TPTP Problem File: SLH0321^1.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/0036_Rank_Argument_General/prob_00197_008981__28139468_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1521 ( 510 unt; 249 typ; 0 def)
% Number of atoms : 3742 (1222 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11225 ( 344 ~; 78 |; 238 &;8900 @)
% ( 0 <=>;1665 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 676 ( 676 >; 0 *; 0 +; 0 <<)
% Number of symbols : 225 ( 222 usr; 16 con; 0-3 aty)
% Number of variables : 3631 ( 269 ^;3265 !; 97 ?;3631 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:49:18.936
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Ovec_Itf__a_J_J_J,type,
set_set_vec_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $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__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
set_vec_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Set__Oset_It__Nat__Onat_J_J,type,
vec_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_Itf__a_J_J,type,
list_vec_a: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
set_char: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $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__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__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (222)
thf(sy_c_Gauss__Jordan__Elimination_Omat__swaprows_001tf__a,type,
gauss_2482569599970757219rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001tf__a,type,
gauss_3598389698021192302_fun_a: mat_a > ( nat > nat ) > nat > $o ).
thf(sy_c_General_Ofilter__min__append_001t__List__Olist_It__Nat__Onat_J,type,
filter8299397450776693369st_nat: ( list_nat > list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_General_Ofilter__min__append_001t__Matrix__Omat_Itf__a_J,type,
filter3550216357054876148_mat_a: ( mat_a > mat_a > $o ) > list_mat_a > list_mat_a > list_mat_a ).
thf(sy_c_General_Ofilter__min__append_001t__Matrix__Ovec_It__Nat__Onat_J,type,
filter8059918229793973476ec_nat: ( vec_nat > vec_nat > $o ) > list_vec_nat > list_vec_nat > list_vec_nat ).
thf(sy_c_General_Ofilter__min__append_001t__Matrix__Ovec_Itf__a_J,type,
filter4530267379147604608_vec_a: ( vec_a > vec_a > $o ) > list_vec_a > list_vec_a > list_vec_a ).
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__Set__Oset_It__Nat__Onat_J,type,
filter1382172366655788959et_nat: ( set_nat > set_nat > $o ) > list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_General_Ofilter__min__append_001tf__a,type,
filter_min_append_a: ( a > a > $o ) > list_a > list_a > list_a ).
thf(sy_c_General_Ofilter__min__aux_001t__Nat__Onat,type,
filter_min_aux_nat: ( nat > nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_General_Oinsert__list_001t__Nat__Onat,type,
insert_list_nat: nat > list_nat > list_nat ).
thf(sy_c_General_Oord__class_Omax__list_001t__Nat__Onat,type,
ord_max_list_nat: list_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
minus_minus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_4779711351438577370et_nat: vec_set_nat > vec_set_nat > vec_set_nat ).
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__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
minus_6230920740010926198_vec_a: set_vec_a > set_vec_a > set_vec_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_Itf__a_J,type,
plus_plus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
plus_plus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_Itf__a_J,type,
plus_plus_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_Incidence__Matrices_Oinc__mat__of_001t__Nat__Onat_001t__Int__Oint,type,
incide1177982898701834729at_int: list_nat > list_set_nat > mat_int ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001t__Nat__Onat,type,
incide6854414339478298687ol_nat: mat_nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001tf__a,type,
incide3034858701194040399_col_a: mat_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Nat__Onat,type,
incide6998539924841383625em_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001tf__a,type,
incide2997380824311827481_mat_a: mat_a > $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_001tf__a,type,
incide7367983062745021296trix_a: mat_a > $o ).
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_001tf__a,type,
incide5355957740755015148lock_a: vec_a > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_List_OListMem_001t__List__Olist_It__Nat__Onat_J,type,
listMem_list_nat: list_nat > list_list_nat > $o ).
thf(sy_c_List_OListMem_001t__Matrix__Omat_Itf__a_J,type,
listMem_mat_a: mat_a > list_mat_a > $o ).
thf(sy_c_List_OListMem_001t__Matrix__Ovec_It__Nat__Onat_J,type,
listMem_vec_nat: vec_nat > list_vec_nat > $o ).
thf(sy_c_List_OListMem_001t__Matrix__Ovec_Itf__a_J,type,
listMem_vec_a: vec_a > list_vec_a > $o ).
thf(sy_c_List_OListMem_001t__Nat__Onat,type,
listMem_nat: nat > list_nat > $o ).
thf(sy_c_List_OListMem_001t__Set__Oset_It__Nat__Onat_J,type,
listMem_set_nat: set_nat > list_set_nat > $o ).
thf(sy_c_List_OListMem_001tf__a,type,
listMem_a: a > list_a > $o ).
thf(sy_c_List_Ocan__select_001t__List__Olist_It__Nat__Onat_J,type,
can_select_list_nat: ( list_nat > $o ) > set_list_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Matrix__Omat_Itf__a_J,type,
can_select_mat_a: ( mat_a > $o ) > set_mat_a > $o ).
thf(sy_c_List_Ocan__select_001t__Matrix__Ovec_It__Nat__Onat_J,type,
can_select_vec_nat: ( vec_nat > $o ) > set_vec_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Matrix__Ovec_Itf__a_J,type,
can_select_vec_a: ( vec_a > $o ) > set_vec_a > $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__Set__Oset_It__Nat__Onat_J,type,
can_select_set_nat: ( set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Odistinct_001t__Matrix__Ovec_It__Nat__Onat_J,type,
distinct_vec_nat: list_vec_nat > $o ).
thf(sy_c_List_Odistinct_001t__Matrix__Ovec_Itf__a_J,type,
distinct_vec_a: list_vec_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Matrix__Omat_Itf__a_J,type,
insert_mat_a: mat_a > list_mat_a > list_mat_a ).
thf(sy_c_List_Oinsert_001t__Matrix__Ovec_It__Nat__Onat_J,type,
insert_vec_nat: vec_nat > list_vec_nat > list_vec_nat ).
thf(sy_c_List_Oinsert_001t__Matrix__Ovec_Itf__a_J,type,
insert_vec_a: vec_a > list_vec_a > list_vec_a ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Matrix__Ovec_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_vec_a_set_nat: ( vec_a > set_nat ) > list_vec_a > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Matrix__Ovec_Itf__a_J,type,
map_nat_vec_a: ( nat > vec_a ) > list_nat > list_vec_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_Itf__a_J,type,
set_mat_a2: list_mat_a > set_mat_a ).
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_Itf__a_J,type,
set_vec_a2: list_vec_a > set_vec_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Matrix__Omat_Itf__a_J,type,
list_ex1_mat_a: ( mat_a > $o ) > list_mat_a > $o ).
thf(sy_c_List_Olist__ex1_001t__Matrix__Ovec_It__Nat__Onat_J,type,
list_ex1_vec_nat: ( vec_nat > $o ) > list_vec_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Matrix__Ovec_Itf__a_J,type,
list_ex1_vec_a: ( vec_a > $o ) > list_vec_a > $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__Set__Oset_It__Nat__Onat_J,type,
list_ex1_set_nat: ( set_nat > $o ) > list_set_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: list_mat_a > mat_a > $o ).
thf(sy_c_List_Omember_001t__Matrix__Ovec_It__Nat__Onat_J,type,
member_vec_nat: list_vec_nat > vec_nat > $o ).
thf(sy_c_List_Omember_001t__Matrix__Ovec_Itf__a_J,type,
member_vec_a: list_vec_a > vec_a > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: list_set_nat > set_nat > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_Onth_001t__Matrix__Ovec_Itf__a_J,type,
nth_vec_a: list_vec_a > nat > vec_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(sy_c_List_Orotate1_001t__Matrix__Ovec_It__Nat__Onat_J,type,
rotate1_vec_nat: list_vec_nat > list_vec_nat ).
thf(sy_c_List_Orotate1_001t__Matrix__Ovec_Itf__a_J,type,
rotate1_vec_a: list_vec_a > list_vec_a ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List__Index_Oindex_001t__Nat__Onat,type,
list_index_nat: list_nat > nat > nat ).
thf(sy_c_List__Index_Oinsert__nth_001t__Nat__Onat,type,
list_insert_nth_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_List__Index_Olast__index_001t__List__Olist_It__Nat__Onat_J,type,
list_l1283075598211768661st_nat: list_list_nat > list_nat > nat ).
thf(sy_c_List__Index_Olast__index_001t__Matrix__Omat_Itf__a_J,type,
list_l7490309137216252624_mat_a: list_mat_a > mat_a > nat ).
thf(sy_c_List__Index_Olast__index_001t__Matrix__Ovec_It__Nat__Onat_J,type,
list_l9074094950320424200ec_nat: list_vec_nat > vec_nat > nat ).
thf(sy_c_List__Index_Olast__index_001t__Matrix__Ovec_Itf__a_J,type,
list_l8470360159308981084_vec_a: list_vec_a > vec_a > 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__Set__Oset_It__Nat__Onat_J,type,
list_l7745475043189873787et_nat: list_set_nat > set_nat > nat ).
thf(sy_c_List__Index_Olast__index_001tf__a,type,
list_last_index_a: list_a > a > nat ).
thf(sy_c_List__Index_Omap__index_H_001t__Nat__Onat_001t__Nat__Onat,type,
list_m6542280337143352873at_nat: nat > ( nat > nat > nat ) > list_nat > list_nat ).
thf(sy_c_List__Index_Oremove__nth_001t__Nat__Onat,type,
list_remove_nth_nat: nat > list_nat > list_nat ).
thf(sy_c_Macaulay__Matrix_Onzrows_001tf__a,type,
macaulay_nzrows_a: mat_a > list_vec_a ).
thf(sy_c_Matrix_Oappend__rows_001tf__a,type,
append_rows_a: mat_a > mat_a > mat_a ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001tf__a,type,
carrier_vec_a: nat > set_vec_a ).
thf(sy_c_Matrix_Ocol_001t__Nat__Onat,type,
col_nat: mat_nat > nat > vec_nat ).
thf(sy_c_Matrix_Ocol_001tf__a,type,
col_a: mat_a > nat > vec_a ).
thf(sy_c_Matrix_Ocols_001t__Nat__Onat,type,
cols_nat: mat_nat > list_vec_nat ).
thf(sy_c_Matrix_Ocols_001tf__a,type,
cols_a: mat_a > list_vec_a ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Nat__Onat,type,
dim_vec_nat: vec_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Set__Oset_It__Nat__Onat_J,type,
dim_vec_set_nat: vec_set_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001tf__a,type,
dim_vec_a: vec_a > nat ).
thf(sy_c_Matrix_Oelements__mat_001t__Nat__Onat,type,
elements_mat_nat: mat_nat > set_nat ).
thf(sy_c_Matrix_Oelements__mat_001tf__a,type,
elements_mat_a: mat_a > set_a ).
thf(sy_c_Matrix_Omat__of__cols_001tf__a,type,
mat_of_cols_a: nat > list_vec_a > mat_a ).
thf(sy_c_Matrix_Omat__of__row_001tf__a,type,
mat_of_row_a: vec_a > mat_a ).
thf(sy_c_Matrix_Omat__of__rows_001tf__a,type,
mat_of_rows_a: nat > list_vec_a > mat_a ).
thf(sy_c_Matrix_Omk__diagonal_001t__Nat__Onat,type,
mk_diagonal_nat: list_nat > mat_nat ).
thf(sy_c_Matrix_Omk__diagonal_001tf__a,type,
mk_diagonal_a: list_a > mat_a ).
thf(sy_c_Matrix_Orow_001t__Nat__Onat,type,
row_nat: mat_nat > nat > vec_nat ).
thf(sy_c_Matrix_Orow_001tf__a,type,
row_a: mat_a > nat > vec_a ).
thf(sy_c_Matrix_Orows_001tf__a,type,
rows_a: mat_a > list_vec_a ).
thf(sy_c_Matrix_Oupdate__mat_001tf__a,type,
update_mat_a: mat_a > product_prod_nat_nat > a > mat_a ).
thf(sy_c_Matrix_Ovec__index_001t__Nat__Onat,type,
vec_index_nat: vec_nat > nat > nat ).
thf(sy_c_Matrix_Ovec__index_001t__Set__Oset_It__Nat__Onat_J,type,
vec_index_set_nat: vec_set_nat > nat > set_nat ).
thf(sy_c_Matrix_Ovec__index_001tf__a,type,
vec_index_a: vec_a > nat > a ).
thf(sy_c_Matrix_Ovec__set_001t__Nat__Onat,type,
vec_set_nat2: vec_nat > set_nat ).
thf(sy_c_Matrix_Ovec__set_001tf__a,type,
vec_set_a: vec_a > set_a ).
thf(sy_c_Matrix_Ozero__vec_001t__Nat__Onat,type,
zero_vec_nat: nat > vec_nat ).
thf(sy_c_Matrix_Ozero__vec_001tf__a,type,
zero_vec_a: nat > vec_a ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Nat__Onat,type,
matrix2751262895470517546ec_nat: nat > vec_nat ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001tf__a_001t__Int__Oint,type,
matrix2825909983993355945_a_int: mat_a > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001tf__a_001tf__a,type,
matrix876235357355983809at_a_a: mat_a > mat_a ).
thf(sy_c_Missing__List_Oadjust__idx,type,
missing_adjust_idx: nat > nat > nat ).
thf(sy_c_Missing__List_Oadjust__idx__rev,type,
missin3815256168798769645dx_rev: nat > nat > nat ).
thf(sy_c_Missing__List_Olist__diff_001t__Nat__Onat,type,
missin818507234016924876ff_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Olist__union_001t__Nat__Onat,type,
missin7861371969718421194on_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Opermut_001t__Nat__Onat,type,
missing_permut_nat: list_nat > ( nat > nat ) > list_nat ).
thf(sy_c_Missing__List_Opermut__aux_001t__Nat__Onat,type,
missin1888654203714970382ux_nat: list_nat > ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_Missing__List_Oremdups__sort_001t__Nat__Onat,type,
missin6101193410121742181rt_nat: list_nat > list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001t__Nat__Onat,type,
missin7175274867594579095th_nat: nat > list_nat > list_nat ).
thf(sy_c_More__Matrix_Otake__cols_001tf__a,type,
more_take_cols_a: mat_a > list_nat > mat_a ).
thf(sy_c_More__Matrix_Otake__cols__var_001tf__a,type,
more_take_cols_var_a: mat_a > list_nat > mat_a ).
thf(sy_c_More__Matrix_Otake__rows_001tf__a,type,
more_take_rows_a: mat_a > list_nat > mat_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Ovec_Itf__a_J_J,type,
size_size_list_vec_a: list_vec_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Nat__Onat,type,
size_size_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
bot_bot_set_vec_a: set_vec_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Ovec_It__Nat__Onat_J,type,
ord_less_vec_nat: vec_nat > vec_nat > $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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_less_set_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_less_set_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_le3318621148231462513_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
ord_le7536782659060323133ec_nat: set_vec_nat > set_vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4791951621262958845_vec_a: set_vec_a > set_vec_a > $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__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__a_J,type,
collect_vec_a: ( vec_a > $o ) > set_vec_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Matrix__Ovec_Itf__a_J,type,
insert_vec_a2: vec_a > set_vec_a > set_vec_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Matrix__Ovec_It__Nat__Onat_J,type,
set_or1228852706299032398ec_nat: vec_nat > vec_nat > set_vec_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_or5134978324852575692_vec_a: set_vec_a > set_vec_a > set_set_vec_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Utility_Omax__list,type,
max_list: list_nat > nat ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a2: mat_a > set_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Nat__Onat_J,type,
member_vec_nat2: vec_nat > set_vec_nat > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__a_J,type,
member_vec_a2: vec_a > set_vec_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
member_set_vec_a: set_vec_a > set_set_vec_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_low,type,
low: nat ).
thf(sy_v_up,type,
up: nat ).
% Relevant facts (1268)
thf(fact_0_assms_I1_J,axiom,
ord_less_nat @ i @ ( dim_row_a @ a2 ) ).
% assms(1)
thf(fact_1_assms_I4_J,axiom,
ord_less_nat @ i @ up ).
% assms(4)
thf(fact_2_assms_I3_J,axiom,
ord_less_eq_nat @ low @ i ).
% assms(3)
thf(fact_3_assms_I5_J,axiom,
( ( ord_less_nat @ l @ low )
| ( ord_less_eq_nat @ up @ l ) ) ).
% assms(5)
thf(fact_4_d,axiom,
distinct_nat @ ( upt @ low @ up ) ).
% d
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_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I: nat,J: nat] : ( set_nat2 @ ( upt @ I @ J ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_7_in__set__member,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( member_mat_a2 @ X @ ( set_mat_a2 @ Xs ) )
= ( member_mat_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_8_in__set__member,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( member_list_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_9_in__set__member,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
= ( member_set_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_10_in__set__member,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_11_in__set__member,axiom,
! [X: vec_a,Xs: list_vec_a] :
( ( member_vec_a2 @ X @ ( set_vec_a2 @ Xs ) )
= ( member_vec_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_12_in__set__member,axiom,
! [X: vec_nat,Xs: list_vec_nat] :
( ( member_vec_nat2 @ X @ ( set_vec_nat2 @ Xs ) )
= ( member_vec_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_13_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_14_list__ex1__iff,axiom,
( list_ex1_mat_a
= ( ^ [P: mat_a > $o,Xs2: list_mat_a] :
? [X2: mat_a] :
( ( member_mat_a2 @ X2 @ ( set_mat_a2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: mat_a] :
( ( ( member_mat_a2 @ Y @ ( set_mat_a2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_15_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P: list_nat > $o,Xs2: list_list_nat] :
? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: list_nat] :
( ( ( member_list_nat2 @ Y @ ( set_list_nat2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_16_list__ex1__iff,axiom,
( list_ex1_set_nat
= ( ^ [P: set_nat > $o,Xs2: list_set_nat] :
? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: set_nat] :
( ( ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_17_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P: a > $o,Xs2: list_a] :
? [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: a] :
( ( ( member_a2 @ Y @ ( set_a2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_18_list__ex1__iff,axiom,
( list_ex1_vec_a
= ( ^ [P: vec_a > $o,Xs2: list_vec_a] :
? [X2: vec_a] :
( ( member_vec_a2 @ X2 @ ( set_vec_a2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: vec_a] :
( ( ( member_vec_a2 @ Y @ ( set_vec_a2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_19_list__ex1__iff,axiom,
( list_ex1_vec_nat
= ( ^ [P: vec_nat > $o,Xs2: list_vec_nat] :
? [X2: vec_nat] :
( ( member_vec_nat2 @ X2 @ ( set_vec_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: vec_nat] :
( ( ( member_vec_nat2 @ Y @ ( set_vec_nat2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_20_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_21_in__set__insert,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( member_mat_a2 @ X @ ( set_mat_a2 @ Xs ) )
=> ( ( insert_mat_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_22_in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat2 @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_23_in__set__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_24_in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_25_in__set__insert,axiom,
! [X: vec_a,Xs: list_vec_a] :
( ( member_vec_a2 @ X @ ( set_vec_a2 @ Xs ) )
=> ( ( insert_vec_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_26_in__set__insert,axiom,
! [X: vec_nat,Xs: list_vec_nat] :
( ( member_vec_nat2 @ X @ ( set_vec_nat2 @ Xs ) )
=> ( ( insert_vec_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_27_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_28_max__list__eq__set,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( ( max_list @ Xs )
= ( max_list @ Ys ) ) ) ).
% max_list_eq_set
thf(fact_29_last__index__eq__index__conv,axiom,
! [X: mat_a,Xs: list_mat_a,Y2: mat_a] :
( ( ( member_mat_a2 @ X @ ( set_mat_a2 @ Xs ) )
| ( member_mat_a2 @ Y2 @ ( set_mat_a2 @ Xs ) ) )
=> ( ( ( list_l7490309137216252624_mat_a @ Xs @ X )
= ( list_l7490309137216252624_mat_a @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_30_last__index__eq__index__conv,axiom,
! [X: list_nat,Xs: list_list_nat,Y2: list_nat] :
( ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
| ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ Xs ) ) )
=> ( ( ( list_l1283075598211768661st_nat @ Xs @ X )
= ( list_l1283075598211768661st_nat @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_31_last__index__eq__index__conv,axiom,
! [X: set_nat,Xs: list_set_nat,Y2: set_nat] :
( ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
| ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ Xs ) ) )
=> ( ( ( list_l7745475043189873787et_nat @ Xs @ X )
= ( list_l7745475043189873787et_nat @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_32_last__index__eq__index__conv,axiom,
! [X: a,Xs: list_a,Y2: a] :
( ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
| ( member_a2 @ Y2 @ ( set_a2 @ Xs ) ) )
=> ( ( ( list_last_index_a @ Xs @ X )
= ( list_last_index_a @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_33_last__index__eq__index__conv,axiom,
! [X: vec_a,Xs: list_vec_a,Y2: vec_a] :
( ( ( member_vec_a2 @ X @ ( set_vec_a2 @ Xs ) )
| ( member_vec_a2 @ Y2 @ ( set_vec_a2 @ Xs ) ) )
=> ( ( ( list_l8470360159308981084_vec_a @ Xs @ X )
= ( list_l8470360159308981084_vec_a @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_34_last__index__eq__index__conv,axiom,
! [X: vec_nat,Xs: list_vec_nat,Y2: vec_nat] :
( ( ( member_vec_nat2 @ X @ ( set_vec_nat2 @ Xs ) )
| ( member_vec_nat2 @ Y2 @ ( set_vec_nat2 @ Xs ) ) )
=> ( ( ( list_l9074094950320424200ec_nat @ Xs @ X )
= ( list_l9074094950320424200ec_nat @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% last_index_eq_index_conv
thf(fact_35_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_36_ListMem__iff,axiom,
( listMem_mat_a
= ( ^ [X2: mat_a,Xs2: list_mat_a] : ( member_mat_a2 @ X2 @ ( set_mat_a2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_37_ListMem__iff,axiom,
( listMem_list_nat
= ( ^ [X2: list_nat,Xs2: list_list_nat] : ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_38_ListMem__iff,axiom,
( listMem_set_nat
= ( ^ [X2: set_nat,Xs2: list_set_nat] : ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_39_ListMem__iff,axiom,
( listMem_a
= ( ^ [X2: a,Xs2: list_a] : ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_40_ListMem__iff,axiom,
( listMem_vec_a
= ( ^ [X2: vec_a,Xs2: list_vec_a] : ( member_vec_a2 @ X2 @ ( set_vec_a2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_41_ListMem__iff,axiom,
( listMem_vec_nat
= ( ^ [X2: vec_nat,Xs2: list_vec_nat] : ( member_vec_nat2 @ X2 @ ( set_vec_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_42_ListMem__iff,axiom,
( listMem_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_43_filter__min__append__minimal,axiom,
! [Xs: list_mat_a,Rel: mat_a > mat_a > $o,Ys: list_mat_a,X: mat_a,Y2: mat_a] :
( ! [X3: mat_a,Y3: mat_a] :
( ( member_mat_a2 @ X3 @ ( set_mat_a2 @ Xs ) )
=> ( ( member_mat_a2 @ Y3 @ ( set_mat_a2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: mat_a,Y3: mat_a] :
( ( member_mat_a2 @ X3 @ ( set_mat_a2 @ Ys ) )
=> ( ( member_mat_a2 @ Y3 @ ( set_mat_a2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_mat_a2 @ X @ ( set_mat_a2 @ ( filter3550216357054876148_mat_a @ Rel @ Xs @ Ys ) ) )
=> ( ( member_mat_a2 @ Y2 @ ( set_mat_a2 @ ( filter3550216357054876148_mat_a @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_44_filter__min__append__minimal,axiom,
! [Xs: list_list_nat,Rel: list_nat > list_nat > $o,Ys: list_list_nat,X: list_nat,Y2: list_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ ( filter8299397450776693369st_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ ( filter8299397450776693369st_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_45_filter__min__append__minimal,axiom,
! [Xs: list_set_nat,Rel: set_nat > set_nat > $o,Ys: list_set_nat,X: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( member_set_nat2 @ Y3 @ ( set_set_nat2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys ) )
=> ( ( member_set_nat2 @ Y3 @ ( set_set_nat2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_set_nat2 @ X @ ( set_set_nat2 @ ( filter1382172366655788959et_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ ( filter1382172366655788959et_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_46_filter__min__append__minimal,axiom,
! [Xs: list_a,Rel: a > a > $o,Ys: list_a,X: a,Y2: a] :
( ! [X3: a,Y3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y3 @ ( set_a2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: a,Y3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ( ( member_a2 @ Y3 @ ( set_a2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_a2 @ X @ ( set_a2 @ ( filter_min_append_a @ Rel @ Xs @ Ys ) ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ ( filter_min_append_a @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_47_filter__min__append__minimal,axiom,
! [Xs: list_vec_a,Rel: vec_a > vec_a > $o,Ys: list_vec_a,X: vec_a,Y2: vec_a] :
( ! [X3: vec_a,Y3: vec_a] :
( ( member_vec_a2 @ X3 @ ( set_vec_a2 @ Xs ) )
=> ( ( member_vec_a2 @ Y3 @ ( set_vec_a2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( member_vec_a2 @ X3 @ ( set_vec_a2 @ Ys ) )
=> ( ( member_vec_a2 @ Y3 @ ( set_vec_a2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_vec_a2 @ X @ ( set_vec_a2 @ ( filter4530267379147604608_vec_a @ Rel @ Xs @ Ys ) ) )
=> ( ( member_vec_a2 @ Y2 @ ( set_vec_a2 @ ( filter4530267379147604608_vec_a @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_48_filter__min__append__minimal,axiom,
! [Xs: list_vec_nat,Rel: vec_nat > vec_nat > $o,Ys: list_vec_nat,X: vec_nat,Y2: vec_nat] :
( ! [X3: vec_nat,Y3: vec_nat] :
( ( member_vec_nat2 @ X3 @ ( set_vec_nat2 @ Xs ) )
=> ( ( member_vec_nat2 @ Y3 @ ( set_vec_nat2 @ Xs ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ! [X3: vec_nat,Y3: vec_nat] :
( ( member_vec_nat2 @ X3 @ ( set_vec_nat2 @ Ys ) )
=> ( ( member_vec_nat2 @ Y3 @ ( set_vec_nat2 @ Ys ) )
=> ( ( Rel @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ) )
=> ( ( member_vec_nat2 @ X @ ( set_vec_nat2 @ ( filter8059918229793973476ec_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( member_vec_nat2 @ Y2 @ ( set_vec_nat2 @ ( filter8059918229793973476ec_nat @ Rel @ Xs @ Ys ) ) )
=> ( ( Rel @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ) ) ).
% filter_min_append_minimal
thf(fact_49_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_50_set__rotate1,axiom,
! [Xs: list_vec_a] :
( ( set_vec_a2 @ ( rotate1_vec_a @ Xs ) )
= ( set_vec_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_51_set__rotate1,axiom,
! [Xs: list_vec_nat] :
( ( set_vec_nat2 @ ( rotate1_vec_nat @ Xs ) )
= ( set_vec_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_52_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_53_assms_I2_J,axiom,
ord_less_nat @ j @ ( dim_col_a @ a2 ) ).
% assms(2)
thf(fact_54_assms_I6_J,axiom,
ord_less_nat @ l @ ( dim_row_a @ a2 ) ).
% assms(6)
thf(fact_55_distinct1__rotate,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_56_distinct__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( insert_nat @ X @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_insert
thf(fact_57_remdups__sort_I3_J,axiom,
! [Xs: list_nat] : ( distinct_nat @ ( missin6101193410121742181rt_nat @ Xs ) ) ).
% remdups_sort(3)
thf(fact_58_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_59_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K: nat] :
? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( F @ K @ I2 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K2: nat] :
? [K3: nat] :
( ( ord_less_eq_nat @ K2 @ K3 )
& ( F @ K3 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_60_distinct__upt,axiom,
! [I4: nat,J2: nat] : ( distinct_nat @ ( upt @ I4 @ J2 ) ) ).
% distinct_upt
thf(fact_61_max__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X @ ( max_list @ Xs ) ) ) ).
% max_list
thf(fact_62_atLeastLessThan__iff,axiom,
! [I4: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat2 @ I4 @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I4 )
& ( ord_less_set_nat @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_63_atLeastLessThan__iff,axiom,
! [I4: set_vec_a,L: set_vec_a,U: set_vec_a] :
( ( member_set_vec_a @ I4 @ ( set_or5134978324852575692_vec_a @ L @ U ) )
= ( ( ord_le4791951621262958845_vec_a @ L @ I4 )
& ( ord_less_set_vec_a @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_64_atLeastLessThan__iff,axiom,
! [I4: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I4 @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I4 )
& ( ord_less_set_a @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_65_atLeastLessThan__iff,axiom,
! [I4: vec_nat,L: vec_nat,U: vec_nat] :
( ( member_vec_nat2 @ I4 @ ( set_or1228852706299032398ec_nat @ L @ U ) )
= ( ( ord_less_eq_vec_nat @ L @ I4 )
& ( ord_less_vec_nat @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_66_atLeastLessThan__iff,axiom,
! [I4: nat,L: nat,U: nat] :
( ( member_nat2 @ I4 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I4 )
& ( ord_less_nat @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_67_ivl__subset,axiom,
! [I4: nat,J2: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J2 @ I4 )
| ( ( ord_less_eq_nat @ M @ I4 )
& ( ord_less_eq_nat @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_68_can__select__set__list__ex1,axiom,
! [P2: vec_a > $o,A: list_vec_a] :
( ( can_select_vec_a @ P2 @ ( set_vec_a2 @ A ) )
= ( list_ex1_vec_a @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_69_can__select__set__list__ex1,axiom,
! [P2: vec_nat > $o,A: list_vec_nat] :
( ( can_select_vec_nat @ P2 @ ( set_vec_nat2 @ A ) )
= ( list_ex1_vec_nat @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_70_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_71_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_72_order__refl,axiom,
! [X: set_vec_a] : ( ord_le4791951621262958845_vec_a @ X @ X ) ).
% order_refl
thf(fact_73_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_74_order__refl,axiom,
! [X: vec_nat] : ( ord_less_eq_vec_nat @ X @ X ) ).
% order_refl
thf(fact_75_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_76_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_77_dual__order_Orefl,axiom,
! [A2: set_vec_a] : ( ord_le4791951621262958845_vec_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_78_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_79_dual__order_Orefl,axiom,
! [A2: vec_nat] : ( ord_less_eq_vec_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_80_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_81_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_82_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_83_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_84_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_85_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A2 = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_86_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_87_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_88_can__select__def,axiom,
( can_select_vec_a
= ( ^ [P: vec_a > $o,A3: set_vec_a] :
? [X2: vec_a] :
( ( member_vec_a2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: vec_a] :
( ( ( member_vec_a2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_89_can__select__def,axiom,
( can_select_mat_a
= ( ^ [P: mat_a > $o,A3: set_mat_a] :
? [X2: mat_a] :
( ( member_mat_a2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: mat_a] :
( ( ( member_mat_a2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_90_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P: list_nat > $o,A3: set_list_nat] :
? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: list_nat] :
( ( ( member_list_nat2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_91_can__select__def,axiom,
( can_select_set_nat
= ( ^ [P: set_nat > $o,A3: set_set_nat] :
? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: set_nat] :
( ( ( member_set_nat2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_92_can__select__def,axiom,
( can_select_a
= ( ^ [P: a > $o,A3: set_a] :
? [X2: a] :
( ( member_a2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: a] :
( ( ( member_a2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_93_can__select__def,axiom,
( can_select_nat
= ( ^ [P: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_94_subset__code_I1_J,axiom,
! [Xs: list_mat_a,B2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( set_mat_a2 @ Xs ) @ B2 )
= ( ! [X2: mat_a] :
( ( member_mat_a2 @ X2 @ ( set_mat_a2 @ Xs ) )
=> ( member_mat_a2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_95_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B2 )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_96_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_97_subset__code_I1_J,axiom,
! [Xs: list_vec_nat,B2: set_vec_nat] :
( ( ord_le7536782659060323133ec_nat @ ( set_vec_nat2 @ Xs ) @ B2 )
= ( ! [X2: vec_nat] :
( ( member_vec_nat2 @ X2 @ ( set_vec_nat2 @ Xs ) )
=> ( member_vec_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_98_subset__code_I1_J,axiom,
! [Xs: list_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ Xs ) @ B2 )
= ( ! [X2: vec_a] :
( ( member_vec_a2 @ X2 @ ( set_vec_a2 @ Xs ) )
=> ( member_vec_a2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_99_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_100_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_101_set__subset__insertI,axiom,
! [Xs: list_vec_nat,X: vec_nat] : ( ord_le7536782659060323133ec_nat @ ( set_vec_nat2 @ Xs ) @ ( set_vec_nat2 @ ( insert_vec_nat @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_102_set__subset__insertI,axiom,
! [Xs: list_vec_a,X: vec_a] : ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ Xs ) @ ( set_vec_a2 @ ( insert_vec_a @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_103_set__subset__insertI,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( insert_a @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_104_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_105_order__antisym__conv,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_106_order__antisym__conv,axiom,
! [Y2: set_vec_a,X: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ Y2 @ X )
=> ( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_107_order__antisym__conv,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_108_order__antisym__conv,axiom,
! [Y2: vec_nat,X: vec_nat] :
( ( ord_less_eq_vec_nat @ Y2 @ X )
=> ( ( ord_less_eq_vec_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_109_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_110_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_111_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_vec_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_115_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_116_ord__le__eq__subst,axiom,
! [A2: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A2: vec_nat,B: vec_nat,F: vec_nat > nat,C: nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: vec_nat,Y4: vec_nat] :
( ( ord_less_eq_vec_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_vec_a,C: set_vec_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le4791951621262958845_vec_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_119_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_120_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_121_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_122_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_124_ord__eq__le__subst,axiom,
! [A2: vec_nat,F: nat > vec_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_vec_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_125_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_126_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_127_ord__eq__le__subst,axiom,
! [A2: nat,F: vec_nat > nat,B: vec_nat,C: vec_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ! [X4: vec_nat,Y4: vec_nat] :
( ( ord_less_eq_vec_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_128_ord__eq__le__subst,axiom,
! [A2: set_vec_a,F: nat > set_vec_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le4791951621262958845_vec_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_129_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_130_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_131_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_132_mem__Collect__eq,axiom,
! [A2: vec_a,P2: vec_a > $o] :
( ( member_vec_a2 @ A2 @ ( collect_vec_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A2: mat_a,P2: mat_a > $o] :
( ( member_mat_a2 @ A2 @ ( collect_mat_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A2: list_nat,P2: list_nat > $o] :
( ( member_list_nat2 @ A2 @ ( collect_list_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( member_set_nat2 @ A2 @ ( collect_set_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A: set_vec_a] :
( ( collect_vec_a
@ ^ [X2: vec_a] : ( member_vec_a2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_mat_a] :
( ( collect_mat_a
@ ^ [X2: mat_a] : ( member_mat_a2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_144_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_145_order__eq__refl,axiom,
! [X: set_nat,Y2: set_nat] :
( ( X = Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_146_order__eq__refl,axiom,
! [X: set_vec_a,Y2: set_vec_a] :
( ( X = Y2 )
=> ( ord_le4791951621262958845_vec_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_147_order__eq__refl,axiom,
! [X: set_a,Y2: set_a] :
( ( X = Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_148_order__eq__refl,axiom,
! [X: vec_nat,Y2: vec_nat] :
( ( X = Y2 )
=> ( ord_less_eq_vec_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_149_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_150_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_vec_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_vec_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A2: vec_nat,B: vec_nat,F: vec_nat > nat,C: nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: vec_nat,Y4: vec_nat] :
( ( ord_less_eq_vec_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_vec_a,C: set_vec_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le4791951621262958845_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le4791951621262958845_vec_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_160_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_161_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_162_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_163_order__subst1,axiom,
! [A2: nat,F: vec_nat > nat,B: vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ! [X4: vec_nat,Y4: vec_nat] :
( ( ord_less_eq_vec_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_164_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_165_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_166_order__subst1,axiom,
! [A2: vec_nat,F: nat > vec_nat,B: nat,C: nat] :
( ( ord_less_eq_vec_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_vec_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_167_order__subst1,axiom,
! [A2: nat,F: set_vec_a > nat,B: set_vec_a,C: set_vec_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ! [X4: set_vec_a,Y4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_168_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_169_order__subst1,axiom,
! [A2: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_170_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_171_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_vec_a,Z: set_vec_a] : ( Y5 = Z ) )
= ( ^ [A4: set_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ B3 )
& ( ord_le4791951621262958845_vec_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_172_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_173_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_nat,Z: vec_nat] : ( Y5 = Z ) )
= ( ^ [A4: vec_nat,B3: vec_nat] :
( ( ord_less_eq_vec_nat @ A4 @ B3 )
& ( ord_less_eq_vec_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_174_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_175_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_176_antisym,axiom,
! [A2: set_vec_a,B: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_177_antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A2: vec_nat,B: vec_nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( ord_less_eq_vec_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_179_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_180_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_181_dual__order_Otrans,axiom,
! [B: set_vec_a,A2: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B @ A2 )
=> ( ( ord_le4791951621262958845_vec_a @ C @ B )
=> ( ord_le4791951621262958845_vec_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_182_dual__order_Otrans,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_183_dual__order_Otrans,axiom,
! [B: vec_nat,A2: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ B @ A2 )
=> ( ( ord_less_eq_vec_nat @ C @ B )
=> ( ord_less_eq_vec_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_184_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_185_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_186_dual__order_Oantisym,axiom,
! [B: set_vec_a,A2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B @ A2 )
=> ( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_187_dual__order_Oantisym,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_188_dual__order_Oantisym,axiom,
! [B: vec_nat,A2: vec_nat] :
( ( ord_less_eq_vec_nat @ B @ A2 )
=> ( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_189_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_190_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_191_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_vec_a,Z: set_vec_a] : ( Y5 = Z ) )
= ( ^ [A4: set_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B3 @ A4 )
& ( ord_le4791951621262958845_vec_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_192_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_193_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: vec_nat,Z: vec_nat] : ( Y5 = Z ) )
= ( ^ [A4: vec_nat,B3: vec_nat] :
( ( ord_less_eq_vec_nat @ B3 @ A4 )
& ( ord_less_eq_vec_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_194_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_195_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_196_order__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_197_order__trans,axiom,
! [X: set_vec_a,Y2: set_vec_a,Z2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
=> ( ( ord_le4791951621262958845_vec_a @ Y2 @ Z2 )
=> ( ord_le4791951621262958845_vec_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_198_order__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_199_order__trans,axiom,
! [X: vec_nat,Y2: vec_nat,Z2: vec_nat] :
( ( ord_less_eq_vec_nat @ X @ Y2 )
=> ( ( ord_less_eq_vec_nat @ Y2 @ Z2 )
=> ( ord_less_eq_vec_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_200_order__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_201_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_202_order_Otrans,axiom,
! [A2: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ord_le4791951621262958845_vec_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_203_order_Otrans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_204_order_Otrans,axiom,
! [A2: vec_nat,B: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ord_less_eq_vec_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_205_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_206_order__antisym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_207_order__antisym,axiom,
! [X: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
=> ( ( ord_le4791951621262958845_vec_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_208_order__antisym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_209_order__antisym,axiom,
! [X: vec_nat,Y2: vec_nat] :
( ( ord_less_eq_vec_nat @ X @ Y2 )
=> ( ( ord_less_eq_vec_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_210_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_211_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_212_ord__le__eq__trans,axiom,
! [A2: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le4791951621262958845_vec_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_213_ord__le__eq__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_214_ord__le__eq__trans,axiom,
! [A2: vec_nat,B: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_vec_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_215_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_216_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
! [A2: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( A2 = B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ord_le4791951621262958845_vec_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_218_ord__eq__le__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( A2 = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_219_ord__eq__le__trans,axiom,
! [A2: vec_nat,B: vec_nat,C: vec_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ord_less_eq_vec_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_220_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_221_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_222_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_vec_a,Z: set_vec_a] : ( Y5 = Z ) )
= ( ^ [X2: set_vec_a,Y: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y )
& ( ord_le4791951621262958845_vec_a @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_223_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
& ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_224_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_nat,Z: vec_nat] : ( Y5 = Z ) )
= ( ^ [X2: vec_nat,Y: vec_nat] :
( ( ord_less_eq_vec_nat @ X2 @ Y )
& ( ord_less_eq_vec_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_225_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_226_le__cases3,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_227_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_228_order__less__imp__not__less,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ~ ( ord_less_set_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_229_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_230_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_231_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_232_order__less__imp__not__eq,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_233_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_234_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_235_order__less__imp__triv,axiom,
! [X: set_nat,Y2: set_nat,P2: $o] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_236_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_237_order__less__not__sym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ~ ( ord_less_set_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_238_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_239_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_240_order__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_241_order__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_242_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_243_order__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_244_order__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_245_order__less__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_246_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_247_order__less__irrefl,axiom,
! [X: set_nat] :
~ ( ord_less_set_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_248_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_249_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_250_ord__less__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_251_ord__less__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_252_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_253_ord__eq__less__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_254_ord__eq__less__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_255_ord__eq__less__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_256_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_257_order__less__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_258_order__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_259_order__less__asym_H,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ord_less_set_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_260_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_261_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_262_order__less__asym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ~ ( ord_less_set_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_263_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_264_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_265_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_266_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_267_order_Ostrict__implies__not__eq,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_268_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_269_dual__order_Ostrict__trans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_270_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_271_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_272_order_Ostrict__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_273_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_274_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_275_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P: nat > $o] :
? [N2: nat] :
( ( P @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_276_dual__order_Oirrefl,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_277_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_278_dual__order_Oasym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ~ ( ord_less_set_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_279_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_280_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_281_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_282_less__induct,axiom,
! [P2: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ).
% less_induct
thf(fact_283_ord__less__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_284_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_285_ord__eq__less__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_286_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_287_order_Oasym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ord_less_set_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_288_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_289_less__imp__neq,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_290_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_291_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_292_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_293_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_294_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_295_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_296_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_297_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_298_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_299_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_300_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M4: nat] :
( ( P2 @ X )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P2 @ M5 )
=> ~ ! [X6: nat] :
( ( P2 @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_301_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K4: nat,B: nat] :
( ( P2 @ K4 )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_302_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_303_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_304_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_305_le__trans,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K4 )
=> ( ord_less_eq_nat @ I4 @ K4 ) ) ) ).
% le_trans
thf(fact_306_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_307_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_308_order__le__imp__less__or__eq,axiom,
! [X: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
=> ( ( ord_less_set_vec_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_309_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_set_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_310_order__le__imp__less__or__eq,axiom,
! [X: vec_nat,Y2: vec_nat] :
( ( ord_less_eq_vec_nat @ X @ Y2 )
=> ( ( ord_less_vec_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_311_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_312_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_313_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_314_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_315_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_316_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_vec_a,C: set_vec_a] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le4791951621262958845_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_vec_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_317_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_vec_a,C: set_vec_a] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_le4791951621262958845_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_vec_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_318_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_319_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_320_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > vec_nat,C: vec_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_vec_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_vec_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_321_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > vec_nat,C: vec_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_vec_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_vec_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_322_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_323_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_324_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_325_order__less__le__subst1,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_326_order__less__le__subst1,axiom,
! [A2: vec_nat,F: nat > vec_nat,B: nat,C: nat] :
( ( ord_less_vec_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_vec_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_327_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_328_order__less__le__subst1,axiom,
! [A2: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_329_order__less__le__subst1,axiom,
! [A2: nat,F: vec_nat > nat,B: vec_nat,C: vec_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ! [X4: vec_nat,Y4: vec_nat] :
( ( ord_less_eq_vec_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_330_order__less__le__subst1,axiom,
! [A2: set_vec_a,F: nat > set_vec_a,B: nat,C: nat] :
( ( ord_less_set_vec_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_vec_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_331_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_332_order__less__le__subst1,axiom,
! [A2: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_333_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_334_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_335_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_336_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_vec_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_vec_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_337_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_338_order__le__less__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_339_order__le__less__subst2,axiom,
! [A2: vec_nat,B: vec_nat,F: vec_nat > nat,C: nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: vec_nat,Y4: vec_nat] :
( ( ord_less_eq_vec_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_340_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_vec_a,C: set_vec_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_vec_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_341_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_342_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_343_order__le__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_344_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_345_order__le__less__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_346_order__le__less__subst1,axiom,
! [A2: set_vec_a,F: nat > set_vec_a,B: nat,C: nat] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_vec_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_347_order__le__less__subst1,axiom,
! [A2: set_vec_a,F: set_nat > set_vec_a,B: set_nat,C: set_nat] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_vec_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_vec_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_348_order__le__less__subst1,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_349_order__le__less__subst1,axiom,
! [A2: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_350_order__le__less__subst1,axiom,
! [A2: vec_nat,F: nat > vec_nat,B: nat,C: nat] :
( ( ord_less_eq_vec_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_vec_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_351_order__le__less__subst1,axiom,
! [A2: vec_nat,F: set_nat > vec_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_vec_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
=> ( ord_less_vec_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_vec_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_352_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_353_order__less__le__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_354_order__less__le__trans,axiom,
! [X: set_vec_a,Y2: set_vec_a,Z2: set_vec_a] :
( ( ord_less_set_vec_a @ X @ Y2 )
=> ( ( ord_le4791951621262958845_vec_a @ Y2 @ Z2 )
=> ( ord_less_set_vec_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_355_order__less__le__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_356_order__less__le__trans,axiom,
! [X: vec_nat,Y2: vec_nat,Z2: vec_nat] :
( ( ord_less_vec_nat @ X @ Y2 )
=> ( ( ord_less_eq_vec_nat @ Y2 @ Z2 )
=> ( ord_less_vec_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_357_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_358_order__le__less__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_359_order__le__less__trans,axiom,
! [X: set_vec_a,Y2: set_vec_a,Z2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
=> ( ( ord_less_set_vec_a @ Y2 @ Z2 )
=> ( ord_less_set_vec_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_360_order__le__less__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_361_order__le__less__trans,axiom,
! [X: vec_nat,Y2: vec_nat,Z2: vec_nat] :
( ( ord_less_eq_vec_nat @ X @ Y2 )
=> ( ( ord_less_vec_nat @ Y2 @ Z2 )
=> ( ord_less_vec_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_362_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_363_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_364_order__neq__le__trans,axiom,
! [A2: set_vec_a,B: set_vec_a] :
( ( A2 != B )
=> ( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( ord_less_set_vec_a @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_365_order__neq__le__trans,axiom,
! [A2: set_a,B: set_a] :
( ( A2 != B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_366_order__neq__le__trans,axiom,
! [A2: vec_nat,B: vec_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ord_less_vec_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_367_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_368_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_369_order__le__neq__trans,axiom,
! [A2: set_vec_a,B: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_vec_a @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_370_order__le__neq__trans,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_371_order__le__neq__trans,axiom,
! [A2: vec_nat,B: vec_nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_vec_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_372_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_373_order__less__imp__le,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_374_order__less__imp__le,axiom,
! [X: set_vec_a,Y2: set_vec_a] :
( ( ord_less_set_vec_a @ X @ Y2 )
=> ( ord_le4791951621262958845_vec_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_375_order__less__imp__le,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_376_order__less__imp__le,axiom,
! [X: vec_nat,Y2: vec_nat] :
( ( ord_less_vec_nat @ X @ Y2 )
=> ( ord_less_eq_vec_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_377_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_378_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_379_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_380_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_381_order__less__le,axiom,
( ord_less_set_vec_a
= ( ^ [X2: set_vec_a,Y: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_382_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_383_order__less__le,axiom,
( ord_less_vec_nat
= ( ^ [X2: vec_nat,Y: vec_nat] :
( ( ord_less_eq_vec_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_384_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_385_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_386_order__le__less,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [X2: set_vec_a,Y: set_vec_a] :
( ( ord_less_set_vec_a @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_387_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_388_order__le__less,axiom,
( ord_less_eq_vec_nat
= ( ^ [X2: vec_nat,Y: vec_nat] :
( ( ord_less_vec_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_389_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_390_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_391_dual__order_Ostrict__implies__order,axiom,
! [B: set_vec_a,A2: set_vec_a] :
( ( ord_less_set_vec_a @ B @ A2 )
=> ( ord_le4791951621262958845_vec_a @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_392_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_393_dual__order_Ostrict__implies__order,axiom,
! [B: vec_nat,A2: vec_nat] :
( ( ord_less_vec_nat @ B @ A2 )
=> ( ord_less_eq_vec_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_394_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_395_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_396_order_Ostrict__implies__order,axiom,
! [A2: set_vec_a,B: set_vec_a] :
( ( ord_less_set_vec_a @ A2 @ B )
=> ( ord_le4791951621262958845_vec_a @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_397_order_Ostrict__implies__order,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_398_order_Ostrict__implies__order,axiom,
! [A2: vec_nat,B: vec_nat] :
( ( ord_less_vec_nat @ A2 @ B )
=> ( ord_less_eq_vec_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_399_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_400_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_401_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_vec_a
= ( ^ [B3: set_vec_a,A4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B3 @ A4 )
& ~ ( ord_le4791951621262958845_vec_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_402_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_403_dual__order_Ostrict__iff__not,axiom,
( ord_less_vec_nat
= ( ^ [B3: vec_nat,A4: vec_nat] :
( ( ord_less_eq_vec_nat @ B3 @ A4 )
& ~ ( ord_less_eq_vec_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_404_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_405_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_406_dual__order_Ostrict__trans2,axiom,
! [B: set_vec_a,A2: set_vec_a,C: set_vec_a] :
( ( ord_less_set_vec_a @ B @ A2 )
=> ( ( ord_le4791951621262958845_vec_a @ C @ B )
=> ( ord_less_set_vec_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_407_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_408_dual__order_Ostrict__trans2,axiom,
! [B: vec_nat,A2: vec_nat,C: vec_nat] :
( ( ord_less_vec_nat @ B @ A2 )
=> ( ( ord_less_eq_vec_nat @ C @ B )
=> ( ord_less_vec_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_409_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_410_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_411_dual__order_Ostrict__trans1,axiom,
! [B: set_vec_a,A2: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B @ A2 )
=> ( ( ord_less_set_vec_a @ C @ B )
=> ( ord_less_set_vec_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_412_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_413_dual__order_Ostrict__trans1,axiom,
! [B: vec_nat,A2: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ B @ A2 )
=> ( ( ord_less_vec_nat @ C @ B )
=> ( ord_less_vec_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_414_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_415_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_416_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_vec_a
= ( ^ [B3: set_vec_a,A4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_417_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_418_dual__order_Ostrict__iff__order,axiom,
( ord_less_vec_nat
= ( ^ [B3: vec_nat,A4: vec_nat] :
( ( ord_less_eq_vec_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_419_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_420_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_421_dual__order_Oorder__iff__strict,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [B3: set_vec_a,A4: set_vec_a] :
( ( ord_less_set_vec_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_422_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_423_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_nat
= ( ^ [B3: vec_nat,A4: vec_nat] :
( ( ord_less_vec_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_424_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_425_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_426_order_Ostrict__iff__not,axiom,
( ord_less_set_vec_a
= ( ^ [A4: set_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ B3 )
& ~ ( ord_le4791951621262958845_vec_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_427_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_428_order_Ostrict__iff__not,axiom,
( ord_less_vec_nat
= ( ^ [A4: vec_nat,B3: vec_nat] :
( ( ord_less_eq_vec_nat @ A4 @ B3 )
& ~ ( ord_less_eq_vec_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_429_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_430_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_431_order_Ostrict__trans2,axiom,
! [A2: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_less_set_vec_a @ A2 @ B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ord_less_set_vec_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_432_order_Ostrict__trans2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_433_order_Ostrict__trans2,axiom,
! [A2: vec_nat,B: vec_nat,C: vec_nat] :
( ( ord_less_vec_nat @ A2 @ B )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ord_less_vec_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_434_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_435_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_436_order_Ostrict__trans1,axiom,
! [A2: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B )
=> ( ( ord_less_set_vec_a @ B @ C )
=> ( ord_less_set_vec_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_437_order_Ostrict__trans1,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_438_order_Ostrict__trans1,axiom,
! [A2: vec_nat,B: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ A2 @ B )
=> ( ( ord_less_vec_nat @ B @ C )
=> ( ord_less_vec_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_439_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_440_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_441_order_Ostrict__iff__order,axiom,
( ord_less_set_vec_a
= ( ^ [A4: set_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_442_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_443_order_Ostrict__iff__order,axiom,
( ord_less_vec_nat
= ( ^ [A4: vec_nat,B3: vec_nat] :
( ( ord_less_eq_vec_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_444_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_445_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_446_order_Oorder__iff__strict,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A4: set_vec_a,B3: set_vec_a] :
( ( ord_less_set_vec_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_447_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_448_order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_nat
= ( ^ [A4: vec_nat,B3: vec_nat] :
( ( ord_less_vec_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_449_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_450_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_451_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_452_less__le__not__le,axiom,
( ord_less_set_vec_a
= ( ^ [X2: set_vec_a,Y: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y )
& ~ ( ord_le4791951621262958845_vec_a @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_453_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
& ~ ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_454_less__le__not__le,axiom,
( ord_less_vec_nat
= ( ^ [X2: vec_nat,Y: vec_nat] :
( ( ord_less_eq_vec_nat @ X2 @ Y )
& ~ ( ord_less_eq_vec_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_455_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_456_antisym__conv2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_457_antisym__conv2,axiom,
! [X: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
=> ( ( ~ ( ord_less_set_vec_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_458_antisym__conv2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ~ ( ord_less_set_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_459_antisym__conv2,axiom,
! [X: vec_nat,Y2: vec_nat] :
( ( ord_less_eq_vec_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_vec_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_460_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_461_antisym__conv1,axiom,
! [X: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_462_antisym__conv1,axiom,
! [X: set_vec_a,Y2: set_vec_a] :
( ~ ( ord_less_set_vec_a @ X @ Y2 )
=> ( ( ord_le4791951621262958845_vec_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_463_antisym__conv1,axiom,
! [X: set_a,Y2: set_a] :
( ~ ( ord_less_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_464_antisym__conv1,axiom,
! [X: vec_nat,Y2: vec_nat] :
( ~ ( ord_less_vec_nat @ X @ Y2 )
=> ( ( ord_less_eq_vec_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_465_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_466_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_467_nless__le,axiom,
! [A2: set_vec_a,B: set_vec_a] :
( ( ~ ( ord_less_set_vec_a @ A2 @ B ) )
= ( ~ ( ord_le4791951621262958845_vec_a @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_468_nless__le,axiom,
! [A2: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_469_nless__le,axiom,
! [A2: vec_nat,B: vec_nat] :
( ( ~ ( ord_less_vec_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_vec_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_470_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_471_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_472_leD,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ~ ( ord_less_set_nat @ X @ Y2 ) ) ).
% leD
thf(fact_473_leD,axiom,
! [Y2: set_vec_a,X: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ Y2 @ X )
=> ~ ( ord_less_set_vec_a @ X @ Y2 ) ) ).
% leD
thf(fact_474_leD,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ~ ( ord_less_set_a @ X @ Y2 ) ) ).
% leD
thf(fact_475_leD,axiom,
! [Y2: vec_nat,X: vec_nat] :
( ( ord_less_eq_vec_nat @ Y2 @ X )
=> ~ ( ord_less_vec_nat @ X @ Y2 ) ) ).
% leD
thf(fact_476_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_477_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_478_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_479_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_480_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_481_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_482_set__obtain__sublist,axiom,
! [S2: set_vec_nat,Ls: list_vec_nat] :
( ( ord_le7536782659060323133ec_nat @ S2 @ ( set_vec_nat2 @ Ls ) )
=> ~ ! [Ss: list_vec_nat] :
( ( distinct_vec_nat @ Ss )
=> ( S2
!= ( set_vec_nat2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_483_set__obtain__sublist,axiom,
! [S2: set_vec_a,Ls: list_vec_a] :
( ( ord_le4791951621262958845_vec_a @ S2 @ ( set_vec_a2 @ Ls ) )
=> ~ ! [Ss: list_vec_a] :
( ( distinct_vec_a @ Ss )
=> ( S2
!= ( set_vec_a2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_484_set__obtain__sublist,axiom,
! [S2: set_a,Ls: list_a] :
( ( ord_less_eq_set_a @ S2 @ ( set_a2 @ Ls ) )
=> ~ ! [Ss: list_a] :
( ( distinct_a @ Ss )
=> ( S2
!= ( set_a2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_485_set__obtain__sublist,axiom,
! [S2: set_nat,Ls: list_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( set_nat2 @ Ls ) )
=> ~ ! [Ss: list_nat] :
( ( distinct_nat @ Ss )
=> ( S2
!= ( set_nat2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_486_subsetI,axiom,
! [A: set_mat_a,B2: set_mat_a] :
( ! [X4: mat_a] :
( ( member_mat_a2 @ X4 @ A )
=> ( member_mat_a2 @ X4 @ B2 ) )
=> ( ord_le3318621148231462513_mat_a @ A @ B2 ) ) ).
% subsetI
thf(fact_487_subsetI,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat2 @ X4 @ A )
=> ( member_list_nat2 @ X4 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_488_subsetI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ A )
=> ( member_set_nat2 @ X4 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_489_subsetI,axiom,
! [A: set_vec_a,B2: set_vec_a] :
( ! [X4: vec_a] :
( ( member_vec_a2 @ X4 @ A )
=> ( member_vec_a2 @ X4 @ B2 ) )
=> ( ord_le4791951621262958845_vec_a @ A @ B2 ) ) ).
% subsetI
thf(fact_490_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A )
=> ( member_a2 @ X4 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_491_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( member_nat2 @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_492_psubsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_493_psubsetI,axiom,
! [A: set_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_vec_a @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_494_psubsetI,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_495_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_496_subset__antisym,axiom,
! [A: set_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B2 )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_497_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_498_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K @ I2 )
=> ( P2 @ I2 ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_499_complete__interval,axiom,
! [A2: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P2 @ A2 )
=> ( ~ ( P2 @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P2 @ X6 ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_500_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A6 ) )
= ( ord_less_nat @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_501_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_502_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_503_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_504_psubsetD,axiom,
! [A: set_vec_a,B2: set_vec_a,C: vec_a] :
( ( ord_less_set_vec_a @ A @ B2 )
=> ( ( member_vec_a2 @ C @ A )
=> ( member_vec_a2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_505_psubsetD,axiom,
! [A: set_mat_a,B2: set_mat_a,C: mat_a] :
( ( ord_less_set_mat_a @ A @ B2 )
=> ( ( member_mat_a2 @ C @ A )
=> ( member_mat_a2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_506_psubsetD,axiom,
! [A: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B2 )
=> ( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_507_psubsetD,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B2 )
=> ( ( member_set_nat2 @ C @ A )
=> ( member_set_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_508_psubsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_509_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_510_psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% psubset_trans
thf(fact_511_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_512_verit__comp__simplify1_I2_J,axiom,
! [A2: set_vec_a] : ( ord_le4791951621262958845_vec_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_513_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_514_verit__comp__simplify1_I2_J,axiom,
! [A2: vec_nat] : ( ord_less_eq_vec_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_515_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_516_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_517_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_518_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_519_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_520_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_521_minf_I2_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_522_minf_I1_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_523_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_524_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_525_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_526_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_527_pinf_I2_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_528_pinf_I1_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_529_verit__comp__simplify1_I1_J,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_530_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_531_subset__iff__psubset__eq,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A3: set_vec_a,B6: set_vec_a] :
( ( ord_less_set_vec_a @ A3 @ B6 )
| ( A3 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_532_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B6: set_a] :
( ( ord_less_set_a @ A3 @ B6 )
| ( A3 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_533_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A3 )
=> ( member_nat2 @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_534_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
=> ( member_nat2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_535_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_536_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_537_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_538_dim__col__take__rows,axiom,
! [A: mat_a,Inds: list_nat] :
( ( dim_col_a @ ( more_take_rows_a @ A @ Inds ) )
= ( dim_col_a @ A ) ) ).
% dim_col_take_rows
thf(fact_539_pivot__fun__monoton,axiom,
! [A: mat_a,F: nat > nat,Nr: nat,I4: nat,K4: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ ( dim_col_a @ A ) )
=> ( ( ( dim_row_a @ A )
= Nr )
=> ( ( ord_less_nat @ I4 @ Nr )
=> ( ( ( ord_less_nat @ K4 @ Nr )
& ( ord_less_nat @ I4 @ K4 ) )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ K4 ) ) ) ) ) ) ).
% pivot_fun_monoton
thf(fact_540_last__index__less,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ord_less_nat @ ( list_last_index_nat @ Xs @ X ) @ N ) ) ) ).
% last_index_less
thf(fact_541_max__list__ge,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X @ ( ord_max_list_nat @ Xs ) ) ) ).
% max_list_ge
thf(fact_542_greaterThanLessThan__iff,axiom,
! [I4: nat,L: nat,U: nat] :
( ( member_nat2 @ I4 @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I4 )
& ( ord_less_nat @ I4 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_543_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_544_greaterThanAtMost__iff,axiom,
! [I4: nat,L: nat,U: nat] :
( ( member_nat2 @ I4 @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I4 )
& ( ord_less_eq_nat @ I4 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_545_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_546_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_547_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_548_size__neq__size__imp__neq,axiom,
! [X: char,Y2: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_549_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_550_Ioc__inj,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( set_or6659071591806873216st_nat @ A2 @ B )
= ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ D @ C ) )
| ( ( A2 = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_551_Ioc__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B ) @ ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_552_size__last__index__conv,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( N
= ( list_last_index_nat @ Xs @ X ) )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ) ).
% size_last_index_conv
thf(fact_553_last__index__size__conv,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( ( list_last_index_nat @ Xs @ X )
= N )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ) ).
% last_index_size_conv
thf(fact_554_last__index__le__size,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( list_last_index_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% last_index_le_size
thf(fact_555_last__index__less__size__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ord_less_nat @ ( list_last_index_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) )
= ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% last_index_less_size_conv
thf(fact_556_pivot__fun__stabilizes,axiom,
! [A: mat_a,F: nat > nat,Nc: nat,I1: nat,I22: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ Nc )
=> ( ( ord_less_eq_nat @ I1 @ I22 )
=> ( ( ord_less_nat @ I22 @ ( dim_row_a @ A ) )
=> ( ( ord_less_eq_nat @ Nc @ ( F @ I1 ) )
=> ( ( F @ I22 )
= Nc ) ) ) ) ) ).
% pivot_fun_stabilizes
thf(fact_557_pivot__fun__mono,axiom,
! [A: mat_a,F: nat > nat,Nc: nat,I1: nat,I22: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ Nc )
=> ( ( ord_less_eq_nat @ I1 @ I22 )
=> ( ( ord_less_nat @ I22 @ ( dim_row_a @ A ) )
=> ( ord_less_eq_nat @ ( F @ I1 ) @ ( F @ I22 ) ) ) ) ) ).
% pivot_fun_mono
thf(fact_558_pivot__funD_I1_J,axiom,
! [A: mat_a,Nr: nat,F: nat > nat,Nc: nat,I4: nat] :
( ( ( dim_row_a @ A )
= Nr )
=> ( ( gauss_3598389698021192302_fun_a @ A @ F @ Nc )
=> ( ( ord_less_nat @ I4 @ Nr )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ Nc ) ) ) ) ).
% pivot_funD(1)
thf(fact_559_pivot__fun__mono__strict,axiom,
! [A: mat_a,F: nat > nat,Nc: nat,I1: nat,I22: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ Nc )
=> ( ( ord_less_nat @ I1 @ I22 )
=> ( ( ord_less_nat @ I22 @ ( dim_row_a @ A ) )
=> ( ( ord_less_nat @ ( F @ I1 ) @ Nc )
=> ( ord_less_nat @ ( F @ I1 ) @ ( F @ I22 ) ) ) ) ) ) ).
% pivot_fun_mono_strict
thf(fact_560_mbs_Ominimal,axiom,
! [X: nat,A: set_nat,P2: nat > $o] :
( ( member_nat2 @ X @ A )
=> ( ( P2 @ X )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_eq_nat @ ( size_size_nat @ X4 ) @ ( size_size_nat @ X ) )
& ( P2 @ X4 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_nat @ ( size_size_nat @ Xa ) @ ( size_size_nat @ X4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_561_mbs_Ominimal,axiom,
! [X: list_nat,A: set_list_nat,P2: list_nat > $o] :
( ( member_list_nat2 @ X @ A )
=> ( ( P2 @ X )
=> ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ X4 ) @ ( size_size_list_nat @ X ) )
& ( P2 @ X4 )
& ! [Xa: list_nat] :
( ( member_list_nat2 @ Xa @ A )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xa ) @ ( size_size_list_nat @ X4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_562_mbs_Ominimal,axiom,
! [X: char,A: set_char,P2: char > $o] :
( ( member_char @ X @ A )
=> ( ( P2 @ X )
=> ? [X4: char] :
( ( member_char @ X4 @ A )
& ( ord_less_eq_nat @ ( size_size_char @ X4 ) @ ( size_size_char @ X ) )
& ( P2 @ X4 )
& ! [Xa: char] :
( ( member_char @ Xa @ A )
=> ( ( ord_less_nat @ ( size_size_char @ Xa ) @ ( size_size_char @ X4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_563_mbs_Oless__not__eq,axiom,
! [X: nat,A: set_nat,Y2: nat] :
( ( member_nat2 @ X @ A )
=> ( ( ord_less_nat @ ( size_size_nat @ X ) @ ( size_size_nat @ Y2 ) )
=> ( X != Y2 ) ) ) ).
% mbs.less_not_eq
thf(fact_564_mbs_Oless__not__eq,axiom,
! [X: list_nat,A: set_list_nat,Y2: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y2 ) )
=> ( X != Y2 ) ) ) ).
% mbs.less_not_eq
thf(fact_565_mbs_Oless__not__eq,axiom,
! [X: char,A: set_char,Y2: char] :
( ( member_char @ X @ A )
=> ( ( ord_less_nat @ ( size_size_char @ X ) @ ( size_size_char @ Y2 ) )
=> ( X != Y2 ) ) ) ).
% mbs.less_not_eq
thf(fact_566_pivot__fun__swaprows,axiom,
! [A: mat_a,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,L: nat,K4: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ Jj )
=> ( ( ( dim_row_a @ A )
= Nr )
=> ( ( ( dim_col_a @ A )
= Nc )
=> ( ( ( F @ L )
= Jj )
=> ( ( ( F @ K4 )
= Jj )
=> ( ( ord_less_nat @ L @ Nr )
=> ( ( ord_less_nat @ K4 @ Nr )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_3598389698021192302_fun_a @ ( gauss_2482569599970757219rows_a @ L @ K4 @ A ) @ F @ Jj ) ) ) ) ) ) ) ) ) ).
% pivot_fun_swaprows
thf(fact_567_index__mat__swaprows_I2_J,axiom,
! [K4: nat,L: nat,A: mat_a] :
( ( dim_row_a @ ( gauss_2482569599970757219rows_a @ K4 @ L @ A ) )
= ( dim_row_a @ A ) ) ).
% index_mat_swaprows(2)
thf(fact_568_index__mat__swaprows_I3_J,axiom,
! [K4: nat,L: nat,A: mat_a] :
( ( dim_col_a @ ( gauss_2482569599970757219rows_a @ K4 @ L @ A ) )
= ( dim_col_a @ A ) ) ).
% index_mat_swaprows(3)
thf(fact_569_length__nzrows,axiom,
! [A: mat_a] : ( ord_less_eq_nat @ ( size_size_list_vec_a @ ( macaulay_nzrows_a @ A ) ) @ ( dim_row_a @ A ) ) ).
% length_nzrows
thf(fact_570_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_571_insert__nth__inverse,axiom,
! [J2: nat,Xs: list_nat,J4: nat,Xs4: list_nat,X: nat] :
( ( ord_less_eq_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ J4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs4 ) )
=> ( ( ( list_insert_nth_nat @ J2 @ X @ Xs )
= ( list_insert_nth_nat @ J4 @ X @ Xs4 ) )
=> ( J2 = J4 ) ) ) ) ) ) ).
% insert_nth_inverse
thf(fact_572_pivot__bound,axiom,
! [A: mat_a,Nr: nat,F: nat > nat,N: nat,I4: nat,J2: nat] :
( ( ( dim_row_a @ A )
= Nr )
=> ( ( gauss_3598389698021192302_fun_a @ A @ F @ N )
=> ( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ Nr )
=> ( ( ( F @ ( plus_plus_nat @ I4 @ J2 ) )
= N )
| ( ord_less_eq_nat @ ( plus_plus_nat @ J2 @ ( F @ I4 ) ) @ ( F @ ( plus_plus_nat @ I4 @ J2 ) ) ) ) ) ) ) ).
% pivot_bound
thf(fact_573_dim__col__take__cols,axiom,
! [Inds: list_nat,A: mat_a] :
( ! [J3: nat] :
( ( member_nat2 @ J3 @ ( set_nat2 @ Inds ) )
=> ( ord_less_nat @ J3 @ ( dim_col_a @ A ) ) )
=> ( ( dim_col_a @ ( more_take_cols_a @ A @ Inds ) )
= ( size_size_list_nat @ Inds ) ) ) ).
% dim_col_take_cols
thf(fact_574_index__less,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ord_less_nat @ ( list_index_nat @ Xs @ X ) @ N ) ) ) ).
% index_less
thf(fact_575_ivl__disj__un__two__touch_I1_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_576_UnCI,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( ~ ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ A ) )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_577_Un__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) )
= ( ( member_nat2 @ C @ A )
| ( member_nat2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_578_nat__add__left__cancel__less,axiom,
! [K4: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K4 @ M ) @ ( plus_plus_nat @ K4 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_579_Un__subset__iff,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C3 )
= ( ( ord_less_eq_set_nat @ A @ C3 )
& ( ord_less_eq_set_nat @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_580_nat__add__left__cancel__le,axiom,
! [K4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K4 @ M ) @ ( plus_plus_nat @ K4 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_581_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_index_nat @ Xs @ X )
= ( list_index_nat @ Xs @ Y2 ) )
= ( X = Y2 ) ) ) ).
% index_eq_index_conv
thf(fact_582_dim__row__take__cols,axiom,
! [A: mat_a,Ls: list_nat] :
( ( dim_row_a @ ( more_take_cols_a @ A @ Ls ) )
= ( dim_row_a @ A ) ) ).
% dim_row_take_cols
thf(fact_583_index__conv__size__if__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( list_index_nat @ Xs @ X )
= ( size_size_list_nat @ Xs ) ) ) ).
% index_conv_size_if_notin
thf(fact_584_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_585_UnE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( ~ ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_586_UnI1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_587_UnI2,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_588_bex__Un,axiom,
! [A: set_nat,B2: set_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( sup_sup_set_nat @ A @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P2 @ X2 ) )
| ? [X2: nat] :
( ( member_nat2 @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_589_ball__Un,axiom,
! [A: set_nat,B2: set_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( P2 @ X2 ) )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_590_Un__assoc,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C3 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_591_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_592_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A3 ) ) ) ).
% Un_commute
thf(fact_593_Un__left__absorb,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_594_Un__left__commute,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C3 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C3 ) ) ) ).
% Un_left_commute
thf(fact_595_atLeastLessThan__add__Un,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( set_or4665077453230672383an_nat @ I4 @ ( plus_plus_nat @ J2 @ K4 ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) @ ( set_or4665077453230672383an_nat @ J2 @ ( plus_plus_nat @ J2 @ K4 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_596_Un__mono,axiom,
! [A: set_nat,C3: set_nat,B2: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ D3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C3 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_597_Un__least,axiom,
! [A: set_nat,C3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_598_Un__upper1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_upper1
thf(fact_599_Un__upper2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_upper2
thf(fact_600_Un__absorb1,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_601_Un__absorb2,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_602_subset__UnE,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B2 )
=> ( C3
!= ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_603_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B6: set_nat] :
( ( sup_sup_set_nat @ A3 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_604_less__add__eq__less,axiom,
! [K4: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K4 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K4 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_605_trans__less__add2,axiom,
! [I4: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_606_trans__less__add1,axiom,
! [I4: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_607_add__less__mono1,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ K4 ) ) ) ).
% add_less_mono1
thf(fact_608_not__add__less2,axiom,
! [J2: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_609_not__add__less1,axiom,
! [I4: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ I4 ) ).
% not_add_less1
thf(fact_610_add__less__mono,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ K4 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_611_add__lessD1,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K4 )
=> ( ord_less_nat @ I4 @ K4 ) ) ).
% add_lessD1
thf(fact_612_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K5: nat] :
( N2
= ( plus_plus_nat @ M2 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_613_trans__le__add2,axiom,
! [I4: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_614_trans__le__add1,axiom,
! [I4: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_615_add__le__mono1,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ K4 ) ) ) ).
% add_le_mono1
thf(fact_616_add__le__mono,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ K4 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_617_le__Suc__ex,axiom,
! [K4: nat,L: nat] :
( ( ord_less_eq_nat @ K4 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K4 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_618_add__leD2,axiom,
! [M: nat,K4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K4 ) @ N )
=> ( ord_less_eq_nat @ K4 @ N ) ) ).
% add_leD2
thf(fact_619_add__leD1,axiom,
! [M: nat,K4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K4 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_620_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_621_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_622_add__leE,axiom,
! [M: nat,K4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K4 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K4 @ N ) ) ) ).
% add_leE
thf(fact_623_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_624_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K4: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K4 ) @ ( F @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_625_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or6659071591806873216st_nat @ M @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_626_index__size__conv,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( ( list_index_nat @ Xs @ X )
= N )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ) ).
% index_size_conv
thf(fact_627_size__index__conv,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( N
= ( list_index_nat @ Xs @ X ) )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ) ).
% size_index_conv
thf(fact_628_index__le__size,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( list_index_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% index_le_size
thf(fact_629_distinct__insert__nth,axiom,
! [Xs: list_nat,X: nat,I4: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( distinct_nat @ ( list_insert_nth_nat @ I4 @ X @ Xs ) ) ) ) ).
% distinct_insert_nth
thf(fact_630_index__less__size__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ord_less_nat @ ( list_index_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) )
= ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% index_less_size_conv
thf(fact_631_filter__min__append__subset,axiom,
! [Rel: nat > nat > $o,Xs: list_nat,Ys: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter1442860272890367977nd_nat @ Rel @ Xs @ Ys ) ) @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% filter_min_append_subset
thf(fact_632_ivl__disj__un__two_I1_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_633_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or5834768355832116004an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_634_le__sup__iff,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_635_le__sup__iff,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_636_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_637_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_638_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_639_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_640_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_641_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_642_set__list__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( missin7861371969718421194on_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_list_union
thf(fact_643_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_644_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_645_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_646_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_647_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_648_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_649_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_650_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J2 )
& ( ord_less_eq_nat @ K4 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_651_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( I4 = J2 )
& ( ord_less_eq_nat @ K4 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_652_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J2 )
& ( K4 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_653_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J2 )
& ( ord_less_nat @ K4 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_654_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( I4 = J2 )
& ( ord_less_nat @ K4 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_655_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J2 )
& ( K4 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_656_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_657_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_658_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_659_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_660_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_661_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_662_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_663_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_664_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_665_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_666_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_667_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_668_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_669_sup_Ocobounded2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_670_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_671_sup_Ocobounded1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_672_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_673_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_674_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_675_sup_OboundedI,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_676_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_677_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_678_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_679_sup__absorb2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_680_sup__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_681_sup__absorb1,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_682_sup__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( sup_sup_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_683_sup_Oabsorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_684_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_685_sup_Oabsorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_686_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_687_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_688_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y2: nat] :
( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_689_sup_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_690_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_691_sup_OorderE,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_692_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_693_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_694_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_695_sup__least,axiom,
! [Y2: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_696_sup__least,axiom,
! [Y2: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_697_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_698_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_699_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_700_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_701_le__supI2,axiom,
! [X: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_702_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_703_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_704_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_705_sup__ge2,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_706_sup__ge2,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_707_sup__ge1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_708_sup__ge1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_709_le__supI,axiom,
! [A2: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_710_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_711_le__supE,axiom,
! [A2: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_712_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_713_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_714_inf__sup__ord_I3_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_715_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_716_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_717_less__supI1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X @ A2 )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_718_less__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_719_less__supI2,axiom,
! [X: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ X @ B )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_720_less__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_721_sup_Oabsorb3,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_722_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_723_sup_Oabsorb4,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_724_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_725_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_set_nat @ B @ A2 )
=> ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_726_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_727_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( A4
= ( sup_sup_set_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_728_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_729_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A2 )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_730_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_731_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_732_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_733_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J2 )
& ( ord_less_nat @ K4 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_734_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J2: nat,K4: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J2 )
& ( ord_less_eq_nat @ K4 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_735_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_736_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_737_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_738_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_739_index__add__mat_I3_J,axiom,
! [A: mat_a,B2: mat_a] :
( ( dim_col_a @ ( plus_plus_mat_a @ A @ B2 ) )
= ( dim_col_a @ B2 ) ) ).
% index_add_mat(3)
thf(fact_740_index__add__mat_I2_J,axiom,
! [A: mat_a,B2: mat_a] :
( ( dim_row_a @ ( plus_plus_mat_a @ A @ B2 ) )
= ( dim_row_a @ B2 ) ) ).
% index_add_mat(2)
thf(fact_741_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_742_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_743_leq__add__left,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y2 @ X ) ) ).
% leq_add_left
thf(fact_744_leq__add__right,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( plus_plus_nat @ X @ Y2 ) ) ).
% leq_add_right
thf(fact_745_le__imp__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ? [C2: nat] :
( B
= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% le_imp_add
thf(fact_746_mk__diagonal__dim_I2_J,axiom,
! [As: list_a] :
( ( dim_col_a @ ( mk_diagonal_a @ As ) )
= ( size_size_list_a @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_747_mk__diagonal__dim_I2_J,axiom,
! [As: list_nat] :
( ( dim_col_nat @ ( mk_diagonal_nat @ As ) )
= ( size_size_list_nat @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_748_mk__diagonal__dim_I1_J,axiom,
! [As: list_a] :
( ( dim_row_a @ ( mk_diagonal_a @ As ) )
= ( size_size_list_a @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_749_mk__diagonal__dim_I1_J,axiom,
! [As: list_nat] :
( ( dim_row_nat @ ( mk_diagonal_nat @ As ) )
= ( size_size_list_nat @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_750_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_751_remove__nth__insert__nth,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_remove_nth_nat @ I4 @ ( list_insert_nth_nat @ I4 @ X @ Xs ) )
= Xs ) ) ).
% remove_nth_insert_nth
thf(fact_752_mat__of__row__dim_I1_J,axiom,
! [Y2: vec_a] :
( ( dim_row_a @ ( mat_of_row_a @ Y2 ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_753_dim__row__append__rows,axiom,
! [A: mat_a,B2: mat_a] :
( ( dim_row_a @ ( append_rows_a @ A @ B2 ) )
= ( plus_plus_nat @ ( dim_row_a @ A ) @ ( dim_row_a @ B2 ) ) ) ).
% dim_row_append_rows
thf(fact_754_dim__update__mat_I2_J,axiom,
! [A: mat_a,Ij: product_prod_nat_nat,A2: a] :
( ( dim_col_a @ ( update_mat_a @ A @ Ij @ A2 ) )
= ( dim_col_a @ A ) ) ).
% dim_update_mat(2)
thf(fact_755_dim__update__mat_I1_J,axiom,
! [A: mat_a,Ij: product_prod_nat_nat,A2: a] :
( ( dim_row_a @ ( update_mat_a @ A @ Ij @ A2 ) )
= ( dim_row_a @ A ) ) ).
% dim_update_mat(1)
thf(fact_756_distinct__remove__nth,axiom,
! [Xs: list_nat,I4: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( list_remove_nth_nat @ I4 @ Xs ) ) ) ).
% distinct_remove_nth
thf(fact_757_set__remove__nth__subset,axiom,
! [J2: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_remove_nth_nat @ J2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove_nth_subset
thf(fact_758_mat__of__rows__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_row_a @ ( mat_of_rows_a @ N @ Vs ) )
= ( size_size_list_vec_a @ Vs ) ) ).
% mat_of_rows_carrier(2)
thf(fact_759_filter__min__aux__subset,axiom,
! [Rel: nat > nat > $o,Xs: list_nat,Ys: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_min_aux_nat @ Rel @ Xs @ Ys ) ) @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% filter_min_aux_subset
thf(fact_760_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_761_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_762_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_763_atLeastAtMost__iff,axiom,
! [I4: nat,L: nat,U: nat] :
( ( member_nat2 @ I4 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I4 )
& ( ord_less_eq_nat @ I4 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_764_set__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( set_nat2 @ ( rotate_nat @ N @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_765_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_766_distinct__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( distinct_nat @ ( rotate_nat @ N @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_rotate
thf(fact_767_mat__of__rows__carrier_I3_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_col_a @ ( mat_of_rows_a @ N @ Vs ) )
= N ) ).
% mat_of_rows_carrier(3)
thf(fact_768_atLeastatMost__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_769_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_770_atLeastatMost__psubset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A2 )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_771_ivl__disj__un__two_I7_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_772_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or6659071591806873216st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_773_filter__min__aux__supset,axiom,
! [Ys: list_nat,Rel: nat > nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Ys ) @ ( set_nat2 @ ( filter_min_aux_nat @ Rel @ Xs @ Ys ) ) ) ).
% filter_min_aux_supset
thf(fact_774_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_775_ivl__disj__un__two__touch_I2_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_776_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_777_ivl__disj__un__two_I4_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or5834768355832116004an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_778_set__take__if__index,axiom,
! [Xs: list_nat,X: nat,I4: nat] :
( ( ord_less_nat @ ( list_index_nat @ Xs @ X ) @ I4 )
=> ( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ I4 @ Xs ) ) ) ) ) ).
% set_take_if_index
thf(fact_779_row__add_I2_J,axiom,
! [I4: nat,A: mat_a,B2: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A ) )
=> ( ( ( dim_row_a @ B2 )
= ( dim_row_a @ A ) )
=> ( ( ( dim_col_a @ B2 )
= ( dim_col_a @ A ) )
=> ( ( row_a @ ( plus_plus_mat_a @ A @ B2 ) @ I4 )
= ( plus_plus_vec_a @ ( row_a @ A @ I4 ) @ ( row_a @ B2 @ I4 ) ) ) ) ) ) ).
% row_add(2)
thf(fact_780_carrier__matD_I1_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a2 @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_a @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_781_carrier__matD_I2_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a2 @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_col_a @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_782_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_783_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_784_take__upt,axiom,
! [I4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I4 @ N ) )
= ( upt @ I4 @ ( plus_plus_nat @ I4 @ M ) ) ) ) ).
% take_upt
thf(fact_785_carrier__matI,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_a @ A )
= Nr )
=> ( ( ( dim_col_a @ A )
= Nc )
=> ( member_mat_a2 @ A @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_786_take__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ! [I3: nat] :
( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_787_distinct__take,axiom,
! [Xs: list_nat,I4: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( take_nat @ I4 @ Xs ) ) ) ).
% distinct_take
thf(fact_788_in__set__takeD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_789_set__take__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_790_index__take__if__set,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( ( list_index_nat @ ( take_nat @ N @ Xs ) @ X )
= ( list_index_nat @ Xs @ X ) ) ) ).
% index_take_if_set
thf(fact_791_index__take__if__index,axiom,
! [Xs: list_nat,X: nat,N: nat] :
( ( ord_less_eq_nat @ ( list_index_nat @ Xs @ X ) @ N )
=> ( ( list_index_nat @ ( take_nat @ N @ Xs ) @ X )
= ( list_index_nat @ Xs @ X ) ) ) ).
% index_take_if_index
thf(fact_792_carrier__mat__triv,axiom,
! [M: mat_a] : ( member_mat_a2 @ M @ ( carrier_mat_a @ ( dim_row_a @ M ) @ ( dim_col_a @ M ) ) ) ).
% carrier_mat_triv
thf(fact_793_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_794_eq__rowI,axiom,
! [B2: mat_a,A: mat_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ B2 ) )
=> ( ( row_a @ A @ I3 )
= ( row_a @ B2 @ I3 ) ) )
=> ( ( ( dim_row_a @ A )
= ( dim_row_a @ B2 ) )
=> ( ( ( dim_col_a @ A )
= ( dim_col_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% eq_rowI
thf(fact_795_index__take,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ I4 @ ( list_index_nat @ Xs @ X ) )
=> ~ ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ I4 @ Xs ) ) ) ) ).
% index_take
thf(fact_796_row__append__rows,axiom,
! [A: mat_a,B2: mat_a,I4: nat] :
( ( ( dim_col_a @ A )
= ( dim_col_a @ B2 ) )
=> ( ( ord_less_nat @ I4 @ ( plus_plus_nat @ ( dim_row_a @ A ) @ ( dim_row_a @ B2 ) ) )
=> ( ( ( ord_less_nat @ I4 @ ( dim_row_a @ A ) )
=> ( ( row_a @ ( append_rows_a @ A @ B2 ) @ I4 )
= ( row_a @ A @ I4 ) ) )
& ( ~ ( ord_less_nat @ I4 @ ( dim_row_a @ A ) )
=> ( ( row_a @ ( append_rows_a @ A @ B2 ) @ I4 )
= ( row_a @ B2 @ ( minus_minus_nat @ I4 @ ( dim_row_a @ A ) ) ) ) ) ) ) ) ).
% row_append_rows
thf(fact_797_row__eq__zero__iff__pivot__fun,axiom,
! [A: mat_a,F: nat > nat,I4: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ ( dim_col_a @ A ) )
=> ( ( ord_less_nat @ I4 @ ( dim_row_a @ A ) )
=> ( ( ( row_a @ A @ I4 )
= ( zero_vec_a @ ( dim_col_a @ A ) ) )
= ( ( F @ I4 )
= ( dim_col_a @ A ) ) ) ) ) ).
% row_eq_zero_iff_pivot_fun
thf(fact_798_row__not__zero__iff__pivot__fun,axiom,
! [A: mat_a,F: nat > nat,I4: nat] :
( ( gauss_3598389698021192302_fun_a @ A @ F @ ( dim_col_a @ A ) )
=> ( ( ord_less_nat @ I4 @ ( dim_row_a @ A ) )
=> ( ( ( row_a @ A @ I4 )
!= ( zero_vec_a @ ( dim_col_a @ A ) ) )
= ( ord_less_nat @ ( F @ I4 ) @ ( dim_col_a @ A ) ) ) ) ) ).
% row_not_zero_iff_pivot_fun
thf(fact_799_insert__nth__remove__nth,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_insert_nth_nat @ I4 @ ( nth_nat @ Xs @ I4 ) @ ( list_remove_nth_nat @ I4 @ Xs ) )
= Xs ) ) ).
% insert_nth_remove_nth
thf(fact_800_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_801_diff__diff__left,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K4 )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J2 @ K4 ) ) ) ).
% diff_diff_left
thf(fact_802_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_803_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_804_Nat_Oadd__diff__assoc,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K4 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K4 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_805_Nat_Oadd__diff__assoc2,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K4 ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K4 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_806_Nat_Odiff__diff__right,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J2 @ K4 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K4 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_807_nth__upt,axiom,
! [I4: nat,K4: nat,J2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ J2 )
=> ( ( nth_nat @ ( upt @ I4 @ J2 ) @ K4 )
= ( plus_plus_nat @ I4 @ K4 ) ) ) ).
% nth_upt
thf(fact_808_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_809_nth__index,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( nth_nat @ Xs @ ( list_index_nat @ Xs @ X ) )
= X ) ) ).
% nth_index
thf(fact_810_nth__last__index,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( nth_nat @ Xs @ ( list_last_index_nat @ Xs @ X ) )
= X ) ) ).
% nth_last_index
thf(fact_811_length__upt,axiom,
! [I4: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I4 @ J2 ) )
= ( minus_minus_nat @ J2 @ I4 ) ) ).
% length_upt
thf(fact_812_index__upt,axiom,
! [M: nat,I4: nat,N: nat] :
( ( ord_less_eq_nat @ M @ I4 )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( list_index_nat @ ( upt @ M @ N ) @ I4 )
= ( minus_minus_nat @ I4 @ M ) ) ) ) ).
% index_upt
thf(fact_813_Nat_Odiff__cancel,axiom,
! [K4: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K4 @ M ) @ ( plus_plus_nat @ K4 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_814_diff__cancel2,axiom,
! [M: nat,K4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K4 ) @ ( plus_plus_nat @ N @ K4 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_815_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_816_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_817_less__imp__diff__less,axiom,
! [J2: nat,K4: nat,N: nat] :
( ( ord_less_nat @ J2 @ K4 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K4 ) ) ).
% less_imp_diff_less
thf(fact_818_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_819_diff__commute,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K4 )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K4 ) @ J2 ) ) ).
% diff_commute
thf(fact_820_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_821_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_822_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_823_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_824_Nat_Odiff__diff__eq,axiom,
! [K4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K4 @ M )
=> ( ( ord_less_eq_nat @ K4 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K4 ) @ ( minus_minus_nat @ N @ K4 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_825_le__diff__iff,axiom,
! [K4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K4 @ M )
=> ( ( ord_less_eq_nat @ K4 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K4 ) @ ( minus_minus_nat @ N @ K4 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_826_eq__diff__iff,axiom,
! [K4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K4 @ M )
=> ( ( ord_less_eq_nat @ K4 @ N )
=> ( ( ( minus_minus_nat @ M @ K4 )
= ( minus_minus_nat @ N @ K4 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_827_le__imp__inv,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( B
= ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% le_imp_inv
thf(fact_828_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_829_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_830_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_831_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_832_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_833_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_834_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_835_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_836_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_837_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_838_add__le__add__imp__diff__le,axiom,
! [I4: nat,K4: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K4 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K4 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K4 ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_839_add__le__imp__le__diff,axiom,
! [I4: nat,K4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ N )
=> ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ N @ K4 ) ) ) ).
% add_le_imp_le_diff
thf(fact_840_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_841_Ex__list__of__length__P,axiom,
! [N: nat,P2: nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X6: nat] : ( P2 @ X6 @ I3 ) )
=> ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_nat @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_842_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z ) )
= ( ^ [Xs2: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys3 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_843_Skolem__list__nth,axiom,
! [K4: nat,P2: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K4 )
=> ? [X7: nat] : ( P2 @ I @ X7 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K4 )
& ! [I: nat] :
( ( ord_less_nat @ I @ K4 )
=> ( P2 @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_844_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_845_less__diff__iff,axiom,
! [K4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K4 @ M )
=> ( ( ord_less_eq_nat @ K4 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K4 ) @ ( minus_minus_nat @ N @ K4 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_846_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_847_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_848_less__diff__conv,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J2 @ K4 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K4 ) @ J2 ) ) ).
% less_diff_conv
thf(fact_849_nzrows__nth__not__zero,axiom,
! [I4: nat,A: mat_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_vec_a @ ( macaulay_nzrows_a @ A ) ) )
=> ( ( nth_vec_a @ ( macaulay_nzrows_a @ A ) @ I4 )
!= ( zero_vec_a @ ( dim_col_a @ A ) ) ) ) ).
% nzrows_nth_not_zero
thf(fact_850_le__diff__conv,axiom,
! [J2: nat,K4: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K4 ) @ I4 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I4 @ K4 ) ) ) ).
% le_diff_conv
thf(fact_851_Nat_Ole__diff__conv2,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J2 @ K4 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K4 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_852_Nat_Odiff__add__assoc,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K4 )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K4 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_853_Nat_Odiff__add__assoc2,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K4 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K4 ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_854_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I4 )
= K4 )
= ( J2
= ( plus_plus_nat @ K4 @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_855_index__first,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ ( list_index_nat @ Xs @ X ) )
=> ( X
!= ( nth_nat @ Xs @ I4 ) ) ) ).
% index_first
thf(fact_856_nth__insert__nth__back,axiom,
! [J2: nat,I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ J2 @ I4 )
=> ( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_insert_nth_nat @ J2 @ X @ Xs ) @ I4 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ I4 @ one_one_nat ) ) ) ) ) ).
% nth_insert_nth_back
thf(fact_857_nth__insert__nth,axiom,
! [I4: nat,Xs: list_nat,J2: nat,X: nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I4 = J2 )
=> ( ( nth_nat @ ( list_insert_nth_nat @ J2 @ X @ Xs ) @ I4 )
= X ) )
& ( ( I4 != J2 )
=> ( ( ( ord_less_nat @ I4 @ J2 )
=> ( ( nth_nat @ ( list_insert_nth_nat @ J2 @ X @ Xs ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) )
& ( ~ ( ord_less_nat @ I4 @ J2 )
=> ( ( nth_nat @ ( list_insert_nth_nat @ J2 @ X @ Xs ) @ I4 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ I4 @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% nth_insert_nth
thf(fact_858_mat__col__eqI,axiom,
! [B2: mat_a,A: mat_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_col_a @ B2 ) )
=> ( ( col_a @ A @ I3 )
= ( col_a @ B2 @ I3 ) ) )
=> ( ( ( dim_row_a @ A )
= ( dim_row_a @ B2 ) )
=> ( ( ( dim_col_a @ A )
= ( dim_col_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% mat_col_eqI
thf(fact_859_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_860_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_861_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_862_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_863_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_864_ex__set__conv__ex__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( P2 @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_865_distinctI,axiom,
! [Xs: list_nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
!= ( nth_nat @ Xs @ J3 ) ) ) ) )
=> ( distinct_nat @ Xs ) ) ).
% distinctI
thf(fact_866_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I != J )
=> ( ( nth_nat @ Xs2 @ I )
!= ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_867_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I4: nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Xs @ J2 ) )
= ( I4 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_868_less__diff__conv2,axiom,
! [K4: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K4 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K4 ) @ I4 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I4 @ K4 ) ) ) ) ).
% less_diff_conv2
thf(fact_869_nth__insert__nth__index__eq,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_insert_nth_nat @ I4 @ X @ Xs ) @ I4 )
= X ) ) ).
% nth_insert_nth_index_eq
thf(fact_870_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [X4: nat] :
( ( ord_less_nat @ X4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X4 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y6 )
= X ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_871_nth__take__prefix,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_872_nth__take__lemma,axiom,
! [K4: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K4 @ ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( ( take_nat @ K4 @ Xs )
= ( take_nat @ K4 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_873_index__eq__iff,axiom,
! [Xs: list_nat,X: nat,I4: nat] :
( ( ( list_index_nat @ Xs @ X )
= I4 )
= ( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I4 )
=> ( ( nth_nat @ Xs @ J )
!= X ) )
& ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% index_eq_iff
thf(fact_874_index__eqI,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( nth_nat @ Xs @ J3 )
!= X ) )
=> ( ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= X ) )
=> ( ( list_index_nat @ Xs @ X )
= I4 ) ) ) ) ).
% index_eqI
thf(fact_875_index__nth__id,axiom,
! [Xs: list_nat,N: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_index_nat @ Xs @ ( nth_nat @ Xs @ N ) )
= N ) ) ) ).
% index_nth_id
thf(fact_876_nth__insert__nth__front,axiom,
! [I4: nat,J2: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_insert_nth_nat @ J2 @ X @ Xs ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ) ).
% nth_insert_nth_front
thf(fact_877_length__remove__nth,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( list_remove_nth_nat @ I4 @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% length_remove_nth
thf(fact_878_obtain__set__list__item,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
!= X ) ) ) ).
% obtain_set_list_item
thf(fact_879_permut__sound,axiom,
! [I4: nat,As: list_nat,F: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As @ F ) @ I4 )
= ( nth_nat @ As @ ( F @ I4 ) ) ) ) ).
% permut_sound
thf(fact_880_set__map__index_H,axiom,
! [X: nat,N: nat,F: nat > nat > nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( list_m6542280337143352873at_nat @ N @ F @ Xs ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( F @ ( plus_plus_nat @ N @ I ) @ ( nth_nat @ Xs @ I ) )
= X ) ) ) ) ).
% set_map_index'
thf(fact_881_permut__aux__sound,axiom,
! [I4: nat,As: list_nat,F: nat > nat,Bs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F @ Bs ) @ I4 )
= ( nth_nat @ Bs @ ( F @ I4 ) ) ) ) ).
% permut_aux_sound
thf(fact_882_Un__Diff__cancel,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_883_Un__Diff__cancel2,axiom,
! [B2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A ) @ A )
= ( sup_sup_set_nat @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_884_ivl__diff,axiom,
! [I4: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I4 @ M ) @ ( set_or4665077453230672383an_nat @ I4 @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_885_length__map__index_H,axiom,
! [N: nat,F: nat > nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( list_m6542280337143352873at_nat @ N @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map_index'
thf(fact_886_Un__Diff,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C3 ) @ ( minus_minus_set_nat @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_887_psubset__imp__ex__mem,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ? [B4: nat] : ( member_nat2 @ B4 @ ( minus_minus_set_nat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_888_Diff__mono,axiom,
! [A: set_nat,C3: set_nat,D3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ D3 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ C3 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_889_Diff__subset,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_890_double__diff,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_891_Diff__partition,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_892_Diff__subset__conv,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C3 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_893_max__list__mono,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ? [Y6: nat] :
( ( member_nat2 @ Y6 @ ( set_nat2 @ Ys ) )
& ( ord_less_eq_nat @ X4 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( max_list @ Xs ) @ ( max_list @ Ys ) ) ) ).
% max_list_mono
thf(fact_894_set__map__index,axiom,
! [X: nat,F: nat > nat > nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( list_m6542280337143352873at_nat @ zero_zero_nat @ F @ Xs ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( F @ I @ ( nth_nat @ Xs @ I ) )
= X ) ) ) ) ).
% set_map_index
thf(fact_895_nth__rows,axiom,
! [I4: nat,A: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A ) )
=> ( ( nth_vec_a @ ( rows_a @ A ) @ I4 )
= ( row_a @ A @ I4 ) ) ) ).
% nth_rows
thf(fact_896_remove__nth__length,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_897_Diff__idemp,axiom,
! [A: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_898_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat2 @ C @ A )
& ~ ( member_nat2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_899_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ~ ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_900_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_901_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_902_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_903_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_904_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_905_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_906_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_907_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_908_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_909_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_910_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_911_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_912_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_913_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_914_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_915_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_916_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_917_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_918_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_919_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_920_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_921_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_922_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_923_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_924_length__rows,axiom,
! [A: mat_a] :
( ( size_size_list_vec_a @ ( rows_a @ A ) )
= ( dim_row_a @ A ) ) ).
% length_rows
thf(fact_925_mat__of__rows__rows,axiom,
! [A: mat_a] :
( ( mat_of_rows_a @ ( dim_col_a @ A ) @ ( rows_a @ A ) )
= A ) ).
% mat_of_rows_rows
thf(fact_926_Ball__at__Least0LessThan__conv,axiom,
! [Xs: list_nat,N: nat,P2: nat > $o] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ ( nth_nat @ Xs @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) ) ) ) ).
% Ball_at_Least0LessThan_conv
thf(fact_927_nth__map__index,axiom,
! [P5: nat,Xs: list_nat,F: nat > nat > nat] :
( ( ord_less_nat @ P5 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_m6542280337143352873at_nat @ zero_zero_nat @ F @ Xs ) @ P5 )
= ( F @ P5 @ ( nth_nat @ Xs @ P5 ) ) ) ) ).
% nth_map_index
thf(fact_928_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_929_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_930_DiffD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_931_DiffD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_932_DiffE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_933_rows__inj,axiom,
! [A: mat_a,B2: mat_a] :
( ( ( dim_col_a @ A )
= ( dim_col_a @ B2 ) )
=> ( ( ( rows_a @ A )
= ( rows_a @ B2 ) )
=> ( A = B2 ) ) ) ).
% rows_inj
thf(fact_934_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_935_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_936_zero__min,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_min
thf(fact_937_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_938_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_939_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_940_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_941_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_942_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_943_rows__carrier,axiom,
! [A: mat_a] : ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ ( rows_a @ A ) ) @ ( carrier_vec_a @ ( dim_col_a @ A ) ) ) ).
% rows_carrier
thf(fact_944_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_945_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_946_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_947_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_948_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_949_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_950_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P2 @ M2 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P2 @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_951_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P2 @ M2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_952_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_953_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_954_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_955_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_956_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_957_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_958_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_959_col__dim,axiom,
! [A: mat_a,I4: nat] : ( member_vec_a2 @ ( col_a @ A @ I4 ) @ ( carrier_vec_a @ ( dim_row_a @ A ) ) ) ).
% col_dim
thf(fact_960_row__carrier,axiom,
! [A: mat_a,I4: nat] : ( member_vec_a2 @ ( row_a @ A @ I4 ) @ ( carrier_vec_a @ ( dim_col_a @ A ) ) ) ).
% row_carrier
thf(fact_961_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_962_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_963_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_964_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_965_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_966_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_967_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_968_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_969_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_970_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_971_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_972_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_973_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_974_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_975_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_976_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_977_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_978_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_979_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_980_less__imp__add__positive,axiom,
! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I4 @ K )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_981_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_982_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P2 @ M2 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P2 @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_983_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P2 @ M2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_984_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_985_take__map__index,axiom,
! [P5: nat,F: nat > nat > nat,Xs: list_nat] :
( ( take_nat @ P5 @ ( list_m6542280337143352873at_nat @ zero_zero_nat @ F @ Xs ) )
= ( list_m6542280337143352873at_nat @ zero_zero_nat @ F @ ( take_nat @ P5 @ Xs ) ) ) ).
% take_map_index
thf(fact_986_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_987_remove__nth__id,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( missin7175274867594579095th_nat @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_988_remove__nth__sound__l,axiom,
! [P5: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ P5 @ N )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P5 )
= ( nth_nat @ Xs @ P5 ) ) ) ).
% remove_nth_sound_l
thf(fact_989_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_990_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_991_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_992_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_993_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_994_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_995_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_996_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_997_nat__diff__split,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_998_nat__diff__split__asm,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_999_map__index__congL,axiom,
! [Xs: list_nat,F: nat > nat > nat] :
( ! [P6: nat] :
( ( ord_less_nat @ P6 @ ( size_size_list_nat @ Xs ) )
=> ( ( F @ P6 @ ( nth_nat @ Xs @ P6 ) )
= ( nth_nat @ Xs @ P6 ) ) )
=> ( ( list_m6542280337143352873at_nat @ zero_zero_nat @ F @ Xs )
= Xs ) ) ).
% map_index_congL
thf(fact_1000_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_nat,P2: nat > nat > $o,P5: nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P2 @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P5 @ As ) ) )
=> ( P2 @ ( nth_nat @ ( missin7175274867594579095th_nat @ P5 @ As ) @ I2 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P5 @ Bs ) @ I2 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_1001_obtain__row__index,axiom,
! [R: vec_a,M4: mat_a] :
( ( member_vec_a2 @ R @ ( set_vec_a2 @ ( rows_a @ M4 ) ) )
=> ~ ! [I3: nat] :
( ( ( row_a @ M4 @ I3 )
= R )
=> ~ ( ord_less_nat @ I3 @ ( dim_row_a @ M4 ) ) ) ) ).
% obtain_row_index
thf(fact_1002_row__prop__cond,axiom,
! [M4: mat_a,P2: vec_a > $o,R: vec_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ M4 ) )
=> ( P2 @ ( row_a @ M4 @ I3 ) ) )
=> ( ( member_vec_a2 @ R @ ( set_vec_a2 @ ( rows_a @ M4 ) ) )
=> ( P2 @ R ) ) ) ).
% row_prop_cond
thf(fact_1003_adjust__idx__rev__nth,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( J2 != I4 )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J2 ) )
= ( nth_nat @ Xs @ J2 ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_1004_set__list__diff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( missin818507234016924876ff_nat @ Xs @ Ys ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_list_diff
thf(fact_1005_adjust__idx__rev__length,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( J2 != I4 )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J2 ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_1006_adjust__idx__nth,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) @ J2 )
= ( nth_nat @ Xs @ ( missing_adjust_idx @ I4 @ J2 ) ) ) ) ).
% adjust_idx_nth
thf(fact_1007_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1008_mat__of__cols__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_row_a @ ( mat_of_cols_a @ N @ Vs ) )
= N ) ).
% mat_of_cols_carrier(2)
thf(fact_1009_mat__of__cols__carrier_I3_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_col_a @ ( mat_of_cols_a @ N @ Vs ) )
= ( size_size_list_vec_a @ Vs ) ) ).
% mat_of_cols_carrier(3)
thf(fact_1010_adjust__idx__i,axiom,
! [I4: nat,J2: nat] :
( ( missing_adjust_idx @ I4 @ J2 )
!= I4 ) ).
% adjust_idx_i
thf(fact_1011_adjust__idx__rev1,axiom,
! [I4: nat,J2: nat] :
( ( missin3815256168798769645dx_rev @ I4 @ ( missing_adjust_idx @ I4 @ J2 ) )
= J2 ) ).
% adjust_idx_rev1
thf(fact_1012_adjust__idx__rev2,axiom,
! [J2: nat,I4: nat] :
( ( J2 != I4 )
=> ( ( missing_adjust_idx @ I4 @ ( missin3815256168798769645dx_rev @ I4 @ J2 ) )
= J2 ) ) ).
% adjust_idx_rev2
thf(fact_1013_adjust__idx__length,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_1014_nth__image,axiom,
! [L: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ L @ ( size_size_list_nat @ Xs ) )
=> ( ( image_nat_nat @ ( nth_nat @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
= ( set_nat2 @ ( take_nat @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_1015_proper__inc__mat__def,axiom,
( incide2997380824311827481_mat_a
= ( ^ [M6: mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_a @ M6 ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_a @ M6 ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_1016_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1017_image__add__0,axiom,
! [S2: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_1018_image__add__atLeastLessThan,axiom,
! [K4: nat,I4: nat,J2: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K4 ) @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ K4 ) ) ) ).
% image_add_atLeastLessThan
thf(fact_1019_image__add__atLeastAtMost,axiom,
! [K4: nat,I4: nat,J2: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K4 ) @ ( set_or1269000886237332187st_nat @ I4 @ J2 ) )
= ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I4 @ K4 ) @ ( plus_plus_nat @ J2 @ K4 ) ) ) ).
% image_add_atLeastAtMost
thf(fact_1020_image__add__greaterThanAtMost,axiom,
! [C: nat,A2: nat,B: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ C ) @ ( set_or6659071591806873216st_nat @ A2 @ B ) )
= ( set_or6659071591806873216st_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_1021_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( member_nat2 @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_1022_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1023_image__Un,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1024_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat2 @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1025_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_1026_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1027_conjugatable__vec__space_Odistinct__map__nth,axiom,
! [Ls: list_nat,Inds: list_nat] :
( ( distinct_nat @ Ls )
=> ( ( distinct_nat @ Inds )
=> ( ! [J3: nat] :
( ( member_nat2 @ J3 @ ( set_nat2 @ Inds ) )
=> ( ord_less_nat @ J3 @ ( size_size_list_nat @ Ls ) ) )
=> ( distinct_nat @ ( map_nat_nat @ ( nth_nat @ Ls ) @ Inds ) ) ) ) ) ).
% conjugatable_vec_space.distinct_map_nth
thf(fact_1028_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_1029_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_1030_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1031_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_1032_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1033_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1034_index__Cons,axiom,
! [X: nat,A2: nat,Xs: list_nat] :
( ( ( X = A2 )
=> ( ( list_index_nat @ ( cons_nat @ X @ Xs ) @ A2 )
= zero_zero_nat ) )
& ( ( X != A2 )
=> ( ( list_index_nat @ ( cons_nat @ X @ Xs ) @ A2 )
= ( plus_plus_nat @ ( list_index_nat @ Xs @ A2 ) @ one_one_nat ) ) ) ) ).
% index_Cons
thf(fact_1035_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_1036_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1037_take__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_1038_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1039_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1040_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z3: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1041_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1042_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z5: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1043_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z5: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1044_distinct__length__2__or__more,axiom,
! [A2: nat,B: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ A2 @ ( cons_nat @ B @ Xs ) ) )
= ( ( A2 != B )
& ( distinct_nat @ ( cons_nat @ A2 @ Xs ) )
& ( distinct_nat @ ( cons_nat @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1045_elem,axiom,
! [X: nat,Xs: list_nat] : ( listMem_nat @ X @ ( cons_nat @ X @ Xs ) ) ).
% elem
thf(fact_1046_ListMem_Ocases,axiom,
! [A1: nat,A22: list_nat] :
( ( listMem_nat @ A1 @ A22 )
=> ( ! [Xs3: list_nat] :
( A22
!= ( cons_nat @ A1 @ Xs3 ) )
=> ~ ! [Xs3: list_nat] :
( ? [Y4: nat] :
( A22
= ( cons_nat @ Y4 @ Xs3 ) )
=> ~ ( listMem_nat @ A1 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_1047_ListMem_Osimps,axiom,
( listMem_nat
= ( ^ [A12: nat,A23: list_nat] :
( ? [X2: nat,Xs2: list_nat] :
( ( A12 = X2 )
& ( A23
= ( cons_nat @ X2 @ Xs2 ) ) )
| ? [X2: nat,Xs2: list_nat,Y: nat] :
( ( A12 = X2 )
& ( A23
= ( cons_nat @ Y @ Xs2 ) )
& ( listMem_nat @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_1048_insert,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( listMem_nat @ X @ Xs )
=> ( listMem_nat @ X @ ( cons_nat @ Y2 @ Xs ) ) ) ).
% insert
thf(fact_1049_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_1050_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1051_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1052_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1053_set__ConsD,axiom,
! [Y2: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1054_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1055_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1056_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1057_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1058_insert__nth_Osimps_I1_J,axiom,
! [X: nat,Xs: list_nat] :
( ( list_insert_nth_nat @ zero_zero_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ).
% insert_nth.simps(1)
thf(fact_1059_remove__nth_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( list_remove_nth_nat @ zero_zero_nat @ ( cons_nat @ X @ Xs ) )
= Xs ) ).
% remove_nth.simps(2)
thf(fact_1060_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_1061_map__nth__eq__conv,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_nat_nat @ F @ Xs )
= Ys )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( nth_nat @ Ys @ I ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_1062_map__index_H__is__ConsD,axiom,
! [N: nat,F: nat > nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( list_m6542280337143352873at_nat @ N @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z3: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs ) )
& ( ( F @ N @ Z3 )
= Y2 )
& ( ( list_m6542280337143352873at_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ F @ Zs )
= Ys ) ) ) ).
% map_index'_is_ConsD
thf(fact_1063_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1064_upt__eq__Cons__conv,axiom,
! [I4: nat,J2: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I4 @ J2 )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I4 @ J2 )
& ( I4 = X )
& ( ( upt @ ( plus_plus_nat @ I4 @ one_one_nat ) @ J2 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1065_nth__map__upt,axiom,
! [I4: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I4 )
= ( F @ ( plus_plus_nat @ M @ I4 ) ) ) ) ).
% nth_map_upt
thf(fact_1066_rows__def,axiom,
( rows_a
= ( ^ [A3: mat_a] : ( map_nat_vec_a @ ( row_a @ A3 ) @ ( upt @ zero_zero_nat @ ( dim_row_a @ A3 ) ) ) ) ) ).
% rows_def
thf(fact_1067_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1068_nth__non__equal__first__eq,axiom,
! [X: nat,Y2: nat,Xs: list_nat,N: nat] :
( ( X != Y2 )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y2 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1069_last__index__Cons,axiom,
! [X: nat,Y2: nat,Xs: list_nat] :
( ( ( X = Y2 )
=> ( ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( list_last_index_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( plus_plus_nat @ ( list_last_index_nat @ Xs @ Y2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( list_last_index_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= zero_zero_nat ) ) ) )
& ( ( X != Y2 )
=> ( ( list_last_index_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( plus_plus_nat @ ( list_last_index_nat @ Xs @ Y2 ) @ one_one_nat ) ) ) ) ).
% last_index_Cons
thf(fact_1070_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1071_index__image,axiom,
! [Xs: list_nat,X8: set_nat] :
( ( distinct_nat @ Xs )
=> ( ( ( set_nat2 @ Xs )
= X8 )
=> ( ( image_nat_nat @ ( list_index_nat @ Xs ) @ X8 )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% index_image
thf(fact_1072_insert__list__def,axiom,
( insert_list_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% insert_list_def
thf(fact_1073_distinct__list__subset__nths,axiom,
! [Ss2: list_nat,Ls: list_nat] :
( ( distinct_nat @ Ss2 )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ss2 ) @ ( set_nat2 @ Ls ) )
=> ~ ! [Ids: list_nat] :
( ( distinct_nat @ Ids )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ids ) @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Ls ) ) )
=> ( Ss2
!= ( map_nat_nat @ ( nth_nat @ Ls ) @ Ids ) ) ) ) ) ) ).
% distinct_list_subset_nths
thf(fact_1074_zero__one__matrix_Omat__ord__inc__sys__point,axiom,
! [Matrix: mat_a,X: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ X @ ( dim_row_a @ Matrix ) )
=> ( ( nth_nat @ ( upt @ zero_zero_nat @ ( dim_row_a @ Matrix ) ) @ X )
= X ) ) ) ).
% zero_one_matrix.mat_ord_inc_sys_point
thf(fact_1075_lessThan__eq__iff,axiom,
! [X: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_1076_lessThan__iff,axiom,
! [I4: nat,K4: nat] :
( ( member_nat2 @ I4 @ ( set_ord_lessThan_nat @ K4 ) )
= ( ord_less_nat @ I4 @ K4 ) ) ).
% lessThan_iff
thf(fact_1077_lessThan__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_1078_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_1079_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_1080_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1081_ivl__disj__un__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_1082_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_1083_zero__one__matrix_Oin__map__col__valid__index__M,axiom,
! [Matrix: mat_a,J2: nat,I4: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( member_nat2 @ I4 @ ( incide5355957740755015148lock_a @ ( col_a @ Matrix @ J2 ) ) )
=> ( ord_less_nat @ I4 @ ( dim_row_a @ Matrix ) ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index_M
thf(fact_1084_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_a,J2: nat,I4: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( ord_less_nat @ I4 @ ( dim_row_a @ Matrix ) )
=> ( ( ( vec_index_a @ ( row_a @ Matrix @ I4 ) @ J2 )
= zero_zero_a )
= ( ( vec_index_a @ ( row_a @ Matrix @ I4 ) @ J2 )
!= one_one_a ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_1085_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_nat,J2: nat,I4: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I4 @ ( dim_row_nat @ Matrix ) )
=> ( ( ( vec_index_nat @ ( row_nat @ Matrix @ I4 ) @ J2 )
= zero_zero_nat )
= ( ( vec_index_nat @ ( row_nat @ Matrix @ I4 ) @ J2 )
!= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_1086_index__zero__vec_I1_J,axiom,
! [I4: nat,N: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( vec_index_nat @ ( zero_vec_nat @ N ) @ I4 )
= zero_zero_nat ) ) ).
% index_zero_vec(1)
thf(fact_1087_zero__one__matrix_Ocol__nth__0__or__1__iff,axiom,
! [Matrix: mat_a,J2: nat,I4: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( ord_less_nat @ I4 @ ( dim_row_a @ Matrix ) )
=> ( ( ( vec_index_a @ ( col_a @ Matrix @ J2 ) @ I4 )
= zero_zero_a )
= ( ( vec_index_a @ ( col_a @ Matrix @ J2 ) @ I4 )
!= one_one_a ) ) ) ) ) ).
% zero_one_matrix.col_nth_0_or_1_iff
thf(fact_1088_zero__one__matrix_Ocol__nth__0__or__1__iff,axiom,
! [Matrix: mat_nat,J2: nat,I4: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I4 @ ( dim_row_nat @ Matrix ) )
=> ( ( ( vec_index_nat @ ( col_nat @ Matrix @ J2 ) @ I4 )
= zero_zero_nat )
= ( ( vec_index_nat @ ( col_nat @ Matrix @ J2 ) @ I4 )
!= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.col_nth_0_or_1_iff
thf(fact_1089_all__ones__index,axiom,
! [I4: nat,N: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( vec_index_nat @ ( matrix2751262895470517546ec_nat @ N ) @ I4 )
= one_one_nat ) ) ).
% all_ones_index
thf(fact_1090_non__empty__col__obtains,axiom,
! [M4: mat_a,J2: nat] :
( ( incide3034858701194040399_col_a @ M4 @ J2 )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ M4 ) )
=> ( ( vec_index_a @ ( col_a @ M4 @ J2 ) @ I3 )
= zero_zero_a ) ) ) ).
% non_empty_col_obtains
thf(fact_1091_non__empty__col__obtains,axiom,
! [M4: mat_nat,J2: nat] :
( ( incide6854414339478298687ol_nat @ M4 @ J2 )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ M4 ) )
=> ( ( vec_index_nat @ ( col_nat @ M4 @ J2 ) @ I3 )
= zero_zero_nat ) ) ) ).
% non_empty_col_obtains
thf(fact_1092_zero__one__matrix_Oblock__nempty__implies__all__zeros,axiom,
! [Matrix: mat_a,J2: nat,I4: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( ( incide5355957740755015148lock_a @ ( col_a @ Matrix @ J2 ) )
= bot_bot_set_nat )
=> ( ( ord_less_nat @ I4 @ ( dim_row_a @ Matrix ) )
=> ( ( vec_index_a @ ( col_a @ Matrix @ J2 ) @ I4 )
= zero_zero_a ) ) ) ) ) ).
% zero_one_matrix.block_nempty_implies_all_zeros
thf(fact_1093_zero__one__matrix_Oblock__nempty__implies__all__zeros,axiom,
! [Matrix: mat_nat,J2: nat,I4: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ( incide3975725477190312290ck_nat @ ( col_nat @ Matrix @ J2 ) )
= bot_bot_set_nat )
=> ( ( ord_less_nat @ I4 @ ( dim_row_nat @ Matrix ) )
=> ( ( vec_index_nat @ ( col_nat @ Matrix @ J2 ) @ I4 )
= zero_zero_nat ) ) ) ) ) ).
% zero_one_matrix.block_nempty_implies_all_zeros
thf(fact_1094_empty__iff,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1095_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1096_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_1097_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_1098_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_1099_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1100_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1101_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1102_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1103_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1104_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1105_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_1106_Un__empty,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1107_atLeastLessThan__empty,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( set_or4665077453230672383an_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_1108_atLeastatMost__empty__iff2,axiom,
! [A2: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1109_atLeastatMost__empty__iff,axiom,
! [A2: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1110_atLeastLessThan__empty__iff2,axiom,
! [A2: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A2 @ B ) )
= ( ~ ( ord_less_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_1111_atLeastLessThan__empty__iff,axiom,
! [A2: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_1112_atLeastatMost__empty,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_1113_Diff__eq__empty__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1114_greaterThanLessThan__empty,axiom,
! [L: nat,K4: nat] :
( ( ord_less_eq_nat @ L @ K4 )
=> ( ( set_or5834768355832116004an_nat @ K4 @ L )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_1115_greaterThanAtMost__empty,axiom,
! [L: nat,K4: nat] :
( ( ord_less_eq_nat @ L @ K4 )
=> ( ( set_or6659071591806873216st_nat @ K4 @ L )
= bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_1116_greaterThanAtMost__empty__iff2,axiom,
! [K4: nat,L: nat] :
( ( bot_bot_set_nat
= ( set_or6659071591806873216st_nat @ K4 @ L ) )
= ( ~ ( ord_less_nat @ K4 @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_1117_greaterThanAtMost__empty__iff,axiom,
! [K4: nat,L: nat] :
( ( ( set_or6659071591806873216st_nat @ K4 @ L )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ K4 @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_1118_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1119_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_1120_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_1121_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1122_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1123_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1124_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1125_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1126_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_1127_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1128_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1129_emptyE,axiom,
! [A2: nat] :
~ ( member_nat2 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1130_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_1131_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat2 @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1132_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat2 @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1133_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1134_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1135_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1136_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1137_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1138_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_1139_set__diff__non__empty__not__subset,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C3 ) )
=> ( ( C3 != bot_bot_set_nat )
=> ( ( A != bot_bot_set_nat )
=> ( ( B2 != bot_bot_set_nat )
=> ~ ( ord_less_eq_set_nat @ A @ C3 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_1140_diff__shunt__var,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1141_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1142_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1143_set__remove__nth,axiom,
! [Xs: list_nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_remove_nth_nat @ J2 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ J2 ) @ bot_bot_set_nat ) ) ) ) ) ).
% set_remove_nth
thf(fact_1144_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat2 @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1145_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B @ A ) )
= ( ( A2 = B )
| ( member_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1146_insert__subset,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
= ( ( member_nat2 @ X @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1147_singletonI,axiom,
! [A2: nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1148_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A: set_nat] :
( ( member_nat2 @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1149_Diff__insert0,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1150_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat2 @ A2 @ B2 ) )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1151_Un__insert__left,axiom,
! [A2: nat,B2: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ B2 ) @ C3 )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1152_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1153_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1154_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1155_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1156_atLeastAtMost__singleton__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B )
= ( insert_nat2 @ C @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1157_atLeastAtMost__singleton,axiom,
! [A2: nat] :
( ( set_or1269000886237332187st_nat @ A2 @ A2 )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_1158_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1159_single__Diff__lessThan,axiom,
! [K4: nat] :
( ( minus_minus_set_nat @ ( insert_nat2 @ K4 @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K4 ) )
= ( insert_nat2 @ K4 @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1160_in__image__insert__iff,axiom,
! [B2: set_set_nat,X: nat,A: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat2 @ C5 @ B2 )
=> ~ ( member_nat2 @ X @ C5 ) )
=> ( ( member_set_nat2 @ A @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X ) @ B2 ) )
= ( ( member_nat2 @ X @ A )
& ( member_set_nat2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1161_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1162_subset__singleton__iff,axiom,
! [X8: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X8 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( ( X8 = bot_bot_set_nat )
| ( X8
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1163_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1164_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1165_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1166_Diff__insert,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1167_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1168_Un__singleton__iff,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1169_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ( insert_nat2 @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( ( A = bot_bot_set_nat )
& ( B2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1170_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat2 @ B @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_1171_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat2 @ B @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1172_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1173_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1174_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_1175_atLeastAtMost__singleton_H,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_1176_subset__insert,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_1177_subset__Diff__insert,axiom,
! [A: set_nat,B2: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ ( insert_nat2 @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C3 ) )
& ~ ( member_nat2 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1178_set__insert__nth,axiom,
! [I4: nat,X: nat,Xs: list_nat] :
( ( set_nat2 @ ( list_insert_nth_nat @ I4 @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_insert_nth
thf(fact_1179_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A: set_nat] :
( ( ( member_nat2 @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) )
& ( ~ ( member_nat2 @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1180_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_1181_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1182_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat2 @ A2 @ B2 )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1183_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat2 @ X @ A )
=> ~ ! [B8: set_nat] :
( ( A
= ( insert_nat2 @ X @ B8 ) )
=> ( member_nat2 @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_1184_insert__ident,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ~ ( member_nat2 @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ A )
= ( insert_nat2 @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_1185_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1186_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat2 @ A2 @ A )
=> ( ~ ( member_nat2 @ B @ B2 )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C6: set_nat] :
( ( A
= ( insert_nat2 @ B @ C6 ) )
& ~ ( member_nat2 @ B @ C6 )
& ( B2
= ( insert_nat2 @ A2 @ C6 ) )
& ~ ( member_nat2 @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1187_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ? [B8: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B8 ) )
& ~ ( member_nat2 @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_1188_Diff__single__insert,axiom,
! [A: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1189_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
= ( ( ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1190_set__insert__list,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_list_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_insert_list
thf(fact_1191_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
= ( ( ( member_nat2 @ X @ B2 )
=> ( ord_less_set_nat @ A @ B2 ) )
& ( ~ ( member_nat2 @ X @ B2 )
=> ( ( ( member_nat2 @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1192_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A4: nat,B3: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B3 ) @ ( insert_nat2 @ B3 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1193_atLeastAtMost__diff__ends,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( insert_nat2 @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) )
= ( set_or5834768355832116004an_nat @ A2 @ B ) ) ).
% atLeastAtMost_diff_ends
thf(fact_1194_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6659071591806873216st_nat
= ( ^ [A4: nat,B3: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B3 ) @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_1195_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat2 @ U @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_1196_ivl__disj__un__singleton_I3_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat2 @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_1197_ivl__disj__un__singleton_I5_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat2 @ L @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_1198_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( insert_nat2 @ U @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_1199_set__nzrows,axiom,
! [A: mat_a] :
( ( set_vec_a2 @ ( macaulay_nzrows_a @ A ) )
= ( minus_6230920740010926198_vec_a @ ( set_vec_a2 @ ( rows_a @ A ) ) @ ( insert_vec_a2 @ ( zero_vec_a @ ( dim_col_a @ A ) ) @ bot_bot_set_vec_a ) ) ) ).
% set_nzrows
thf(fact_1200_zero__one__matrix_Orow__elems__ss01,axiom,
! [Matrix: mat_a,I4: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ I4 @ ( dim_row_a @ Matrix ) )
=> ( ord_less_eq_set_a @ ( vec_set_a @ ( row_a @ Matrix @ I4 ) ) @ ( insert_a2 @ zero_zero_a @ ( insert_a2 @ one_one_a @ bot_bot_set_a ) ) ) ) ) ).
% zero_one_matrix.row_elems_ss01
thf(fact_1201_zero__one__matrix_Orow__elems__ss01,axiom,
! [Matrix: mat_nat,I4: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ I4 @ ( dim_row_nat @ Matrix ) )
=> ( ord_less_eq_set_nat @ ( vec_set_nat2 @ ( row_nat @ Matrix @ I4 ) ) @ ( insert_nat2 @ zero_zero_nat @ ( insert_nat2 @ one_one_nat @ bot_bot_set_nat ) ) ) ) ) ).
% zero_one_matrix.row_elems_ss01
thf(fact_1202_zero__one__matrix_Onon__empty__col__01,axiom,
! [Matrix: mat_a,J2: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( incide3034858701194040399_col_a @ Matrix @ J2 )
= ( member_a2 @ one_one_a @ ( vec_set_a @ ( col_a @ Matrix @ J2 ) ) ) ) ) ) ).
% zero_one_matrix.non_empty_col_01
thf(fact_1203_zero__one__matrix_Onon__empty__col__01,axiom,
! [Matrix: mat_nat,J2: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( incide6854414339478298687ol_nat @ Matrix @ J2 )
= ( member_nat2 @ one_one_nat @ ( vec_set_nat2 @ ( col_nat @ Matrix @ J2 ) ) ) ) ) ) ).
% zero_one_matrix.non_empty_col_01
thf(fact_1204_zero__one__matrix_Oblock__nempty__implies__no__one,axiom,
! [Matrix: mat_a,J2: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( ( incide5355957740755015148lock_a @ ( col_a @ Matrix @ J2 ) )
= bot_bot_set_nat )
=> ~ ( member_a2 @ one_one_a @ ( vec_set_a @ ( col_a @ Matrix @ J2 ) ) ) ) ) ) ).
% zero_one_matrix.block_nempty_implies_no_one
thf(fact_1205_zero__one__matrix_Oblock__nempty__implies__no__one,axiom,
! [Matrix: mat_nat,J2: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ( incide3975725477190312290ck_nat @ ( col_nat @ Matrix @ J2 ) )
= bot_bot_set_nat )
=> ~ ( member_nat2 @ one_one_nat @ ( vec_set_nat2 @ ( col_nat @ Matrix @ J2 ) ) ) ) ) ) ).
% zero_one_matrix.block_nempty_implies_no_one
thf(fact_1206_zero__one__matrix_Oone__implies__block__nempty,axiom,
! [Matrix: mat_a,J2: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( member_a2 @ one_one_a @ ( vec_set_a @ ( col_a @ Matrix @ J2 ) ) )
=> ( ( incide5355957740755015148lock_a @ ( col_a @ Matrix @ J2 ) )
!= bot_bot_set_nat ) ) ) ) ).
% zero_one_matrix.one_implies_block_nempty
thf(fact_1207_zero__one__matrix_Oone__implies__block__nempty,axiom,
! [Matrix: mat_nat,J2: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( member_nat2 @ one_one_nat @ ( vec_set_nat2 @ ( col_nat @ Matrix @ J2 ) ) )
=> ( ( incide3975725477190312290ck_nat @ ( col_nat @ Matrix @ J2 ) )
!= bot_bot_set_nat ) ) ) ) ).
% zero_one_matrix.one_implies_block_nempty
thf(fact_1208_zero__one__matrix_Ocol__elems__ss01,axiom,
! [Matrix: mat_a,J2: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ord_less_eq_set_a @ ( vec_set_a @ ( col_a @ Matrix @ J2 ) ) @ ( insert_a2 @ zero_zero_a @ ( insert_a2 @ one_one_a @ bot_bot_set_a ) ) ) ) ) ).
% zero_one_matrix.col_elems_ss01
thf(fact_1209_zero__one__matrix_Ocol__elems__ss01,axiom,
! [Matrix: mat_nat,J2: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ord_less_eq_set_nat @ ( vec_set_nat2 @ ( col_nat @ Matrix @ J2 ) ) @ ( insert_nat2 @ zero_zero_nat @ ( insert_nat2 @ one_one_nat @ bot_bot_set_nat ) ) ) ) ) ).
% zero_one_matrix.col_elems_ss01
thf(fact_1210_set__update__distinct,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1211_length__list__update,axiom,
! [Xs: list_nat,I4: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I4 @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_1212_nth__list__update__neq,axiom,
! [I4: nat,J2: nat,Xs: list_nat,X: nat] :
( ( I4 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_1213_list__update__id,axiom,
! [Xs: list_nat,I4: nat] :
( ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_1214_list__update__beyond,axiom,
! [Xs: list_nat,I4: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I4 )
=> ( ( list_update_nat @ Xs @ I4 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1215_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,Y2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y2 ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_1216_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ I4 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1217_set__swap,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I4 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1218_distinct__swap,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I4 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_1219_set__update__subset__insert,axiom,
! [Xs: list_nat,I4: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I4 @ X ) ) @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1220_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1221_parallel__list__update,axiom,
! [N: nat,R: nat > nat > $o,P5: list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ! [Xs3: list_nat,I3: nat,Y4: nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
=> ( ( ord_less_nat @ I3 @ N )
=> ( ( R @ ( nth_nat @ Xs3 @ I3 ) @ Y4 )
=> ( ( P5 @ Xs3 )
=> ( P5 @ ( list_update_nat @ Xs3 @ I3 @ Y4 ) ) ) ) ) )
=> ( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( P5 @ Xs )
=> ( ( ( size_size_list_nat @ Ys )
= N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( R @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
=> ( P5 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_1222_list__update__same__conv,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I4 @ X )
= Xs )
= ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1223_nth__list__update,axiom,
! [I4: nat,Xs: list_nat,J2: nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I4 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ J2 )
= X ) )
& ( ( I4 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1224_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y2 )
= ( cons_nat @ Y2 @ Xs ) ) ).
% list_update_code(2)
thf(fact_1225_set__update__subsetI,axiom,
! [Xs: list_nat,A: set_nat,X: nat,I4: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
=> ( ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I4 @ X ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1226_take__update__swap,axiom,
! [M: nat,Xs: list_nat,N: nat,X: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_1227_index__update__if__diff2,axiom,
! [N: nat,Xs: list_nat,X: nat,Y2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( X
!= ( nth_nat @ Xs @ N ) )
=> ( ( X != Y2 )
=> ( ( list_index_nat @ ( list_update_nat @ Xs @ N @ Y2 ) @ X )
= ( list_index_nat @ Xs @ X ) ) ) ) ) ).
% index_update_if_diff2
thf(fact_1228_index__swap__if__distinct,axiom,
! [Xs: list_nat,I4: nat,J2: nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( X
= ( nth_nat @ Xs @ I4 ) )
=> ( ( list_index_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I4 ) ) @ X )
= J2 ) )
& ( ( X
!= ( nth_nat @ Xs @ I4 ) )
=> ( ( ( X
= ( nth_nat @ Xs @ J2 ) )
=> ( ( list_index_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I4 ) ) @ X )
= I4 ) )
& ( ( X
!= ( nth_nat @ Xs @ J2 ) )
=> ( ( list_index_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I4 ) ) @ X )
= ( list_index_nat @ Xs @ X ) ) ) ) ) ) ) ) ) ).
% index_swap_if_distinct
thf(fact_1229_distinct__list__update,axiom,
! [Xs: list_nat,A2: nat,I4: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat2 @ A2 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ I4 ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs @ I4 @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_1230_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1231_zero__one__matrix_Omap__col__to__block__wf,axiom,
! [Matrix: mat_a,C: vec_a] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( member_vec_a2 @ C @ ( set_vec_a2 @ ( cols_a @ Matrix ) ) )
=> ( ord_less_eq_set_nat @ ( incide5355957740755015148lock_a @ C ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_row_a @ Matrix ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_wf
thf(fact_1232_cols__length,axiom,
! [A: mat_a] :
( ( size_size_list_vec_a @ ( cols_a @ A ) )
= ( dim_col_a @ A ) ) ).
% cols_length
thf(fact_1233_mat__of__cols__cols,axiom,
! [A: mat_a] :
( ( mat_of_cols_a @ ( dim_row_a @ A ) @ ( cols_a @ A ) )
= A ) ).
% mat_of_cols_cols
thf(fact_1234_cols__nth,axiom,
! [I4: nat,A: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_col_a @ A ) )
=> ( ( nth_vec_a @ ( cols_a @ A ) @ I4 )
= ( col_a @ A @ I4 ) ) ) ).
% cols_nth
thf(fact_1235_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat,Y4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( ( member_nat2 @ Y4 @ A )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1236_obtain__col__index,axiom,
! [C: vec_a,M4: mat_a] :
( ( member_vec_a2 @ C @ ( set_vec_a2 @ ( cols_a @ M4 ) ) )
=> ~ ! [J3: nat] :
( ( ( col_a @ M4 @ J3 )
= C )
=> ~ ( ord_less_nat @ J3 @ ( dim_col_a @ M4 ) ) ) ) ).
% obtain_col_index
thf(fact_1237_col__prop__cond,axiom,
! [M4: mat_a,P2: vec_a > $o,C: vec_a] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_a @ M4 ) )
=> ( P2 @ ( col_a @ M4 @ J3 ) ) )
=> ( ( member_vec_a2 @ C @ ( set_vec_a2 @ ( cols_a @ M4 ) ) )
=> ( P2 @ C ) ) ) ).
% col_prop_cond
thf(fact_1238_cols__dim,axiom,
! [A: mat_a] : ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ ( cols_a @ A ) ) @ ( carrier_vec_a @ ( dim_row_a @ A ) ) ) ).
% cols_dim
thf(fact_1239_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X2: nat] :
( A3
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1240_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X4: nat] :
( A
!= ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_1241_zero__one__matrix_Omat__ord__inc__sys__block,axiom,
! [Matrix: mat_a,J2: nat] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ Matrix ) )
=> ( ( nth_set_nat @ ( map_vec_a_set_nat @ incide5355957740755015148lock_a @ ( cols_a @ Matrix ) ) @ J2 )
= ( incide5355957740755015148lock_a @ ( col_a @ Matrix @ J2 ) ) ) ) ) ).
% zero_one_matrix.mat_ord_inc_sys_block
thf(fact_1242_cols__def,axiom,
( cols_a
= ( ^ [A3: mat_a] : ( map_nat_vec_a @ ( col_a @ A3 ) @ ( upt @ zero_zero_nat @ ( dim_col_a @ A3 ) ) ) ) ) ).
% cols_def
thf(fact_1243_take__cols__var__def,axiom,
( more_take_cols_var_a
= ( ^ [A3: mat_a,Inds2: list_nat] : ( mat_of_cols_a @ ( dim_row_a @ A3 ) @ ( map_nat_vec_a @ ( nth_vec_a @ ( cols_a @ A3 ) ) @ Inds2 ) ) ) ) ).
% take_cols_var_def
thf(fact_1244_zero__one__matrix_Omat__ord__inc__sys__N,axiom,
! [Matrix: mat_a] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( ( incide1177982898701834729at_int @ ( upt @ zero_zero_nat @ ( dim_row_a @ Matrix ) ) @ ( map_vec_a_set_nat @ incide5355957740755015148lock_a @ ( cols_a @ Matrix ) ) )
= ( matrix2825909983993355945_a_int @ Matrix ) ) ) ).
% zero_one_matrix.mat_ord_inc_sys_N
thf(fact_1245_lift__01__mat__simp_I1_J,axiom,
! [M4: mat_a] :
( ( dim_row_a @ ( matrix876235357355983809at_a_a @ M4 ) )
= ( dim_row_a @ M4 ) ) ).
% lift_01_mat_simp(1)
thf(fact_1246_lift__01__mat__simp_I2_J,axiom,
! [M4: mat_a] :
( ( dim_col_a @ ( matrix876235357355983809at_a_a @ M4 ) )
= ( dim_col_a @ M4 ) ) ).
% lift_01_mat_simp(2)
thf(fact_1247_vec__contains__col__elements__mat,axiom,
! [J2: nat,M4: mat_nat,A2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_nat @ M4 ) )
=> ( ( member_nat2 @ A2 @ ( vec_set_nat2 @ ( col_nat @ M4 @ J2 ) ) )
=> ( member_nat2 @ A2 @ ( elements_mat_nat @ M4 ) ) ) ) ).
% vec_contains_col_elements_mat
thf(fact_1248_vec__contains__col__elements__mat,axiom,
! [J2: nat,M4: mat_a,A2: a] :
( ( ord_less_nat @ J2 @ ( dim_col_a @ M4 ) )
=> ( ( member_a2 @ A2 @ ( vec_set_a @ ( col_a @ M4 @ J2 ) ) )
=> ( member_a2 @ A2 @ ( elements_mat_a @ M4 ) ) ) ) ).
% vec_contains_col_elements_mat
thf(fact_1249_vec__contains__row__elements__mat,axiom,
! [I4: nat,M4: mat_nat,A2: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ M4 ) )
=> ( ( member_nat2 @ A2 @ ( vec_set_nat2 @ ( row_nat @ M4 @ I4 ) ) )
=> ( member_nat2 @ A2 @ ( elements_mat_nat @ M4 ) ) ) ) ).
% vec_contains_row_elements_mat
thf(fact_1250_vec__contains__row__elements__mat,axiom,
! [I4: nat,M4: mat_a,A2: a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ M4 ) )
=> ( ( member_a2 @ A2 @ ( vec_set_a @ ( row_a @ M4 @ I4 ) ) )
=> ( member_a2 @ A2 @ ( elements_mat_a @ M4 ) ) ) ) ).
% vec_contains_row_elements_mat
thf(fact_1251_col__elems__subset__mat,axiom,
! [I4: nat,N4: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_col_a @ N4 ) )
=> ( ord_less_eq_set_a @ ( vec_set_a @ ( col_a @ N4 @ I4 ) ) @ ( elements_mat_a @ N4 ) ) ) ).
% col_elems_subset_mat
thf(fact_1252_row__elems__subset__mat,axiom,
! [I4: nat,N4: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ N4 ) )
=> ( ord_less_eq_set_a @ ( vec_set_a @ ( row_a @ N4 @ I4 ) ) @ ( elements_mat_a @ N4 ) ) ) ).
% row_elems_subset_mat
thf(fact_1253_zero__one__matrix__def,axiom,
( incide4966654671090901726ix_nat
= ( ^ [Matrix2: mat_nat] : ( ord_less_eq_set_nat @ ( elements_mat_nat @ Matrix2 ) @ ( insert_nat2 @ zero_zero_nat @ ( insert_nat2 @ one_one_nat @ bot_bot_set_nat ) ) ) ) ) ).
% zero_one_matrix_def
thf(fact_1254_zero__one__matrix_Oelems01,axiom,
! [Matrix: mat_nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ord_less_eq_set_nat @ ( elements_mat_nat @ Matrix ) @ ( insert_nat2 @ zero_zero_nat @ ( insert_nat2 @ one_one_nat @ bot_bot_set_nat ) ) ) ) ).
% zero_one_matrix.elems01
thf(fact_1255_zero__one__matrix_Ointro,axiom,
! [Matrix: mat_nat] :
( ( ord_less_eq_set_nat @ ( elements_mat_nat @ Matrix ) @ ( insert_nat2 @ zero_zero_nat @ ( insert_nat2 @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( incide4966654671090901726ix_nat @ Matrix ) ) ).
% zero_one_matrix.intro
thf(fact_1256_zero__one__matrix_Omat__is__ordered__incidence__sys,axiom,
! [Matrix: mat_a] :
( ( incide7367983062745021296trix_a @ Matrix )
=> ( incide6998539924841383625em_nat @ ( upt @ zero_zero_nat @ ( dim_row_a @ Matrix ) ) @ ( map_vec_a_set_nat @ incide5355957740755015148lock_a @ ( cols_a @ Matrix ) ) ) ) ).
% zero_one_matrix.mat_is_ordered_incidence_sys
thf(fact_1257_zero__one__matrix_Omap__col__to__block__elem__not,axiom,
! [Matrix: mat_nat,C: vec_nat,I4: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( member_vec_nat2 @ C @ ( set_vec_nat2 @ ( cols_nat @ Matrix ) ) )
=> ( ( ord_less_nat @ I4 @ ( dim_vec_nat @ C ) )
=> ( ( ~ ( member_nat2 @ I4 @ ( incide3975725477190312290ck_nat @ C ) ) )
= ( ( vec_index_nat @ C @ I4 )
= zero_zero_nat ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem_not
thf(fact_1258_dim__col,axiom,
! [A: mat_a,I4: nat] :
( ( dim_vec_a @ ( col_a @ A @ I4 ) )
= ( dim_row_a @ A ) ) ).
% dim_col
thf(fact_1259_index__row_I2_J,axiom,
! [A: mat_a,I4: nat] :
( ( dim_vec_a @ ( row_a @ A @ I4 ) )
= ( dim_col_a @ A ) ) ).
% index_row(2)
thf(fact_1260_mat__of__row__dim_I2_J,axiom,
! [Y2: vec_a] :
( ( dim_col_a @ ( mat_of_row_a @ Y2 ) )
= ( dim_vec_a @ Y2 ) ) ).
% mat_of_row_dim(2)
thf(fact_1261_index__minus__vec_I1_J,axiom,
! [I4: nat,V_2: vec_nat,V_1: vec_nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_nat @ V_2 ) )
=> ( ( vec_index_nat @ ( minus_minus_vec_nat @ V_1 @ V_2 ) @ I4 )
= ( minus_minus_nat @ ( vec_index_nat @ V_1 @ I4 ) @ ( vec_index_nat @ V_2 @ I4 ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1262_index__minus__vec_I1_J,axiom,
! [I4: nat,V_2: vec_set_nat,V_1: vec_set_nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_set_nat @ V_2 ) )
=> ( ( vec_index_set_nat @ ( minus_4779711351438577370et_nat @ V_1 @ V_2 ) @ I4 )
= ( minus_minus_set_nat @ ( vec_index_set_nat @ V_1 @ I4 ) @ ( vec_index_set_nat @ V_2 @ I4 ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1263_index__add__vec_I1_J,axiom,
! [I4: nat,V_2: vec_nat,V_1: vec_nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_nat @ V_2 ) )
=> ( ( vec_index_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ I4 )
= ( plus_plus_nat @ ( vec_index_nat @ V_1 @ I4 ) @ ( vec_index_nat @ V_2 @ I4 ) ) ) ) ).
% index_add_vec(1)
thf(fact_1264_vec__setE,axiom,
! [A2: nat,V: vec_nat] :
( ( member_nat2 @ A2 @ ( vec_set_nat2 @ V ) )
=> ~ ! [I3: nat] :
( ( ( vec_index_nat @ V @ I3 )
= A2 )
=> ~ ( ord_less_nat @ I3 @ ( dim_vec_nat @ V ) ) ) ) ).
% vec_setE
thf(fact_1265_vec__setI,axiom,
! [V: vec_nat,I4: nat,A2: nat] :
( ( ( vec_index_nat @ V @ I4 )
= A2 )
=> ( ( ord_less_nat @ I4 @ ( dim_vec_nat @ V ) )
=> ( member_nat2 @ A2 @ ( vec_set_nat2 @ V ) ) ) ) ).
% vec_setI
thf(fact_1266_vec__contains__obtains__index,axiom,
! [A2: nat,V: vec_nat] :
( ( member_nat2 @ A2 @ ( vec_set_nat2 @ V ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_nat @ V ) )
=> ( ( vec_index_nat @ V @ I3 )
!= A2 ) ) ) ).
% vec_contains_obtains_index
thf(fact_1267_less__eq__vec__def,axiom,
( ord_less_eq_vec_nat
= ( ^ [V2: vec_nat,W: vec_nat] :
( ( ( dim_vec_nat @ V2 )
= ( dim_vec_nat @ W ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ W ) )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V2 @ I ) @ ( vec_index_nat @ W @ I ) ) ) ) ) ) ).
% less_eq_vec_def
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_nat2 @ i @ ( set_nat2 @ ( upt @ low @ up ) ) ).
%------------------------------------------------------------------------------