_vec_b,N2: multiset_vec_b] :
( ( ( add_mset_vec_b @ X @ M )
= N2 )
=> ( member_vec_b2 @ X @ ( set_mset_vec_b @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_168_union__single__eq__member,axiom,
! [X: real,M: multiset_real,N2: multiset_real] :
( ( ( add_mset_real @ X @ M )
= N2 )
=> ( member_real2 @ X @ ( set_mset_real @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_169_union__single__eq__member,axiom,
! [X: nat,M: multiset_nat,N2: multiset_nat] :
( ( ( add_mset_nat @ X @ M )
= N2 )
=> ( member_nat2 @ X @ ( set_mset_nat @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_170_insert__noteq__member,axiom,
! [B: vec_b,B2: multiset_vec_b,C2: vec_b,C3: multiset_vec_b] :
( ( ( add_mset_vec_b @ B @ B2 )
= ( add_mset_vec_b @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_vec_b2 @ C2 @ ( set_mset_vec_b @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_171_insert__noteq__member,axiom,
! [B: real,B2: multiset_real,C2: real,C3: multiset_real] :
( ( ( add_mset_real @ B @ B2 )
= ( add_mset_real @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_real2 @ C2 @ ( set_mset_real @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_172_insert__noteq__member,axiom,
! [B: nat,B2: multiset_nat,C2: nat,C3: multiset_nat] :
( ( ( add_mset_nat @ B @ B2 )
= ( add_mset_nat @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_nat2 @ C2 @ ( set_mset_nat @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_173_multi__member__split,axiom,
! [X: vec_b,M: multiset_vec_b] :
( ( member_vec_b2 @ X @ ( set_mset_vec_b @ M ) )
=> ? [A4: multiset_vec_b] :
( M
= ( add_mset_vec_b @ X @ A4 ) ) ) ).
% multi_member_split
thf(fact_174_multi__member__split,axiom,
! [X: real,M: multiset_real] :
( ( member_real2 @ X @ ( set_mset_real @ M ) )
=> ? [A4: multiset_real] :
( M
= ( add_mset_real @ X @ A4 ) ) ) ).
% multi_member_split
thf(fact_175_multi__member__split,axiom,
! [X: nat,M: multiset_nat] :
( ( member_nat2 @ X @ ( set_mset_nat @ M ) )
=> ? [A4: multiset_nat] :
( M
= ( add_mset_nat @ X @ A4 ) ) ) ).
% multi_member_split
thf(fact_176_mset__add,axiom,
! [A3: vec_b,A: multiset_vec_b] :
( ( member_vec_b2 @ A3 @ ( set_mset_vec_b @ A ) )
=> ~ ! [B3: multiset_vec_b] :
( A
!= ( add_mset_vec_b @ A3 @ B3 ) ) ) ).
% mset_add
thf(fact_177_mset__add,axiom,
! [A3: real,A: multiset_real] :
( ( member_real2 @ A3 @ ( set_mset_real @ A ) )
=> ~ ! [B3: multiset_real] :
( A
!= ( add_mset_real @ A3 @ B3 ) ) ) ).
% mset_add
thf(fact_178_mset__add,axiom,
! [A3: nat,A: multiset_nat] :
( ( member_nat2 @ A3 @ ( set_mset_nat @ A ) )
=> ~ ! [B3: multiset_nat] :
( A
!= ( add_mset_nat @ A3 @ B3 ) ) ) ).
% mset_add
thf(fact_179_multi__member__last,axiom,
! [X: vec_b] : ( member_vec_b2 @ X @ ( set_mset_vec_b @ ( add_mset_vec_b @ X @ zero_z1737153457778485697_vec_b ) ) ) ).
% multi_member_last
thf(fact_180_multi__member__last,axiom,
! [X: real] : ( member_real2 @ X @ ( set_mset_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) ) ).
% multi_member_last
thf(fact_181_multi__member__last,axiom,
! [X: nat] : ( member_nat2 @ X @ ( set_mset_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) ).
% multi_member_last
thf(fact_182_less__multiset__doubletons,axiom,
! [Y2: nat,T: nat,S: nat,X: nat] :
( ( ( ord_less_nat @ Y2 @ T )
| ( ord_less_nat @ Y2 @ S ) )
=> ( ( ( ord_less_nat @ X @ T )
| ( ord_less_nat @ X @ S ) )
=> ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ Y2 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ T @ ( add_mset_nat @ S @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_183_less__multiset__doubletons,axiom,
! [Y2: int,T: int,S: int,X: int] :
( ( ( ord_less_int @ Y2 @ T )
| ( ord_less_int @ Y2 @ S ) )
=> ( ( ( ord_less_int @ X @ T )
| ( ord_less_int @ X @ S ) )
=> ( ord_le1599922481286804176et_int @ ( add_mset_int @ Y2 @ ( add_mset_int @ X @ zero_z3170743180189231877et_int ) ) @ ( add_mset_int @ T @ ( add_mset_int @ S @ zero_z3170743180189231877et_int ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_184_less__multiset__doubletons,axiom,
! [Y2: real,T: real,S: real,X: real] :
( ( ( ord_less_real @ Y2 @ T )
| ( ord_less_real @ Y2 @ S ) )
=> ( ( ( ord_less_real @ X @ T )
| ( ord_less_real @ X @ S ) )
=> ( ord_le7573655249420395216t_real @ ( add_mset_real @ Y2 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) @ ( add_mset_real @ T @ ( add_mset_real @ S @ zero_z8811559133707751557t_real ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_185_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_186_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_187_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_188_lift__01__vec__simp_I1_J,axiom,
! [V: vec_b] :
( ( dim_vec_b @ ( matrix7059812428859951221ec_b_b @ V ) )
= ( dim_vec_b @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_189_lift__01__vec__simp_I1_J,axiom,
! [V: vec_int] :
( ( dim_vec_b @ ( matrix1865072738833226460_int_b @ V ) )
= ( dim_vec_int @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_190_lift__01__vec__simp_I1_J,axiom,
! [V: vec_b] :
( ( dim_vec_int @ ( matrix1311240063772730166_b_int @ V ) )
= ( dim_vec_b @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_191_lift__01__vec__simp_I1_J,axiom,
! [V: vec_int] :
( ( dim_vec_int @ ( matrix8301520909418075407nt_int @ V ) )
= ( dim_vec_int @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_192_all__ones__vec__smult,axiom,
! [I: nat,N: nat,K: int] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_int @ ( smult_vec_int @ K @ ( matrix2748772424961467270ec_int @ N ) ) @ I )
= K ) ) ).
% all_ones_vec_smult
thf(fact_193_mset__le__single__right__iff,axiom,
! [M: multiset_nat,Y2: nat] :
( ( ord_le6602235886369790592et_nat @ M @ ( add_mset_nat @ Y2 @ zero_z7348594199698428585et_nat ) )
= ( ( M
= ( add_mset_nat @ Y2 @ zero_z7348594199698428585et_nat ) )
| ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ X2 @ Y2 ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_194_mset__le__single__right__iff,axiom,
! [M: multiset_int,Y2: int] :
( ( ord_le2424384866860593884et_int @ M @ ( add_mset_int @ Y2 @ zero_z3170743180189231877et_int ) )
= ( ( M
= ( add_mset_int @ Y2 @ zero_z3170743180189231877et_int ) )
| ! [X2: int] :
( ( member_int @ X2 @ ( set_mset_int @ M ) )
=> ( ord_less_int @ X2 @ Y2 ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_195_mset__le__single__right__iff,axiom,
! [M: multiset_real,Y2: real] :
( ( ord_le2426415917361421532t_real @ M @ ( add_mset_real @ Y2 @ zero_z8811559133707751557t_real ) )
= ( ( M
= ( add_mset_real @ Y2 @ zero_z8811559133707751557t_real ) )
| ! [X2: real] :
( ( member_real2 @ X2 @ ( set_mset_real @ M ) )
=> ( ord_less_real @ X2 @ Y2 ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_196_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_197_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_198_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_199_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_200_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_201_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_202_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_203_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_204_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_205_one__smult__vec,axiom,
! [V: vec_int] :
( ( smult_vec_int @ one_one_int @ V )
= V ) ).
% one_smult_vec
thf(fact_206_one__smult__vec,axiom,
! [V: vec_real] :
( ( smult_vec_real @ one_one_real @ V )
= V ) ).
% one_smult_vec
thf(fact_207_scalar__vec__one,axiom,
! [V: vec_nat] :
( ( smult_vec_nat @ one_one_nat @ V )
= V ) ).
% scalar_vec_one
thf(fact_208_scalar__vec__one,axiom,
! [V: vec_int] :
( ( smult_vec_int @ one_one_int @ V )
= V ) ).
% scalar_vec_one
thf(fact_209_scalar__vec__one,axiom,
! [V: vec_real] :
( ( smult_vec_real @ one_one_real @ V )
= V ) ).
% scalar_vec_one
thf(fact_210_index__smult__vec_I2_J,axiom,
! [A3: int,V: vec_int] :
( ( dim_vec_int @ ( smult_vec_int @ A3 @ V ) )
= ( dim_vec_int @ V ) ) ).
% index_smult_vec(2)
thf(fact_211_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_212_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_213_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_214_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_215_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_216_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_217_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_218_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_219_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_220_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_221_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_222_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_223_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_224_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_225_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_226_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_227_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_228_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_229_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_230_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_231_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_232_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_233_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_234_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_235_mset__le__single__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_le6602235886369790592et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y2 @ zero_z7348594199698428585et_nat ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% mset_le_single_iff
thf(fact_236_mset__le__single__iff,axiom,
! [X: int,Y2: int] :
( ( ord_le2424384866860593884et_int @ ( add_mset_int @ X @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ Y2 @ zero_z3170743180189231877et_int ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% mset_le_single_iff
thf(fact_237_mset__le__single__iff,axiom,
! [X: real,Y2: real] :
( ( ord_le2426415917361421532t_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) @ ( add_mset_real @ Y2 @ zero_z8811559133707751557t_real ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% mset_le_single_iff
thf(fact_238_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_239_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_240_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_241_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_242_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_243_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_244_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_245_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_246_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_247_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_248_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_249_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_250_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_251_less__eq__vec__def,axiom,
( ord_less_eq_vec_nat
= ( ^ [V2: vec_nat,W2: vec_nat] :
( ( ( dim_vec_nat @ V2 )
= ( dim_vec_nat @ W2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_nat @ W2 ) )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V2 @ I3 ) @ ( vec_index_nat @ W2 @ I3 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_252_less__eq__vec__def,axiom,
( ord_less_eq_vec_int
= ( ^ [V2: vec_int,W2: vec_int] :
( ( ( dim_vec_int @ V2 )
= ( dim_vec_int @ W2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_int @ W2 ) )
=> ( ord_less_eq_int @ ( vec_index_int @ V2 @ I3 ) @ ( vec_index_int @ W2 @ I3 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_253_less__eq__vec__def,axiom,
( ord_less_eq_vec_real
= ( ^ [V2: vec_real,W2: vec_real] :
( ( ( dim_vec_real @ V2 )
= ( dim_vec_real @ W2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_real @ W2 ) )
=> ( ord_less_eq_real @ ( vec_index_real @ V2 @ I3 ) @ ( vec_index_real @ W2 @ I3 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_254_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_255_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_256_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_257_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_258_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_259_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_260_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_261_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_262_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_263_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_264_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_265_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_266_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_267_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_268_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_269_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_270_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_271_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_272_multiset__induct__min,axiom,
! [P2: multiset_nat > $o,M: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X5: nat,M4: multiset_nat] :
( ( P2 @ M4 )
=> ( ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_mset_nat @ M4 ) )
=> ( ord_less_eq_nat @ X5 @ Xa ) )
=> ( P2 @ ( add_mset_nat @ X5 @ M4 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_min
thf(fact_273_multiset__induct__min,axiom,
! [P2: multiset_int > $o,M: multiset_int] :
( ( P2 @ zero_z3170743180189231877et_int )
=> ( ! [X5: int,M4: multiset_int] :
( ( P2 @ M4 )
=> ( ! [Xa: int] :
( ( member_int @ Xa @ ( set_mset_int @ M4 ) )
=> ( ord_less_eq_int @ X5 @ Xa ) )
=> ( P2 @ ( add_mset_int @ X5 @ M4 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_min
thf(fact_274_multiset__induct__min,axiom,
! [P2: multiset_real > $o,M: multiset_real] :
( ( P2 @ zero_z8811559133707751557t_real )
=> ( ! [X5: real,M4: multiset_real] :
( ( P2 @ M4 )
=> ( ! [Xa: real] :
( ( member_real2 @ Xa @ ( set_mset_real @ M4 ) )
=> ( ord_less_eq_real @ X5 @ Xa ) )
=> ( P2 @ ( add_mset_real @ X5 @ M4 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_min
thf(fact_275_multiset__induct__max,axiom,
! [P2: multiset_nat > $o,M: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X5: nat,M4: multiset_nat] :
( ( P2 @ M4 )
=> ( ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_mset_nat @ M4 ) )
=> ( ord_less_eq_nat @ Xa @ X5 ) )
=> ( P2 @ ( add_mset_nat @ X5 @ M4 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_max
thf(fact_276_multiset__induct__max,axiom,
! [P2: multiset_int > $o,M: multiset_int] :
( ( P2 @ zero_z3170743180189231877et_int )
=> ( ! [X5: int,M4: multiset_int] :
( ( P2 @ M4 )
=> ( ! [Xa: int] :
( ( member_int @ Xa @ ( set_mset_int @ M4 ) )
=> ( ord_less_eq_int @ Xa @ X5 ) )
=> ( P2 @ ( add_mset_int @ X5 @ M4 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_max
thf(fact_277_multiset__induct__max,axiom,
! [P2: multiset_real > $o,M: multiset_real] :
( ( P2 @ zero_z8811559133707751557t_real )
=> ( ! [X5: real,M4: multiset_real] :
( ( P2 @ M4 )
=> ( ! [Xa: real] :
( ( member_real2 @ Xa @ ( set_mset_real @ M4 ) )
=> ( ord_less_eq_real @ Xa @ X5 ) )
=> ( P2 @ ( add_mset_real @ X5 @ M4 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_max
thf(fact_278_list__of__mset__exi,axiom,
! [M2: multiset_vec_b] :
? [L: list_vec_b] :
( M2
= ( mset_vec_b @ L ) ) ).
% list_of_mset_exi
thf(fact_279_list__of__mset__exi,axiom,
! [M2: multiset_real] :
? [L: list_real] :
( M2
= ( mset_real @ L ) ) ).
% list_of_mset_exi
thf(fact_280_list__of__mset__exi,axiom,
! [M2: multiset_nat] :
? [L: list_nat] :
( M2
= ( mset_nat @ L ) ) ).
% list_of_mset_exi
thf(fact_281_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_282_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_283_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_284_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_285_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_286_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_287_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( ( vec_index_b @ ( matrix7059812428859951221ec_b_b @ V ) @ I )
= ( matrix4781043112069605324ne_b_b @ ( vec_index_b @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_288_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_b @ ( matrix1865072738833226460_int_b @ V ) @ I )
= ( matrix6038540757728371653_int_b @ ( vec_index_int @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_289_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( ( vec_index_int @ ( matrix1311240063772730166_b_int @ V ) @ I )
= ( matrix5484708082667875359_b_int @ ( vec_index_b @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_290_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ ( matrix8301520909418075407nt_int @ V ) @ I )
= ( matrix1697308990001484774nt_int @ ( vec_index_int @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_291_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_292_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_293_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_294_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_295_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_296_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_297_minf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_298_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_299_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_300_minf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_301_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_302_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_303_of__zero__neq__one__0__iff,axiom,
! [X: b] :
( ( ( matrix4781043112069605324ne_b_b @ X )
= zero_zero_b )
= ( X = zero_zero_b ) ) ).
% of_zero_neq_one_0_iff
thf(fact_304_of__zero__neq__one__0__iff,axiom,
! [X: nat] :
( ( ( matrix8283685725398817569_nat_b @ X )
= zero_zero_b )
= ( X = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_305_of__zero__neq__one__0__iff,axiom,
! [X: int] :
( ( ( matrix6038540757728371653_int_b @ X )
= zero_zero_b )
= ( X = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_306_of__zero__neq__one__0__iff,axiom,
! [X: real] :
( ( ( matrix6537263852557659589real_b @ X )
= zero_zero_b )
= ( X = zero_zero_real ) ) ).
% of_zero_neq_one_0_iff
thf(fact_307_of__zero__neq__one__0__iff,axiom,
! [X: b] :
( ( ( matrix5487198553176925635_b_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_b ) ) ).
% of_zero_neq_one_0_iff
thf(fact_308_of__zero__neq__one__0__iff,axiom,
! [X: nat] :
( ( ( matrix700445748609480494at_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_309_of__zero__neq__one__0__iff,axiom,
! [X: int] :
( ( ( matrix1699799460510535050nt_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_310_of__zero__neq__one__0__iff,axiom,
! [X: real] :
( ( ( matrix4086780077301154698al_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_real ) ) ).
% of_zero_neq_one_0_iff
thf(fact_311_of__zero__neq__one__0__iff,axiom,
! [X: b] :
( ( ( matrix5484708082667875359_b_int @ X )
= zero_zero_int )
= ( X = zero_zero_b ) ) ).
% of_zero_neq_one_0_iff
thf(fact_312_of__zero__neq__one__0__iff,axiom,
! [X: nat] :
( ( ( matrix697955278100430218at_int @ X )
= zero_zero_int )
= ( X = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_313_of__zero__neq__one__0,axiom,
( ( matrix4781043112069605324ne_b_b @ zero_zero_b )
= zero_zero_b ) ).
% of_zero_neq_one_0
thf(fact_314_of__zero__neq__one__0,axiom,
( ( matrix5487198553176925635_b_nat @ zero_zero_b )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_315_of__zero__neq__one__0,axiom,
( ( matrix5484708082667875359_b_int @ zero_zero_b )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_316_of__zero__neq__one__0,axiom,
( ( matrix2280091663418064671b_real @ zero_zero_b )
= zero_zero_real ) ).
% of_zero_neq_one_0
thf(fact_317_of__zero__neq__one__0,axiom,
( ( matrix8283685725398817569_nat_b @ zero_zero_nat )
= zero_zero_b ) ).
% of_zero_neq_one_0
thf(fact_318_of__zero__neq__one__0,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_319_of__zero__neq__one__0,axiom,
( ( matrix697955278100430218at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_320_of__zero__neq__one__0,axiom,
( ( matrix8742843541027031818t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_zero_neq_one_0
thf(fact_321_of__zero__neq__one__0,axiom,
( ( matrix6038540757728371653_int_b @ zero_zero_int )
= zero_zero_b ) ).
% of_zero_neq_one_0
thf(fact_322_of__zero__neq__one__0,axiom,
( ( matrix1699799460510535050nt_nat @ zero_zero_int )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_323_of__zero__hom_Ohom__0__iff,axiom,
! [X: b] :
( ( ( matrix4781043112069605324ne_b_b @ X )
= zero_zero_b )
= ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_324_of__zero__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( matrix8283685725398817569_nat_b @ X )
= zero_zero_b )
= ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_325_of__zero__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( matrix6038540757728371653_int_b @ X )
= zero_zero_b )
= ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_326_of__zero__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( matrix6537263852557659589real_b @ X )
= zero_zero_b )
= ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_327_of__zero__hom_Ohom__0__iff,axiom,
! [X: b] :
( ( ( matrix5487198553176925635_b_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_328_of__zero__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( matrix700445748609480494at_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_329_of__zero__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( matrix1699799460510535050nt_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_330_of__zero__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( matrix4086780077301154698al_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_331_of__zero__hom_Ohom__0__iff,axiom,
! [X: b] :
( ( ( matrix5484708082667875359_b_int @ X )
= zero_zero_int )
= ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_332_of__zero__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( matrix697955278100430218at_int @ X )
= zero_zero_int )
= ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_333_of__zero__hom_Ohom__zero,axiom,
( ( matrix4781043112069605324ne_b_b @ zero_zero_b )
= zero_zero_b ) ).
% of_zero_hom.hom_zero
thf(fact_334_of__zero__hom_Ohom__zero,axiom,
( ( matrix5487198553176925635_b_nat @ zero_zero_b )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_335_of__zero__hom_Ohom__zero,axiom,
( ( matrix5484708082667875359_b_int @ zero_zero_b )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_336_of__zero__hom_Ohom__zero,axiom,
( ( matrix2280091663418064671b_real @ zero_zero_b )
= zero_zero_real ) ).
% of_zero_hom.hom_zero
thf(fact_337_of__zero__hom_Ohom__zero,axiom,
( ( matrix8283685725398817569_nat_b @ zero_zero_nat )
= zero_zero_b ) ).
% of_zero_hom.hom_zero
thf(fact_338_of__zero__hom_Ohom__zero,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_339_of__zero__hom_Ohom__zero,axiom,
( ( matrix697955278100430218at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_340_of__zero__hom_Ohom__zero,axiom,
( ( matrix8742843541027031818t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_zero_hom.hom_zero
thf(fact_341_of__zero__hom_Ohom__zero,axiom,
( ( matrix6038540757728371653_int_b @ zero_zero_int )
= zero_zero_b ) ).
% of_zero_hom.hom_zero
thf(fact_342_of__zero__hom_Ohom__zero,axiom,
( ( matrix1699799460510535050nt_nat @ zero_zero_int )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_343_of__zero__neq__one__1,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_344_of__zero__neq__one__1,axiom,
( ( matrix697955278100430218at_int @ one_one_nat )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_345_of__zero__neq__one__1,axiom,
( ( matrix8742843541027031818t_real @ one_one_nat )
= one_one_real ) ).
% of_zero_neq_one_1
thf(fact_346_of__zero__neq__one__1,axiom,
( ( matrix1699799460510535050nt_nat @ one_one_int )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_347_of__zero__neq__one__1,axiom,
( ( matrix1697308990001484774nt_int @ one_one_int )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_348_of__zero__neq__one__1,axiom,
( ( matrix1706393078865277798t_real @ one_one_int )
= one_one_real ) ).
% of_zero_neq_one_1
thf(fact_349_of__zero__neq__one__1,axiom,
( ( matrix4086780077301154698al_nat @ one_one_real )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_350_of__zero__neq__one__1,axiom,
( ( matrix4084289606792104422al_int @ one_one_real )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_351_of__zero__neq__one__1,axiom,
( ( matrix3070681271257819494l_real @ one_one_real )
= one_one_real ) ).
% of_zero_neq_one_1
thf(fact_352_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_353_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix697955278100430218at_int @ one_one_nat )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_354_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix8742843541027031818t_real @ one_one_nat )
= one_one_real ) ).
% of_inj_on_01_hom.hom_one
thf(fact_355_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1699799460510535050nt_nat @ one_one_int )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_356_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1697308990001484774nt_int @ one_one_int )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_357_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1706393078865277798t_real @ one_one_int )
= one_one_real ) ).
% of_inj_on_01_hom.hom_one
thf(fact_358_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix4086780077301154698al_nat @ one_one_real )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_359_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix4084289606792104422al_int @ one_one_real )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_360_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix3070681271257819494l_real @ one_one_real )
= one_one_real ) ).
% of_inj_on_01_hom.hom_one
thf(fact_361_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_362_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_363_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ B ) )
=> ? [X5: nat] :
( ( P2 @ X5 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_364_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_365_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_366_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_367_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_368_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_369_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_370_conj__le__cong,axiom,
! [X: int,X7: int,P2: $o,P3: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_371_imp__le__cong,axiom,
! [X: int,X7: int,P2: $o,P3: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_372_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_373_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_374_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_375_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_376_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_377_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_378_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_379_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_380_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_381_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_382_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_383_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_384_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_385_subset__code_I1_J,axiom,
! [Xs: list_vec_b,B2: set_vec_b] :
( ( ord_le4862985661309304830_vec_b @ ( set_vec_b2 @ Xs ) @ B2 )
= ( ! [X2: vec_b] :
( ( member_vec_b2 @ X2 @ ( set_vec_b2 @ Xs ) )
=> ( member_vec_b2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_386_subset__code_I1_J,axiom,
! [Xs: list_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
= ( ! [X2: real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ( member_real2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_387_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_388_of__zero__hom_Ohom__0,axiom,
! [X: b] :
( ( ( matrix4781043112069605324ne_b_b @ X )
= zero_zero_b )
=> ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0
thf(fact_389_of__zero__hom_Ohom__0,axiom,
! [X: nat] :
( ( ( matrix8283685725398817569_nat_b @ X )
= zero_zero_b )
=> ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_390_of__zero__hom_Ohom__0,axiom,
! [X: int] :
( ( ( matrix6038540757728371653_int_b @ X )
= zero_zero_b )
=> ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_391_of__zero__hom_Ohom__0,axiom,
! [X: real] :
( ( ( matrix6537263852557659589real_b @ X )
= zero_zero_b )
=> ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0
thf(fact_392_of__zero__hom_Ohom__0,axiom,
! [X: b] :
( ( ( matrix5487198553176925635_b_nat @ X )
= zero_zero_nat )
=> ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0
thf(fact_393_of__zero__hom_Ohom__0,axiom,
! [X: nat] :
( ( ( matrix700445748609480494at_nat @ X )
= zero_zero_nat )
=> ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_394_of__zero__hom_Ohom__0,axiom,
! [X: int] :
( ( ( matrix1699799460510535050nt_nat @ X )
= zero_zero_nat )
=> ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_395_of__zero__hom_Ohom__0,axiom,
! [X: real] :
( ( ( matrix4086780077301154698al_nat @ X )
= zero_zero_nat )
=> ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0
thf(fact_396_of__zero__hom_Ohom__0,axiom,
! [X: b] :
( ( ( matrix5484708082667875359_b_int @ X )
= zero_zero_int )
=> ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0
thf(fact_397_of__zero__hom_Ohom__0,axiom,
! [X: nat] :
( ( ( matrix697955278100430218at_int @ X )
= zero_zero_int )
=> ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_398_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_399_of__zero__neq__one__def,axiom,
( matrix4781043112069605324ne_b_b
= ( ^ [X2: b] : ( if_b @ ( X2 = zero_zero_b ) @ zero_zero_b @ one_one_b ) ) ) ).
% of_zero_neq_one_def
thf(fact_400_of__zero__neq__one__def,axiom,
( matrix5487198553176925635_b_nat
= ( ^ [X2: b] : ( if_nat @ ( X2 = zero_zero_b ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_401_of__zero__neq__one__def,axiom,
( matrix5484708082667875359_b_int
= ( ^ [X2: b] : ( if_int @ ( X2 = zero_zero_b ) @ zero_zero_int @ one_one_int ) ) ) ).
% of_zero_neq_one_def
thf(fact_402_of__zero__neq__one__def,axiom,
( matrix2280091663418064671b_real
= ( ^ [X2: b] : ( if_real @ ( X2 = zero_zero_b ) @ zero_zero_real @ one_one_real ) ) ) ).
% of_zero_neq_one_def
thf(fact_403_of__zero__neq__one__def,axiom,
( matrix8283685725398817569_nat_b
= ( ^ [X2: nat] : ( if_b @ ( X2 = zero_zero_nat ) @ zero_zero_b @ one_one_b ) ) ) ).
% of_zero_neq_one_def
thf(fact_404_of__zero__neq__one__def,axiom,
( matrix700445748609480494at_nat
= ( ^ [X2: nat] : ( if_nat @ ( X2 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_405_of__zero__neq__one__def,axiom,
( matrix697955278100430218at_int
= ( ^ [X2: nat] : ( if_int @ ( X2 = zero_zero_nat ) @ zero_zero_int @ one_one_int ) ) ) ).
% of_zero_neq_one_def
thf(fact_406_of__zero__neq__one__def,axiom,
( matrix8742843541027031818t_real
= ( ^ [X2: nat] : ( if_real @ ( X2 = zero_zero_nat ) @ zero_zero_real @ one_one_real ) ) ) ).
% of_zero_neq_one_def
thf(fact_407_of__zero__neq__one__def,axiom,
( matrix6038540757728371653_int_b
= ( ^ [X2: int] : ( if_b @ ( X2 = zero_zero_int ) @ zero_zero_b @ one_one_b ) ) ) ).
% of_zero_neq_one_def
thf(fact_408_of__zero__neq__one__def,axiom,
( matrix1699799460510535050nt_nat
= ( ^ [X2: int] : ( if_nat @ ( X2 = zero_zero_int ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_409_set__subset__insertI,axiom,
! [Xs: list_vec_b,X: vec_b] : ( ord_le4862985661309304830_vec_b @ ( set_vec_b2 @ Xs ) @ ( set_vec_b2 @ ( insert_vec_b @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_410_set__subset__insertI,axiom,
! [Xs: list_real,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ ( set_real2 @ ( insert_real @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_411_set__subset__insertI,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( insert_nat @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_412_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_413_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_414_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_415_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_416_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_417_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_418_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_419_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_420_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_421_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_422_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_423_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_424_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_425_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_426_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_427_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_428_ord__eq__le__subst,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_429_ord__eq__le__subst,axiom,
! [A3: real,F: nat > real,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_430_ord__eq__le__subst,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_431_ord__eq__le__subst,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
! [A3: real,F: int > real,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_ord__eq__le__subst,axiom,
! [A3: nat,F: real > nat,B: real,C2: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_434_ord__eq__le__subst,axiom,
! [A3: int,F: real > int,B: real,C2: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_ord__eq__le__subst,axiom,
! [A3: real,F: real > real,B: real,C2: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_436_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_437_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_438_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_439_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_440_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_441_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_442_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_443_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_444_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_445_order__subst2,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_446_order__subst2,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_447_order__subst2,axiom,
! [A3: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_448_order__subst2,axiom,
! [A3: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_449_order__subst2,axiom,
! [A3: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_450_order__subst2,axiom,
! [A3: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_451_order__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_452_order__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_453_order__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_454_order__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_455_order__subst1,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_456_order__subst1,axiom,
! [A3: int,F: real > int,B: real,C2: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_457_order__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_458_order__subst1,axiom,
! [A3: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_459_order__subst1,axiom,
! [A3: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_460_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_461_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_462_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_463_antisym,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_464_antisym,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_465_antisym,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_466_dual__order_Otrans,axiom,
! [B: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_467_dual__order_Otrans,axiom,
! [B: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_468_dual__order_Otrans,axiom,
! [B: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_469_dual__order_Oantisym,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_470_dual__order_Oantisym,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_471_dual__order_Oantisym,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_472_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_473_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_474_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_475_linorder__wlog,axiom,
! [P2: nat > nat > $o,A3: nat,B: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
=> ( P2 @ A6 @ B5 ) )
=> ( ! [A6: nat,B5: nat] :
( ( P2 @ B5 @ A6 )
=> ( P2 @ A6 @ B5 ) )
=> ( P2 @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_476_linorder__wlog,axiom,
! [P2: int > int > $o,A3: int,B: int] :
( ! [A6: int,B5: int] :
( ( ord_less_eq_int @ A6 @ B5 )
=> ( P2 @ A6 @ B5 ) )
=> ( ! [A6: int,B5: int] :
( ( P2 @ B5 @ A6 )
=> ( P2 @ A6 @ B5 ) )
=> ( P2 @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_477_linorder__wlog,axiom,
! [P2: real > real > $o,A3: real,B: real] :
( ! [A6: real,B5: real] :
( ( ord_less_eq_real @ A6 @ B5 )
=> ( P2 @ A6 @ B5 ) )
=> ( ! [A6: real,B5: real] :
( ( P2 @ B5 @ A6 )
=> ( P2 @ A6 @ B5 ) )
=> ( P2 @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_478_order__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_479_order__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_480_order__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_481_order_Otrans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_482_order_Otrans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_483_order_Otrans,axiom,
! [A3: real,B: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_484_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_485_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_486_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_487_ord__le__eq__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_488_ord__le__eq__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_489_ord__le__eq__trans,axiom,
! [A3: real,B: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_490_ord__eq__le__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( A3 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_491_ord__eq__le__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( A3 = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_492_ord__eq__le__trans,axiom,
! [A3: real,B: real,C2: real] :
( ( A3 = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_493_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_494_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_495_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_496_le__cases3,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_497_le__cases3,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_498_le__cases3,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_499_nle__le,axiom,
! [A3: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B ) )
= ( ( ord_less_eq_nat @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_500_nle__le,axiom,
! [A3: int,B: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B ) )
= ( ( ord_less_eq_int @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_501_nle__le,axiom,
! [A3: real,B: real] :
( ( ~ ( ord_less_eq_real @ A3 @ B ) )
= ( ( ord_less_eq_real @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_502_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_503_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_504_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_505_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_506_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_507_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_508_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_509_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_510_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_511_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_512_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_513_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_514_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P2: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_515_order__less__imp__triv,axiom,
! [X: int,Y2: int,P2: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_516_order__less__imp__triv,axiom,
! [X: real,Y2: real,P2: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_517_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_518_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_519_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_520_order__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_521_order__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_522_order__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_523_order__less__subst2,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_524_order__less__subst2,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_525_order__less__subst2,axiom,
! [A3: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_526_order__less__subst2,axiom,
! [A3: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_527_order__less__subst2,axiom,
! [A3: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_528_order__less__subst2,axiom,
! [A3: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_529_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_530_order__less__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_531_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_532_order__less__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_533_order__less__subst1,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_534_order__less__subst1,axiom,
! [A3: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_535_order__less__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_536_order__less__subst1,axiom,
! [A3: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_537_order__less__subst1,axiom,
! [A3: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_538_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_539_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_540_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_541_ord__less__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_542_ord__less__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_543_ord__less__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_544_ord__less__eq__subst,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_545_ord__less__eq__subst,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_546_ord__less__eq__subst,axiom,
! [A3: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_547_ord__less__eq__subst,axiom,
! [A3: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_548_ord__less__eq__subst,axiom,
! [A3: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_549_ord__less__eq__subst,axiom,
! [A3: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_550_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_551_ord__eq__less__subst,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_552_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_553_ord__eq__less__subst,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_554_ord__eq__less__subst,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_555_ord__eq__less__subst,axiom,
! [A3: real,F: int > real,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_556_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B: real,C2: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_557_ord__eq__less__subst,axiom,
! [A3: int,F: real > int,B: real,C2: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_558_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B: real,C2: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_559_order__less__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_560_order__less__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_561_order__less__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_562_order__less__asym_H,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ~ ( ord_less_nat @ B @ A3 ) ) ).
% order_less_asym'
thf(fact_563_order__less__asym_H,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ~ ( ord_less_int @ B @ A3 ) ) ).
% order_less_asym'
thf(fact_564_order__less__asym_H,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ~ ( ord_less_real @ B @ A3 ) ) ).
% order_less_asym'
thf(fact_565_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_566_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_567_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_568_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_569_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_570_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_571_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_572_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_573_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_574_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( A3 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_575_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
=> ( A3 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_576_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
=> ( A3 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_577_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( A3 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_578_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ( A3 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_579_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ( A3 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_580_dual__order_Ostrict__trans,axiom,
! [B: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_581_dual__order_Ostrict__trans,axiom,
! [B: int,A3: int,C2: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_582_dual__order_Ostrict__trans,axiom,
! [B: real,A3: real,C2: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_583_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_584_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_585_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_586_order_Ostrict__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_587_order_Ostrict__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_588_order_Ostrict__trans,axiom,
! [A3: real,B: real,C2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_589_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A3: nat,B: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_nat @ A6 @ B5 )
=> ( P2 @ A6 @ B5 ) )
=> ( ! [A6: nat] : ( P2 @ A6 @ A6 )
=> ( ! [A6: nat,B5: nat] :
( ( P2 @ B5 @ A6 )
=> ( P2 @ A6 @ B5 ) )
=> ( P2 @ A3 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_590_linorder__less__wlog,axiom,
! [P2: int > int > $o,A3: int,B: int] :
( ! [A6: int,B5: int] :
( ( ord_less_int @ A6 @ B5 )
=> ( P2 @ A6 @ B5 ) )
=> ( ! [A6: int] : ( P2 @ A6 @ A6 )
=> ( ! [A6: int,B5: int] :
( ( P2 @ B5 @ A6 )
=> ( P2 @ A6 @ B5 ) )
=> ( P2 @ A3 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_591_linorder__less__wlog,axiom,
! [P2: real > real > $o,A3: real,B: real] :
( ! [A6: real,B5: real] :
( ( ord_less_real @ A6 @ B5 )
=> ( P2 @ A6 @ B5 ) )
=> ( ! [A6: real] : ( P2 @ A6 @ A6 )
=> ( ! [A6: real,B5: real] :
( ( P2 @ B5 @ A6 )
=> ( P2 @ A6 @ B5 ) )
=> ( P2 @ A3 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_592_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X8: nat] : ( P4 @ X8 ) )
= ( ^ [P: nat > $o] :
? [N4: nat] :
( ( P @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_593_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_594_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_595_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_596_dual__order_Oasym,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
=> ~ ( ord_less_nat @ A3 @ B ) ) ).
% dual_order.asym
thf(fact_597_dual__order_Oasym,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
=> ~ ( ord_less_int @ A3 @ B ) ) ).
% dual_order.asym
thf(fact_598_dual__order_Oasym,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
=> ~ ( ord_less_real @ A3 @ B ) ) ).
% dual_order.asym
thf(fact_599_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_600_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_601_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_602_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_603_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_604_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_605_less__induct,axiom,
! [P2: nat > $o,A3: nat] :
( ! [X5: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X5 )
=> ( P2 @ Y5 ) )
=> ( P2 @ X5 ) )
=> ( P2 @ A3 ) ) ).
% less_induct
thf(fact_606_ord__less__eq__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_607_ord__less__eq__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_608_ord__less__eq__trans,axiom,
! [A3: real,B: real,C2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_609_ord__eq__less__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( A3 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_610_ord__eq__less__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( A3 = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_611_ord__eq__less__trans,axiom,
! [A3: real,B: real,C2: real] :
( ( A3 = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_612_order_Oasym,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ~ ( ord_less_nat @ B @ A3 ) ) ).
% order.asym
thf(fact_613_order_Oasym,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ~ ( ord_less_int @ B @ A3 ) ) ).
% order.asym
thf(fact_614_order_Oasym,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ~ ( ord_less_real @ B @ A3 ) ) ).
% order.asym
thf(fact_615_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_616_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_617_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_618_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y2 ) ) ) ).
% dense
thf(fact_619_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_620_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_621_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_622_lt__ex,axiom,
! [X: int] :
? [Y6: int] : ( ord_less_int @ Y6 @ X ) ).
% lt_ex
thf(fact_623_lt__ex,axiom,
! [X: real] :
? [Y6: real] : ( ord_less_real @ Y6 @ X ) ).
% lt_ex
thf(fact_624_pinf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P3 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_625_pinf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P3 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_626_pinf_I1_J,axiom,
! [P2: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P3 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_627_pinf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P3 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_628_pinf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P3 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_629_pinf_I2_J,axiom,
! [P2: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P3 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_630_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_631_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_632_pinf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_633_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_634_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_635_pinf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_636_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_637_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_638_pinf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_639_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_640_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_641_pinf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_642_minf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P3 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_643_minf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P3 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_644_minf_I1_J,axiom,
! [P2: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P3 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_645_minf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P3 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_646_minf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P3 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_647_minf_I2_J,axiom,
! [P2: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( P2 @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P3 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_648_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_649_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_650_minf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_651_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_652_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_653_minf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_654_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_655_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_656_minf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_657_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_658_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_659_minf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z2 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_660_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_661_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_662_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_663_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_664_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_665_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_666_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_667_order__less__le__subst2,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_668_order__less__le__subst2,axiom,
! [A3: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_669_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_670_order__less__le__subst2,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_671_order__less__le__subst2,axiom,
! [A3: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_672_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_673_order__less__le__subst2,axiom,
! [A3: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_674_order__less__le__subst2,axiom,
! [A3: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_675_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_676_order__less__le__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_677_order__less__le__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_678_order__less__le__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_679_order__less__le__subst1,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_680_order__less__le__subst1,axiom,
! [A3: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_681_order__less__le__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_682_order__less__le__subst1,axiom,
! [A3: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_683_order__less__le__subst1,axiom,
! [A3: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_684_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_685_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_686_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_687_order__le__less__subst2,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_688_order__le__less__subst2,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_689_order__le__less__subst2,axiom,
! [A3: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_690_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_691_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_692_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_eq_real @ X5 @ Y6 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_693_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_694_order__le__less__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_695_order__le__less__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_696_order__le__less__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_697_order__le__less__subst1,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_698_order__le__less__subst1,axiom,
! [A3: int,F: real > int,B: real,C2: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_699_order__le__less__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_700_order__le__less__subst1,axiom,
! [A3: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_701_order__le__less__subst1,axiom,
! [A3: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X5: real,Y6: real] :
( ( ord_less_real @ X5 @ Y6 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y6 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_702_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_703_order__less__le__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_704_order__less__le__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_705_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_706_order__le__less__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_707_order__le__less__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_708_order__neq__le__trans,axiom,
! [A3: nat,B: nat] :
( ( A3 != B )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_709_order__neq__le__trans,axiom,
! [A3: int,B: int] :
( ( A3 != B )
=> ( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_710_order__neq__le__trans,axiom,
! [A3: real,B: real] :
( ( A3 != B )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_711_order__le__neq__trans,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_712_order__le__neq__trans,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_713_order__le__neq__trans,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_714_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_715_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_716_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_717_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_718_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_719_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_720_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_721_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_722_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_723_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_724_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_725_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_726_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_727_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_728_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_729_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( ord_less_eq_nat @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_730_dual__order_Ostrict__implies__order,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
=> ( ord_less_eq_int @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_731_dual__order_Ostrict__implies__order,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
=> ( ord_less_eq_real @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_732_order_Ostrict__implies__order,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ord_less_eq_nat @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_733_order_Ostrict__implies__order,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_eq_int @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_734_order_Ostrict__implies__order,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_eq_real @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_735_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_736_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_737_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_738_dual__order_Ostrict__trans2,axiom,
! [B: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_739_dual__order_Ostrict__trans2,axiom,
! [B: int,A3: int,C2: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_740_dual__order_Ostrict__trans2,axiom,
! [B: real,A3: real,C2: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_741_dual__order_Ostrict__trans1,axiom,
! [B: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_742_dual__order_Ostrict__trans1,axiom,
! [B: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_743_dual__order_Ostrict__trans1,axiom,
! [B: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_744_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_745_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_746_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_747_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_748_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_749_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_750_dense__le__bounded,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W3: real] :
( ( ord_less_real @ X @ W3 )
=> ( ( ord_less_real @ W3 @ Y2 )
=> ( ord_less_eq_real @ W3 @ Z3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_751_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y2: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W3: real] :
( ( ord_less_real @ Z3 @ W3 )
=> ( ( ord_less_real @ W3 @ X )
=> ( ord_less_eq_real @ Y2 @ W3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_752_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_753_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_754_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_755_order_Ostrict__trans2,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_756_order_Ostrict__trans2,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_757_order_Ostrict__trans2,axiom,
! [A3: real,B: real,C2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_758_order_Ostrict__trans1,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_759_order_Ostrict__trans1,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_760_order_Ostrict__trans1,axiom,
! [A3: real,B: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_761_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_762_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_763_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_764_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_765_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_766_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_767_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_768_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_769_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_770_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_771_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_772_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_773_dense__le,axiom,
! [Y2: real,Z3: real] :
( ! [X5: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ( ord_less_eq_real @ X5 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_le
thf(fact_774_dense__ge,axiom,
! [Z3: real,Y2: real] :
( ! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_eq_real @ Y2 @ X5 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_ge
thf(fact_775_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_776_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_777_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_778_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_779_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_780_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_781_nless__le,axiom,
! [A3: nat,B: nat] :
( ( ~ ( ord_less_nat @ A3 @ B ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_782_nless__le,axiom,
! [A3: int,B: int] :
( ( ~ ( ord_less_int @ A3 @ B ) )
= ( ~ ( ord_less_eq_int @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_783_nless__le,axiom,
! [A3: real,B: real] :
( ( ~ ( ord_less_real @ A3 @ B ) )
= ( ~ ( ord_less_eq_real @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_784_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_785_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_786_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_787_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_788_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_789_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_790_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_791_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_792_pinf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_793_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_794_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_795_pinf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X6: real] :
( ( ord_less_real @ Z2 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_796_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_797_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_798_unit__vecs__first__distinct,axiom,
! [I: nat,J: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ N )
=> ~ ( member_vec_b2 @ ( unit_vec_b @ N @ J ) @ ( set_vec_b2 @ ( unit_vecs_first_b @ N @ I ) ) ) ) ) ).
% unit_vecs_first_distinct
thf(fact_799_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_800_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_801_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( F @ K2 @ I5 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_802_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P2 @ I5 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_803_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq_int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq_int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_804_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_805_verit__comp__simplify1_I2_J,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_806_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_807_verit__la__disequality,axiom,
! [A3: nat,B: nat] :
( ( A3 = B )
| ~ ( ord_less_eq_nat @ A3 @ B )
| ~ ( ord_less_eq_nat @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_808_verit__la__disequality,axiom,
! [A3: int,B: int] :
( ( A3 = B )
| ~ ( ord_less_eq_int @ A3 @ B )
| ~ ( ord_less_eq_int @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_809_verit__la__disequality,axiom,
! [A3: real,B: real] :
( ( A3 = B )
| ~ ( ord_less_eq_real @ A3 @ B )
| ~ ( ord_less_eq_real @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_810_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_811_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_812_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_813_int__if,axiom,
! [P2: $o,A3: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A3 @ B ) )
= ( semiri1314217659103216013at_int @ A3 ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A3 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_814_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_815_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A7 ) )
= ( ord_less_nat @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_816_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A7 ) )
= ( ord_less_int @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_817_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A7 ) )
= ( ord_less_real @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_818_subsetI,axiom,
! [A: set_vec_b,B2: set_vec_b] :
( ! [X5: vec_b] :
( ( member_vec_b2 @ X5 @ A )
=> ( member_vec_b2 @ X5 @ B2 ) )
=> ( ord_le4862985661309304830_vec_b @ A @ B2 ) ) ).
% subsetI
thf(fact_819_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X5: nat] :
( ( member_nat2 @ X5 @ A )
=> ( member_nat2 @ X5 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_820_subsetI,axiom,
! [A: set_real,B2: set_real] :
( ! [X5: real] :
( ( member_real2 @ X5 @ A )
=> ( member_real2 @ X5 @ B2 ) )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ).
% subsetI
thf(fact_821_sum__vec__one__zero,axiom,
! [V: vec_int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_vec_int @ V ) )
=> ( ord_less_eq_int @ ( vec_index_int @ V @ I2 ) @ one_one_int ) )
=> ( ord_less_eq_int @ ( matrix3634415343793898042ec_int @ V ) @ ( semiri1314217659103216013at_int @ ( dim_vec_int @ V ) ) ) ) ).
% sum_vec_one_zero
thf(fact_822_complete__interval,axiom,
! [A3: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A3 @ B )
=> ( ( P2 @ A3 )
=> ( ~ ( P2 @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A3 @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A3 @ X6 )
& ( ord_less_nat @ X6 @ C ) )
=> ( P2 @ X6 ) )
& ! [D: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A3 @ X5 )
& ( ord_less_nat @ X5 @ D ) )
=> ( P2 @ X5 ) )
=> ( ord_less_eq_nat @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_823_complete__interval,axiom,
! [A3: int,B: int,P2: int > $o] :
( ( ord_less_int @ A3 @ B )
=> ( ( P2 @ A3 )
=> ( ~ ( P2 @ B )
=> ? [C: int] :
( ( ord_less_eq_int @ A3 @ C )
& ( ord_less_eq_int @ C @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A3 @ X6 )
& ( ord_less_int @ X6 @ C ) )
=> ( P2 @ X6 ) )
& ! [D: int] :
( ! [X5: int] :
( ( ( ord_less_eq_int @ A3 @ X5 )
& ( ord_less_int @ X5 @ D ) )
=> ( P2 @ X5 ) )
=> ( ord_less_eq_int @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_824_complete__interval,axiom,
! [A3: real,B: real,P2: real > $o] :
( ( ord_less_real @ A3 @ B )
=> ( ( P2 @ A3 )
=> ( ~ ( P2 @ B )
=> ? [C: real] :
( ( ord_less_eq_real @ A3 @ C )
& ( ord_less_eq_real @ C @ B )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A3 @ X6 )
& ( ord_less_real @ X6 @ C ) )
=> ( P2 @ X6 ) )
& ! [D: real] :
( ! [X5: real] :
( ( ( ord_less_eq_real @ A3 @ X5 )
& ( ord_less_real @ X5 @ D ) )
=> ( P2 @ X5 ) )
=> ( ord_less_eq_real @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_825_zero__min,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_min
thf(fact_826_ex__gt__or__lt,axiom,
! [A3: real] :
? [B5: real] :
( ( ord_less_real @ A3 @ B5 )
| ( ord_less_real @ B5 @ A3 ) ) ).
% ex_gt_or_lt
thf(fact_827_subset__iff,axiom,
( ord_le4862985661309304830_vec_b
= ( ^ [A2: set_vec_b,B7: set_vec_b] :
! [T2: vec_b] :
( ( member_vec_b2 @ T2 @ A2 )
=> ( member_vec_b2 @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_828_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B7: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A2 )
=> ( member_nat2 @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_829_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B7: set_real] :
! [T2: real] :
( ( member_real2 @ T2 @ A2 )
=> ( member_real2 @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_830_subset__eq,axiom,
( ord_le4862985661309304830_vec_b
= ( ^ [A2: set_vec_b,B7: set_vec_b] :
! [X2: vec_b] :
( ( member_vec_b2 @ X2 @ A2 )
=> ( member_vec_b2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_831_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B7: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( member_nat2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_832_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B7: set_real] :
! [X2: real] :
( ( member_real2 @ X2 @ A2 )
=> ( member_real2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_833_subsetD,axiom,
! [A: set_vec_b,B2: set_vec_b,C2: vec_b] :
( ( ord_le4862985661309304830_vec_b @ A @ B2 )
=> ( ( member_vec_b2 @ C2 @ A )
=> ( member_vec_b2 @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_834_subsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat2 @ C2 @ A )
=> ( member_nat2 @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_835_subsetD,axiom,
! [A: set_real,B2: set_real,C2: real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_real2 @ C2 @ A )
=> ( member_real2 @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_836_in__mono,axiom,
! [A: set_vec_b,B2: set_vec_b,X: vec_b] :
( ( ord_le4862985661309304830_vec_b @ A @ B2 )
=> ( ( member_vec_b2 @ X @ A )
=> ( member_vec_b2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_837_in__mono,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat2 @ X @ A )
=> ( member_nat2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_838_in__mono,axiom,
! [A: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_real2 @ X @ A )
=> ( member_real2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_839_mset__list__of__mset,axiom,
! [M2: multiset_vec_b] :
( ( mset_vec_b @ ( multis2398011375292950429_vec_b @ M2 ) )
= M2 ) ).
% mset_list_of_mset
thf(fact_840_mset__list__of__mset,axiom,
! [M2: multiset_real] :
( ( mset_real @ ( multis3585236986679824865t_real @ M2 ) )
= M2 ) ).
% mset_list_of_mset
thf(fact_841_mset__list__of__mset,axiom,
! [M2: multiset_nat] :
( ( mset_nat @ ( multis105632648212199813et_nat @ M2 ) )
= M2 ) ).
% mset_list_of_mset
thf(fact_842_index__smult__vec_I1_J,axiom,
! [I: nat,V: vec_nat,A3: nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ V ) )
=> ( ( vec_index_nat @ ( smult_vec_nat @ A3 @ V ) @ I )
= ( times_times_nat @ A3 @ ( vec_index_nat @ V @ I ) ) ) ) ).
% index_smult_vec(1)
thf(fact_843_index__smult__vec_I1_J,axiom,
! [I: nat,V: vec_int,A3: int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ ( smult_vec_int @ A3 @ V ) @ I )
= ( times_times_int @ A3 @ ( vec_index_int @ V @ I ) ) ) ) ).
% index_smult_vec(1)
thf(fact_844_index__smult__vec_I1_J,axiom,
! [I: nat,V: vec_real,A3: real] :
( ( ord_less_nat @ I @ ( dim_vec_real @ V ) )
=> ( ( vec_index_real @ ( smult_vec_real @ A3 @ V ) @ I )
= ( times_times_real @ A3 @ ( vec_index_real @ V @ I ) ) ) ) ).
% index_smult_vec(1)
thf(fact_845_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_846_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_847_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_848_frac__eq,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= X )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% frac_eq
thf(fact_849_neg__equal__iff__equal,axiom,
! [A3: int,B: int] :
( ( ( uminus_uminus_int @ A3 )
= ( uminus_uminus_int @ B ) )
= ( A3 = B ) ) ).
% neg_equal_iff_equal
thf(fact_850_neg__equal__iff__equal,axiom,
! [A3: real,B: real] :
( ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B ) )
= ( A3 = B ) ) ).
% neg_equal_iff_equal
thf(fact_851_add_Oinverse__inverse,axiom,
! [A3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_852_add_Oinverse__inverse,axiom,
! [A3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_853_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_854_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_855_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_856_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_857_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_858_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_859_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_860_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_861_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_862_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_863_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_864_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_865_mult__eq__0__iff,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_866_mult__eq__0__iff,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_867_mult__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_868_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_869_mult__cancel__left,axiom,
! [C2: int,A3: int,B: int] :
( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_870_mult__cancel__left,axiom,
! [C2: real,A3: real,B: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_871_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_872_mult__cancel__right,axiom,
! [A3: int,C2: int,B: int] :
( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_873_mult__cancel__right,axiom,
! [A3: real,C2: real,B: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_874_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_875_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_876_neg__0__equal__iff__equal,axiom,
! [A3: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A3 ) )
= ( zero_zero_int = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_877_neg__0__equal__iff__equal,axiom,
! [A3: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A3 ) )
= ( zero_zero_real = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_878_neg__equal__0__iff__equal,axiom,
! [A3: int] :
( ( ( uminus_uminus_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_879_neg__equal__0__iff__equal,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_880_equal__neg__zero,axiom,
! [A3: int] :
( ( A3
= ( uminus_uminus_int @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_881_equal__neg__zero,axiom,
! [A3: real] :
( ( A3
= ( uminus_uminus_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_882_neg__equal__zero,axiom,
! [A3: int] :
( ( ( uminus_uminus_int @ A3 )
= A3 )
= ( A3 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_883_neg__equal__zero,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= A3 )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_884_neg__le__iff__le,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ B ) ) ).
% neg_le_iff_le
thf(fact_885_neg__le__iff__le,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ B ) ) ).
% neg_le_iff_le
thf(fact_886_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_887_mult_Oright__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.right_neutral
thf(fact_888_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_889_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_890_mult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% mult_1
thf(fact_891_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_892_div__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% div_0
thf(fact_893_div__0,axiom,
! [A3: int] :
( ( divide_divide_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% div_0
thf(fact_894_div__0,axiom,
! [A3: real] :
( ( divide_divide_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% div_0
thf(fact_895_div__by__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_896_div__by__0,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_897_div__by__0,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_898_neg__less__iff__less,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ B ) ) ).
% neg_less_iff_less
thf(fact_899_neg__less__iff__less,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ B ) ) ).
% neg_less_iff_less
thf(fact_900_mult__minus__right,axiom,
! [A3: int,B: int] :
( ( times_times_int @ A3 @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A3 @ B ) ) ) ).
% mult_minus_right
thf(fact_901_mult__minus__right,axiom,
! [A3: real,B: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B ) ) ) ).
% mult_minus_right
thf(fact_902_minus__mult__minus,axiom,
! [A3: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A3 @ B ) ) ).
% minus_mult_minus
thf(fact_903_minus__mult__minus,axiom,
! [A3: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A3 @ B ) ) ).
% minus_mult_minus
thf(fact_904_mult__minus__left,axiom,
! [A3: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A3 @ B ) ) ) ).
% mult_minus_left
thf(fact_905_mult__minus__left,axiom,
! [A3: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B ) ) ) ).
% mult_minus_left
thf(fact_906_div__by__1,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ one_one_nat )
= A3 ) ).
% div_by_1
thf(fact_907_div__by__1,axiom,
! [A3: int] :
( ( divide_divide_int @ A3 @ one_one_int )
= A3 ) ).
% div_by_1
thf(fact_908_div__by__1,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ one_one_real )
= A3 ) ).
% div_by_1
thf(fact_909_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_910_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_911_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_912_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_913_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_914_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_915_neg__less__eq__nonneg,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_916_neg__less__eq__nonneg,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_917_less__eq__neg__nonpos,axiom,
! [A3: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_918_less__eq__neg__nonpos,axiom,
! [A3: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_919_neg__le__0__iff__le,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_920_neg__le__0__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_921_neg__0__le__iff__le,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_922_neg__0__le__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_923_mult__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ( times_times_int @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_924_mult__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ( times_times_real @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_925_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_926_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_927_mult__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ( times_times_int @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_928_mult__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ( times_times_real @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_929_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_930_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_931_less__neg__neg,axiom,
! [A3: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_932_less__neg__neg,axiom,
! [A3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_933_neg__less__pos,axiom,
! [A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% neg_less_pos
thf(fact_934_neg__less__pos,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_pos
thf(fact_935_neg__0__less__iff__less,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_936_neg__0__less__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_937_neg__less__0__iff__less,axiom,
! [A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_938_neg__less__0__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_939_nonzero__mult__div__cancel__left,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B ) @ A3 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_940_nonzero__mult__div__cancel__left,axiom,
! [A3: int,B: int] :
( ( A3 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ B ) @ A3 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_941_nonzero__mult__div__cancel__left,axiom,
! [A3: real,B: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B ) @ A3 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_942_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A3: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B ) @ B )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_943_nonzero__mult__div__cancel__right,axiom,
! [B: int,A3: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A3 @ B ) @ B )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_944_nonzero__mult__div__cancel__right,axiom,
! [B: real,A3: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B ) @ B )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_945_div__self,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ A3 @ A3 )
= one_one_nat ) ) ).
% div_self
thf(fact_946_div__self,axiom,
! [A3: int] :
( ( A3 != zero_zero_int )
=> ( ( divide_divide_int @ A3 @ A3 )
= one_one_int ) ) ).
% div_self
thf(fact_947_div__self,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ).
% div_self
thf(fact_948_mult__minus1,axiom,
! [Z3: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z3 )
= ( uminus_uminus_int @ Z3 ) ) ).
% mult_minus1
thf(fact_949_mult__minus1,axiom,
! [Z3: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z3 )
= ( uminus_uminus_real @ Z3 ) ) ).
% mult_minus1
thf(fact_950_mult__minus1__right,axiom,
! [Z3: int] :
( ( times_times_int @ Z3 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z3 ) ) ).
% mult_minus1_right
thf(fact_951_mult__minus1__right,axiom,
! [Z3: real] :
( ( times_times_real @ Z3 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z3 ) ) ).
% mult_minus1_right
thf(fact_952_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_953_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_954_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_955_frac__1,axiom,
( ( archim2898591450579166408c_real @ one_one_real )
= zero_zero_real ) ).
% frac_1
thf(fact_956_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_957_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_958_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_959_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_960_zless__nat__conj,axiom,
! [W: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_961_index__uminus__vec_I1_J,axiom,
! [I: nat,V: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ ( uminus8720015189474472720ec_int @ V ) @ I )
= ( uminus_uminus_int @ ( vec_index_int @ V @ I ) ) ) ) ).
% index_uminus_vec(1)
thf(fact_962_index__uminus__vec_I1_J,axiom,
! [I: nat,V: vec_real] :
( ( ord_less_nat @ I @ ( dim_vec_real @ V ) )
=> ( ( vec_index_real @ ( uminus8989278663012614928c_real @ V ) @ I )
= ( uminus_uminus_real @ ( vec_index_real @ V @ I ) ) ) ) ).
% index_uminus_vec(1)
thf(fact_963_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_964_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_965_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_966_mult_Oleft__commute,axiom,
! [B: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_967_mult_Oleft__commute,axiom,
! [B: int,A3: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A3 @ C2 ) )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_968_mult_Oleft__commute,axiom,
! [B: real,A3: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_969_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_970_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_971_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_972_minus__equation__iff,axiom,
! [A3: int,B: int] :
( ( ( uminus_uminus_int @ A3 )
= B )
= ( ( uminus_uminus_int @ B )
= A3 ) ) ).
% minus_equation_iff
thf(fact_973_minus__equation__iff,axiom,
! [A3: real,B: real] :
( ( ( uminus_uminus_real @ A3 )
= B )
= ( ( uminus_uminus_real @ B )
= A3 ) ) ).
% minus_equation_iff
thf(fact_974_equation__minus__iff,axiom,
! [A3: int,B: int] :
( ( A3
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_975_equation__minus__iff,axiom,
! [A3: real,B: real] :
( ( A3
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_976_mult_Oassoc,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_977_mult_Oassoc,axiom,
! [A3: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_978_mult_Oassoc,axiom,
! [A3: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_979_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_980_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_981_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_982_minus__mult__commute,axiom,
! [A3: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B )
= ( times_times_int @ A3 @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_983_minus__mult__commute,axiom,
! [A3: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B )
= ( times_times_real @ A3 @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_984_square__eq__iff,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ A3 )
= ( times_times_int @ B @ B ) )
= ( ( A3 = B )
| ( A3
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_985_square__eq__iff,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ A3 )
= ( times_times_real @ B @ B ) )
= ( ( A3 = B )
| ( A3
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_986_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_987_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_988_nat__mult__distrib,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_989_nat__mult__distrib__neg,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z5 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_990_int__ops_I7_J,axiom,
! [A3: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A3 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_991_int__ops_I8_J,axiom,
! [A3: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A3 @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_992_le__imp__neg__le,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_993_le__imp__neg__le,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_994_minus__le__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A3 ) ) ).
% minus_le_iff
thf(fact_995_minus__le__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A3 ) ) ).
% minus_le_iff
thf(fact_996_le__minus__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_minus_iff
thf(fact_997_le__minus__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A3 ) ) ) ).
% le_minus_iff
thf(fact_998_verit__negate__coefficient_I2_J,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_999_verit__negate__coefficient_I2_J,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1000_less__minus__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A3 ) ) ) ).
% less_minus_iff
thf(fact_1001_less__minus__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A3 ) ) ) ).
% less_minus_iff
thf(fact_1002_minus__less__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A3 ) ) ).
% minus_less_iff
thf(fact_1003_minus__less__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A3 ) ) ).
% minus_less_iff
thf(fact_1004_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1005_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_1006_mult__not__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1007_mult__not__zero,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1008_mult__not__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1009_divisors__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1010_divisors__zero,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1011_divisors__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1012_no__zero__divisors,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1013_no__zero__divisors,axiom,
! [A3: int,B: int] :
( ( A3 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A3 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1014_no__zero__divisors,axiom,
! [A3: real,B: real] :
( ( A3 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A3 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1015_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_1016_mult__left__cancel,axiom,
! [C2: int,A3: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_1017_mult__left__cancel,axiom,
! [C2: real,A3: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_1018_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_1019_mult__right__cancel,axiom,
! [C2: int,A3: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_1020_mult__right__cancel,axiom,
! [C2: real,A3: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_1021_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1022_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1023_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1024_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_1025_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_1026_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_1027_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1028_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1029_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1030_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1031_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_1032_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_1033_int__cases2,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1034_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1035_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1036_verit__less__mono__div__int2,axiom,
! [A: int,B2: int,N: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1037_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1038_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_1039_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X8: nat] : ( P4 @ X8 ) )
= ( ^ [P: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1040_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X8: nat] : ( P4 @ X8 ) )
= ( ^ [P: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1041_eq__nat__nat__iff,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z5 ) )
= ( Z3 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1042_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1043_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M6: nat] :
( ( ord_less_nat @ zero_zero_nat @ M6 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1044_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_1045_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_1046_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1047_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1048_nat__mono__iff,axiom,
! [Z3: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1049_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z3: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1050_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1051_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1052_int__eq__iff,axiom,
! [M2: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z3 )
= ( ( M2
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1053_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1054_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1055_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1056_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1057_nat__less__eq__zless,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1058_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1059_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1060_nat__le__eq__zle,axiom,
! [W: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1061_split__nat,axiom,
! [P2: nat > $o,I: int] :
( ( P2 @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P2 @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1062_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1063_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1064_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_1065_ln__less__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1066_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1067_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1068_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1069_ln__inj__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% ln_inj_iff
thf(fact_1070_ln__le__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1071_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1072_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_1073_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1074_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1075_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1076_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1077_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real2 @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X5: real] :
( ( member_real2 @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Z4 ) )
=> ? [Y6: real] :
( ! [X6: real] :
( ( member_real2 @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y6 ) )
& ! [Z4: real] :
( ! [X5: real] :
( ( member_real2 @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Z4 ) )
=> ( ord_less_eq_real @ Y6 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1078_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1079_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1080_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1081_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1082_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1083_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1084_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1085_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1086_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1087_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1088_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_1089_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1090_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1091_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1092_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1093_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1094_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1095_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1096_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1097_zdiv__int,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_1098_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1099_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1100_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1101_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1102_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1103_zdiv__mono1,axiom,
! [A3: int,A7: int,B: int] :
( ( ord_less_eq_int @ A3 @ A7 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A7 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1104_zdiv__mono2,axiom,
! [A3: int,B6: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_eq_int @ B6 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A3 @ B6 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1105_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1106_zdiv__mono1__neg,axiom,
! [A3: int,A7: int,B: int] :
( ( ord_less_eq_int @ A3 @ A7 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1107_zdiv__mono2__neg,axiom,
! [A3: int,B6: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_eq_int @ B6 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B6 ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1108_div__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
= ( ( K = zero_zero_int )
| ( L2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L2 ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1109_div__nonneg__neg__le0,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1110_div__nonpos__pos__le0,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1111_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1112_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1113_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1114_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B ) )
= ( ( ord_less_eq_int @ B @ A3 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1115_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1116_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1117_nat__div__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y2 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib
thf(fact_1118_nat__div__distrib_H,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y2 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib'
thf(fact_1119_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1120_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1121_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1122_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1123_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1124_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1125_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1126_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1127_ln__root,axiom,
! [N: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_1128_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1129_real__root__ge__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ one_one_real @ Y2 ) ) ) ).
% real_root_ge_1_iff
thf(fact_1130_real__root__eq__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y2 ) )
= ( X = Y2 ) ) ) ).
% real_root_eq_iff
thf(fact_1131_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_1132_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1133_real__root__less__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ).
% real_root_less_iff
thf(fact_1134_real__root__le__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% real_root_le_iff
thf(fact_1135_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1136_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1137_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1138_real__root__gt__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y2 ) )
= ( ord_less_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_gt_0_iff
thf(fact_1139_real__root__ge__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_ge_0_iff
thf(fact_1140_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1141_real__root__gt__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y2 ) )
= ( ord_less_real @ one_one_real @ Y2 ) ) ) ).
% real_root_gt_1_iff
thf(fact_1142_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1143_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_1144_real__root__less__mono,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_less_mono
thf(fact_1145_real__root__le__mono,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_le_mono
thf(fact_1146_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1147_real__root__strict__decreasing,axiom,
! [N: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1148_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1149_real__root__strict__increasing,axiom,
! [N: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N2 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1150_real__root__decreasing,axiom,
! [N: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1151_real__root__increasing,axiom,
! [N: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N2 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1152_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_1153_log__base__root,axiom,
! [N: nat,B: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log @ ( root @ N @ B ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% log_base_root
thf(fact_1154_powr__gt__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A3 ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_1155_powr__nonneg__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ ( powr_real @ A3 @ X ) @ zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_1156_powr__less__cancel__iff,axiom,
! [X: real,A3: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B ) )
= ( ord_less_real @ A3 @ B ) ) ) ).
% powr_less_cancel_iff
thf(fact_1157_log__one,axiom,
! [A3: real] :
( ( log @ A3 @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_1158_powr__eq__one__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ( powr_real @ A3 @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_1159_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_1160_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr_real @ X @ one_one_real )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_1161_powr__le__cancel__iff,axiom,
! [X: real,A3: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B ) )
= ( ord_less_eq_real @ A3 @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_1162_zero__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A3 @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_1163_log__less__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A3 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_1164_one__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ ( log @ A3 @ X ) )
= ( ord_less_real @ A3 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_1165_log__less__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A3 @ X ) @ one_one_real )
= ( ord_less_real @ X @ A3 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_1166_log__less__cancel__iff,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_1167_log__eq__one,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ A3 @ A3 )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_1168_zero__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A3 @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1169_log__le__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A3 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1170_one__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A3 @ X ) )
= ( ord_less_eq_real @ A3 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1171_log__le__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A3 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A3 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1172_log__le__cancel__iff,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1173_log__powr__cancel,axiom,
! [A3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ A3 @ ( powr_real @ A3 @ Y2 ) )
= Y2 ) ) ) ).
% log_powr_cancel
thf(fact_1174_powr__log__cancel,axiom,
! [A3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ A3 @ ( log @ A3 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_1175_log__base__powr,axiom,
! [A3: real,B: real,X: real] :
( ( A3 != zero_zero_real )
=> ( ( log @ ( powr_real @ A3 @ B ) @ X )
= ( divide_divide_real @ ( log @ A3 @ X ) @ B ) ) ) ).
% log_base_powr
thf(fact_1176_log__def,axiom,
( log
= ( ^ [A5: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A5 ) ) ) ) ).
% log_def
thf(fact_1177_powr__mono,axiom,
! [A3: real,B: real,X: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_mono
thf(fact_1178_less__log__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ ( log @ B @ X ) )
= ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_1179_log__less__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ B @ X ) @ Y2 )
= ( ord_less_real @ X @ ( powr_real @ B @ Y2 ) ) ) ) ) ).
% log_less_iff
thf(fact_1180_powr__non__neg,axiom,
! [A3: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A3 @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_1181_less__powr__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( powr_real @ B @ Y2 ) )
= ( ord_less_real @ ( log @ B @ X ) @ Y2 ) ) ) ) ).
% less_powr_iff
thf(fact_1182_powr__less__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X )
= ( ord_less_real @ Y2 @ ( log @ B @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_1183_powr__less__mono2__neg,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( powr_real @ Y2 @ A3 ) @ ( powr_real @ X @ A3 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1184_powr__le__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X )
= ( ord_less_eq_real @ Y2 @ ( log @ B @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_1185_le__powr__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y2 ) )
= ( ord_less_eq_real @ ( log @ B @ X ) @ Y2 ) ) ) ) ).
% le_powr_iff
thf(fact_1186_log__le__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y2 )
= ( ord_less_eq_real @ X @ ( powr_real @ B @ Y2 ) ) ) ) ) ).
% log_le_iff
thf(fact_1187_le__log__iff,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ ( log @ B @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_1188_powr__less__cancel,axiom,
! [X: real,A3: real,B: real] :
( ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% powr_less_cancel
thf(fact_1189_powr__less__mono,axiom,
! [A3: real,B: real,X: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_less_mono
thf(fact_1190_powr__ge__pzero,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y2 ) ) ).
% powr_ge_pzero
thf(fact_1191_powr__mono2,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).
% powr_mono2
thf(fact_1192_powr__powr__swap,axiom,
! [X: real,A3: real,B: real] :
( ( powr_real @ ( powr_real @ X @ A3 ) @ B )
= ( powr_real @ ( powr_real @ X @ B ) @ A3 ) ) ).
% powr_powr_swap
thf(fact_1193_powr__powr,axiom,
! [X: real,A3: real,B: real] :
( ( powr_real @ ( powr_real @ X @ A3 ) @ B )
= ( powr_real @ X @ ( times_times_real @ A3 @ B ) ) ) ).
% powr_powr
thf(fact_1194_log__powr,axiom,
! [X: real,B: real,Y2: real] :
( ( X != zero_zero_real )
=> ( ( log @ B @ ( powr_real @ X @ Y2 ) )
= ( times_times_real @ Y2 @ ( log @ B @ X ) ) ) ) ).
% log_powr
thf(fact_1195_powr__less__mono2,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1196_powr__mono2_H,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ Y2 @ A3 ) @ ( powr_real @ X @ A3 ) ) ) ) ) ).
% powr_mono2'
thf(fact_1197_gr__one__powr,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y2 ) ) ) ) ).
% gr_one_powr
thf(fact_1198_powr__inj,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ( powr_real @ A3 @ X )
= ( powr_real @ A3 @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% powr_inj
thf(fact_1199_powr__le1,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_1200_powr__mono__both,axiom,
! [A3: real,B: real,X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y2 @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1201_ge__one__powr__ge__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A3 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1202_powr__divide,axiom,
! [X: real,Y2: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( divide_divide_real @ X @ Y2 ) @ A3 )
= ( divide_divide_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).
% powr_divide
thf(fact_1203_powr__mult,axiom,
! [X: real,Y2: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( times_times_real @ X @ Y2 ) @ A3 )
= ( times_times_real @ ( powr_real @ X @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_1204_divide__powr__uminus,axiom,
! [A3: real,B: real,C2: real] :
( ( divide_divide_real @ A3 @ ( powr_real @ B @ C2 ) )
= ( times_times_real @ A3 @ ( powr_real @ B @ ( uminus_uminus_real @ C2 ) ) ) ) ).
% divide_powr_uminus
thf(fact_1205_ln__powr,axiom,
! [X: real,Y2: real] :
( ( X != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X @ Y2 ) )
= ( times_times_real @ Y2 @ ( ln_ln_real @ X ) ) ) ) ).
% ln_powr
thf(fact_1206_log__base__change,axiom,
! [A3: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ B @ X )
= ( divide_divide_real @ ( log @ A3 @ X ) @ ( log @ A3 @ B ) ) ) ) ) ).
% log_base_change
thf(fact_1207_powr__neg__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X ) ) ) ).
% powr_neg_one
thf(fact_1208_ln__powr__bound,axiom,
! [X: real,A3: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A3 ) @ A3 ) ) ) ) ).
% ln_powr_bound
thf(fact_1209_ln__powr__bound2,axiom,
! [X: real,A3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A3 ) @ ( times_times_real @ ( powr_real @ A3 @ A3 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_1210_log__eq__div__ln__mult__log,axiom,
! [A3: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A3 @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A3 ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_1211_log__root,axiom,
! [N: nat,A3: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( log @ B @ ( root @ N @ A3 ) )
= ( divide_divide_real @ ( log @ B @ A3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_1212_log__minus__eq__powr,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ ( log @ B @ X ) @ Y2 )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y2 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_1213_log__of__power__le,axiom,
! [M2: nat,B: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_1214_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_1215_log__pow__cancel,axiom,
! [A3: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( log @ A3 @ ( power_power_real @ A3 @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_1216_real__arch__pow,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1217_real__arch__pow__inv,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1218_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_1219_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [X5: real] :
( ( ord_less_real @ zero_zero_real @ X5 )
& ( ( power_power_real @ X5 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X5 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1220_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1221_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1222_ln__div,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y2 ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).
% ln_div
thf(fact_1223_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_1224_log__of__power__eq,axiom,
! [M2: nat,B: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ).
% log_of_power_eq
thf(fact_1225_less__log__of__power,axiom,
! [B: real,N: nat,M2: real] :
( ( ord_less_real @ ( power_power_real @ B @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).
% less_log_of_power
thf(fact_1226_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_1227_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_1228_real__root__pos__unique,axiom,
! [N: nat,Y2: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( power_power_real @ Y2 @ N )
= X )
=> ( ( root @ N @ X )
= Y2 ) ) ) ) ).
% real_root_pos_unique
thf(fact_1229_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1230_ln__diff__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y2 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y2 ) @ Y2 ) ) ) ) ).
% ln_diff_le
thf(fact_1231_log__divide,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( A3 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( log @ A3 @ ( divide_divide_real @ X @ Y2 ) )
= ( minus_minus_real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y2 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1232_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1233_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1234_le__log__of__power,axiom,
! [B: real,N: nat,M2: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_1235_log__base__pow,axiom,
! [A3: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( log @ ( power_power_real @ A3 @ N ) @ X )
= ( divide_divide_real @ ( log @ A3 @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1236_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1237_log__nat__power,axiom,
! [X: real,B: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% log_nat_power
thf(fact_1238_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1239_log__of__power__less,axiom,
! [M2: nat,B: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_1240_minus__log__eq__powr,axiom,
! [B: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ Y2 @ ( log @ B @ X ) )
= ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y2 ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_1241_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1242_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1243_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1244_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_1245_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1246_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1247_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1248_zle__diff1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1249_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_1250_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1251_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1252_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1253_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1254_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1255_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1256_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1257_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1258_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1259_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1260_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1261_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1262_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M6: nat,N3: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1263_nat__power__eq,axiom,
! [Z3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( power_power_int @ Z3 @ N ) )
= ( power_power_nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1264_nat__diff__distrib_H,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y2 ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1265_nat__diff__distrib,axiom,
! [Z5: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z3 )
=> ( ( nat2 @ ( minus_minus_int @ Z3 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1266_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
% Helper facts (9)
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X: b,Y2: b] :
( ( if_b @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X: b,Y2: b] :
( ( if_b @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_vec_b2 @ c @ ( set_vec_b2 @ ( cols_b @ matrix ) ) ).
%------------------------------------------------------------------------------