TPTP Problem File: SLH0322^1.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/0038_Fishers_Inequality/prob_00134_007705__28202650_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1374 ( 600 unt; 160 typ; 0 def)
% Number of atoms : 3146 (1372 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 12908 ( 221 ~; 61 |; 137 &;11268 @)
% ( 0 <=>;1221 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 311 ( 311 >; 0 *; 0 +; 0 <<)
% Number of symbols : 144 ( 141 usr; 13 con; 0-4 aty)
% Number of variables : 3022 ( 66 ^;2893 !; 63 ?;3022 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:49:42.433
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Omat_It__Nat__Onat_J_J,type,
mat_mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Omat_It__Int__Oint_J_J,type,
mat_mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
set_mat_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Set__Oset_It__Nat__Onat_J_J,type,
mat_set_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__Set__Oset_Itf__a_J_J,type,
list_set_a: $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__Set__Oset_It__Int__Oint_J,type,
set_int: $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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (141)
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001tf__a,type,
design2964366272795260673oint_a: set_a > a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001tf__a,type,
design108908007054065099oint_a: set_a > a > set_a ).
thf(sy_c_Determinant_Odet_001t__Int__Oint,type,
det_int: mat_int > int ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001t__Nat__Onat_001t__Nat__Onat,type,
dual_o7421254750609082426at_nat: list_nat > list_set_nat > list_nat > list_set_nat > $o ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001t__Nat__Onat_001tf__a,type,
dual_o2119597811525285140_nat_a: list_nat > list_set_nat > list_a > list_set_a > $o ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001tf__a_001t__Nat__Onat,type,
dual_o7311038347201774518_a_nat: list_a > list_set_a > list_nat > list_set_nat > $o ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001tf__a_001tf__a,type,
dual_o5859382014055506072al_a_a: list_a > list_set_a > list_a > list_set_a > $o ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Int__Oint_J,type,
minus_minus_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Matrix__Omat_It__Int__Oint_J_J,type,
minus_5217014392749413595at_int: mat_mat_int > mat_mat_int > mat_mat_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Matrix__Omat_It__Nat__Onat_J_J,type,
minus_7707097537681140351at_nat: mat_mat_nat > mat_mat_nat > mat_mat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Nat__Onat_J,type,
minus_minus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_6025431102612251238et_nat: mat_set_nat > mat_set_nat > mat_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__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Int__Oint_J,type,
plus_plus_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Matrix__Omat_It__Int__Oint_J_J,type,
plus_p473217906811894667at_int: mat_mat_int > mat_mat_int > mat_mat_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Matrix__Omat_It__Nat__Onat_J_J,type,
plus_p2963301051743621423at_nat: mat_mat_nat > mat_mat_nat > mat_mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Int__Oint_J,type,
times_times_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Int__Oint_J,type,
uminus5233555848319219356at_int: mat_int > mat_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Matrix__Omat_It__Int__Oint_J_J,type,
uminus1241013589067180107at_int: mat_mat_int > mat_mat_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > 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_Oinc__mat__of_001tf__a_001t__Int__Oint,type,
incide7016682120514301311_a_int: list_a > list_set_a > mat_int ).
thf(sy_c_Incidence__Matrices_Omat__rep__num_001t__Int__Oint,type,
incide7000514267430604580um_int: mat_int > nat > nat ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001t__Int__Oint,type,
incide6851923868969248411ol_int: mat_int > nat > $o ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001t__Nat__Onat,type,
incide6854414339478298687ol_nat: mat_nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__bibd_001t__Nat__Onat,type,
incide4338384322530710165bd_nat: list_nat > list_set_nat > nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__bibd_001tf__a,type,
incide4817766913905363833bibd_a: list_a > list_set_a > nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__block__design_001tf__a,type,
incide5219153079875704461sign_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__constant__rep_001t__Nat__Onat,type,
incide6328174194974408527ep_nat: list_nat > list_set_nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__constant__rep_001tf__a,type,
incide6922509864216205631_rep_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__design_001t__Nat__Onat,type,
incide8999572217031194378gn_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__design_001tf__a,type,
incide2848671379600480836sign_a: list_a > list_set_a > $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_Oordered__incidence__system_001tf__a,type,
incide1624170830610365509stem_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incomplete__design_001tf__a,type,
incide1377962018248667206sign_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__pairwise__balance_001t__Nat__Onat,type,
incide3388802471754236788ce_nat: list_nat > list_set_nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__pairwise__balance_001tf__a,type,
incide6880889959311561818ance_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__proper__design_001t__Nat__Onat,type,
incide1001368407746664282gn_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__proper__design_001tf__a,type,
incide3676903341588786676sign_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001t__Int__Oint,type,
incide294466202882093137at_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001t__Nat__Onat,type,
incide296956673391143413at_nat: mat_nat > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Int__Oint,type,
incide4964164200581851450ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Nat__Onat,type,
incide4966654671090901726ix_nat: mat_nat > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix__int,type,
incide8301514189696901506ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix__ring__1_001t__Int__Oint,type,
incide6080938071136783841_1_int: mat_int > $o ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Int__Oint,type,
carrier_mat_int: nat > nat > set_mat_int ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Int__Oint,type,
diagonal_mat_int: mat_int > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Nat__Onat,type,
diagonal_mat_nat: mat_nat > $o ).
thf(sy_c_Matrix_Odim__col_001t__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Matrix__Omat_It__Int__Oint_J,type,
dim_col_mat_int: mat_mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Matrix__Omat_It__Nat__Onat_J,type,
dim_col_mat_nat: mat_mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Set__Oset_It__Nat__Onat_J,type,
dim_col_set_nat: mat_set_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Int__Oint,type,
dim_row_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Omat_It__Int__Oint_J,type,
dim_row_mat_int: mat_mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Omat_It__Nat__Onat_J,type,
dim_row_mat_nat: mat_mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Set__Oset_It__Nat__Onat_J,type,
dim_row_set_nat: mat_set_nat > nat ).
thf(sy_c_Matrix_Oindex__mat_001t__Int__Oint,type,
index_mat_int: mat_int > product_prod_nat_nat > int ).
thf(sy_c_Matrix_Oindex__mat_001t__Matrix__Omat_It__Int__Oint_J,type,
index_mat_mat_int: mat_mat_int > product_prod_nat_nat > mat_int ).
thf(sy_c_Matrix_Oindex__mat_001t__Matrix__Omat_It__Nat__Onat_J,type,
index_mat_mat_nat: mat_mat_nat > product_prod_nat_nat > mat_nat ).
thf(sy_c_Matrix_Oindex__mat_001t__Nat__Onat,type,
index_mat_nat: mat_nat > product_prod_nat_nat > nat ).
thf(sy_c_Matrix_Oindex__mat_001t__Set__Oset_It__Nat__Onat_J,type,
index_mat_set_nat: mat_set_nat > product_prod_nat_nat > set_nat ).
thf(sy_c_Matrix_Otranspose__mat_001t__Int__Oint,type,
transpose_mat_int: mat_int > mat_int ).
thf(sy_c_Matrix_Otranspose__mat_001t__Nat__Onat,type,
transpose_mat_nat: mat_nat > mat_nat ).
thf(sy_c_Matrix_Oupdate__mat_001t__Int__Oint,type,
update_mat_int: mat_int > product_prod_nat_nat > int > mat_int ).
thf(sy_c_Matrix_Oupdate__mat_001t__Nat__Onat,type,
update_mat_nat: mat_nat > product_prod_nat_nat > nat > mat_nat ).
thf(sy_c_Matrix_Oupper__triangular_001t__Int__Oint,type,
upper_triangular_int: mat_int > $o ).
thf(sy_c_Matrix_Oupper__triangular_001t__Nat__Onat,type,
upper_triangular_nat: mat_nat > $o ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001t__Int__Oint,type,
matrix8485685120660989714at_int: nat > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001t__Nat__Onat,type,
matrix8488175591170039990at_nat: nat > mat_nat ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Int__Oint_001t__Int__Oint,type,
matrix323868623736973467nt_int: mat_int > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Int__Oint_001t__Nat__Onat,type,
matrix326359094246023743nt_nat: mat_int > mat_nat ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Nat__Onat_001t__Int__Oint,type,
matrix8547886948690694719at_int: mat_nat > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Nat__Onat_001t__Nat__Onat,type,
matrix8550377419199744995at_nat: mat_nat > mat_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Omat_It__Int__Oint_J,type,
ord_less_mat_int: mat_int > mat_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Omat_It__Nat__Onat_J,type,
ord_less_mat_nat: mat_nat > mat_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__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Int__Oint_J,type,
ord_less_eq_mat_int: mat_int > mat_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Matrix__Omat_It__Int__Oint_J_J,type,
ord_le6155007575904973794at_int: mat_mat_int > mat_mat_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Matrix__Omat_It__Nat__Onat_J_J,type,
ord_le8645090720836700550at_nat: mat_mat_nat > mat_mat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Nat__Onat_J,type,
ord_less_eq_mat_nat: mat_nat > mat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le4000661125347319327et_nat: mat_set_nat > mat_set_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__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Rank__Argument__General_Oadd__multiple__cols_001t__Int__Oint,type,
rank_A5092470159319574270ls_int: int > nat > list_nat > mat_int > mat_int ).
thf(sy_c_Rank__Argument__General_Oadd__multiple__cols_001t__Nat__Onat,type,
rank_A5094960629828624546ls_nat: nat > nat > list_nat > mat_nat > mat_nat ).
thf(sy_c_Rank__Argument__General_Oadd__row__to__multiple_001t__Int__Oint,type,
rank_A6931195264251906052le_int: int > list_nat > nat > mat_int > mat_int ).
thf(sy_c_Rank__Argument__General_Oadd__row__to__multiple_001t__Nat__Onat,type,
rank_A6933685734760956328le_nat: nat > list_nat > nat > mat_nat > mat_nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Int__Oint_J,type,
collect_mat_int: ( mat_int > $o ) > set_mat_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Matrix__Omat_It__Int__Oint_J,type,
set_or2787914382489358134at_int: mat_int > mat_int > set_mat_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Matrix__Omat_It__Nat__Onat_J,type,
set_or6965765401998554842at_nat: mat_nat > mat_nat > set_mat_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__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Int__Oint_J,type,
member_mat_int: mat_int > set_mat_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_C____,type,
c: mat_int ).
thf(sy_v_D____,type,
d: mat_int ).
thf(sy_v__092_060B_062s,type,
b_s: list_set_a ).
thf(sy_v__092_060Lambda_062,type,
lambda: nat ).
thf(sy_v__092_060V_062s,type,
v_s: list_a ).
thf(sy_v__092_060k_062,type,
k: nat ).
thf(sy_v_i,type,
i: nat ).
% Relevant facts (1209)
thf(fact_0_ine0,axiom,
! [I: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ one_one_nat @ ( dim_row_int @ d ) ) )
=> ( I != zero_zero_nat ) ) ).
% ine0
thf(fact_1_D__def,axiom,
( d
= ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ c ) ) @ c ) ) ).
% D_def
thf(fact_2_t__design__min__v,axiom,
ord_less_nat @ one_one_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% t_design_min_v
thf(fact_3_incomplete,axiom,
ord_less_nat @ k @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% incomplete
thf(fact_4_necessary__condition__one,axiom,
( ( times_times_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
= ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ).
% necessary_condition_one
thf(fact_5_v__non__zero,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% v_non_zero
thf(fact_6_rep__not__zero,axiom,
( ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
!= zero_zero_nat ) ).
% rep_not_zero
thf(fact_7_index__lt__replication,axiom,
ord_less_nat @ lambda @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% index_lt_replication
thf(fact_8_ordered__bibd__axioms,axiom,
incide4817766913905363833bibd_a @ v_s @ b_s @ k @ lambda ).
% ordered_bibd_axioms
thf(fact_9_r__gzero,axiom,
ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% r_gzero
thf(fact_10__092_060open_062_092_060And_062i_O_Ai_A_092_060in_062_A_1231_O_O_060dim__row_AD_125_A_092_060Longrightarrow_062_Ai_A_060_Adim__row_A_IN_A_K_AN_092_060_094sup_062T_J_092_060close_062,axiom,
! [I: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ one_one_nat @ ( dim_row_int @ d ) ) )
=> ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) ).
% \<open>\<And>i. i \<in> {1..<dim_row D} \<Longrightarrow> i < dim_row (N * N\<^sup>T)\<close>
thf(fact_11_ordered__constant__rep__axioms,axiom,
incide6922509864216205631_rep_a @ v_s @ b_s @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% ordered_constant_rep_axioms
thf(fact_12_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_13_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_14_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_15_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_16_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_17_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_18_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_19_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_20_del__point__order,axiom,
! [P: a] :
( ( member_a @ P @ ( set_a2 @ v_s ) )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) )
= ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ).
% del_point_order
thf(fact_21_transpose__N__mult__diag,axiom,
! [I: nat,J: nat] :
( ( I = J )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ J @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( index_mat_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ) ) ) ).
% transpose_N_mult_diag
thf(fact_22_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_23_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_24_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_25_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_26_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_27_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_28_del__invalid__point,axiom,
! [P: a] :
( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
=> ( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P )
= ( set_a2 @ v_s ) ) ) ).
% del_invalid_point
thf(fact_29_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_30_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_31_C__def,axiom,
( c
= ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% C_def
thf(fact_32_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_33_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_34_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_35_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_36_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_37_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_38_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_39_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_40_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_41_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_42_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_43_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_44_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_45_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_46_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_47_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_48_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_49_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_50_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_51_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_52_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_53_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: mat_int,P2: mat_int > $o] :
( ( member_mat_int @ A @ ( collect_mat_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_mat_int] :
( ( collect_mat_int
@ ^ [X: mat_int] : ( member_mat_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__cong,axiom,
! [P2: mat_int > $o,Q: mat_int > $o] :
( ! [X2: mat_int] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_mat_int @ P2 )
= ( collect_mat_int @ Q ) ) ) ).
% Collect_cong
thf(fact_62_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_63_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_64_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_65_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_66_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_67_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_68_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_69_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_70_transpose__N__mult__dim_I1_J,axiom,
( ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% transpose_N_mult_dim(1)
thf(fact_71_transpose__N__mult__dim_I2_J,axiom,
( ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% transpose_N_mult_dim(2)
thf(fact_72_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_73_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_74_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_75_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_76_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_77_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_78_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_79_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_80_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_81_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_82_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_83_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_84_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_85_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_86_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_87_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_88_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_89_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_90_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_91_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_92_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_93_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_94_transpose__N__mult__off__diag,axiom,
! [I: nat,J: nat] :
( ( I != J )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ J @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( index_mat_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ lambda ) ) ) ) ) ).
% transpose_N_mult_off_diag
thf(fact_95_transform__N__step2__vals_I3_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( I = J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ lambda ) ) ) ) ) ) ) ).
% transform_N_step2_vals(3)
thf(fact_96_mat__is__proper,axiom,
incide294466202882093137at_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% mat_is_proper
thf(fact_97_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_98_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_99_ordered__incomplete__design__axioms,axiom,
incide1377962018248667206sign_a @ v_s @ b_s @ k ).
% ordered_incomplete_design_axioms
thf(fact_100_dim__row__is__v,axiom,
( ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% dim_row_is_v
thf(fact_101_transform__N__step1__vals_I3_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ lambda ) ) ) ) ) ) ).
% transform_N_step1_vals(3)
thf(fact_102_transform__N__step2__vals_I2_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ lambda ) ) ) ) ) ) ).
% transform_N_step2_vals(2)
thf(fact_103_transform__N__step1__vals_I1_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ) ) ) ) ).
% transform_N_step1_vals(1)
thf(fact_104_transform__N__step1__vals_I2_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ lambda ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% transform_N_step1_vals(2)
thf(fact_105_transform__N__step1__vals_I4_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( I = J )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ lambda ) ) ) ) ) ) ) ) ).
% transform_N_step1_vals(4)
thf(fact_106_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_107_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_108_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_109_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_110_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_111_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_112_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_113_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_114_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_115_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_116_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_117_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_118_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_119_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_120_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_121_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_122_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_123_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_124_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_125_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_126_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_127_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_128_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_129_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_130_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_131_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_132_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_133_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_134_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_135_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_136_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_137_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_138_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_139_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_140_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_141_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_142_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_143_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_144_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_145_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_146_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_147_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_148_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_149_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_150_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_151_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_152_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_153_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_154_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_155_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_156_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_157_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_158_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_159_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_160_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_161_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_162_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_163_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_164_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_165_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_166_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_167_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_168_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_169_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_170_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_171_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_172_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_173_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_174_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_175_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_176_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_177_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_178_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_179_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_180_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_181_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_182_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_183_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_184_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_185_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_186_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_187_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_188_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_189_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_190_mult__of__nat__commute,axiom,
! [X3: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_191_mult__of__nat__commute,axiom,
! [X3: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_192_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_193_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_194_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_195_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_196_is__dualD1,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs )
=> ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ).
% is_dualD1
thf(fact_197_is__dualD1,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs )
=> ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ).
% is_dualD1
thf(fact_198_all__cols__non__empty,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( incide6851923868969248411ol_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J ) ) ).
% all_cols_non_empty
thf(fact_199_incidence__mat__rep__num,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( incide7000514267430604580um_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ I )
= ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% incidence_mat_rep_num
thf(fact_200_ordered__block__design__axioms,axiom,
incide5219153079875704461sign_a @ v_s @ b_s @ k ).
% ordered_block_design_axioms
thf(fact_201_ordered__pairwise__balance__axioms,axiom,
incide6880889959311561818ance_a @ v_s @ b_s @ lambda ).
% ordered_pairwise_balance_axioms
thf(fact_202_ordered__design__axioms,axiom,
incide2848671379600480836sign_a @ v_s @ b_s ).
% ordered_design_axioms
thf(fact_203_lift__mat__01__index__iff_I2_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ( index_mat_nat @ ( matrix326359094246023743nt_nat @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat )
= ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ) ).
% lift_mat_01_index_iff(2)
thf(fact_204_lift__mat__01__index__iff_I2_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ( index_mat_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int )
= ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ) ).
% lift_mat_01_index_iff(2)
thf(fact_205_lift__mat__01__index__iff_I1_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ( index_mat_nat @ ( matrix326359094246023743nt_nat @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat )
= ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ).
% lift_mat_01_index_iff(1)
thf(fact_206_lift__mat__01__index__iff_I1_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ( index_mat_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int )
= ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ).
% lift_mat_01_index_iff(1)
thf(fact_207_zero__one__matrix__ring__1__axioms,axiom,
incide6080938071136783841_1_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% zero_one_matrix_ring_1_axioms
thf(fact_208_add__multiple__cols__index__unchanged,axiom,
! [I: nat,A2: mat_nat,J: nat,K: nat,A: nat,Ls: list_nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( K != J )
=> ( ( index_mat_nat @ ( rank_A5094960629828624546ls_nat @ A @ K @ Ls @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_multiple_cols_index_unchanged
thf(fact_209_add__multiple__cols__index__unchanged,axiom,
! [I: nat,A2: mat_int,J: nat,K: nat,A: int,Ls: list_nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( K != J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ A @ K @ Ls @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_multiple_cols_index_unchanged
thf(fact_210_index__transpose__mat_I1_J,axiom,
! [I: nat,A2: mat_nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_col_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_row_nat @ A2 ) )
=> ( ( index_mat_nat @ ( transpose_mat_nat @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ J @ I ) ) ) ) ) ).
% index_transpose_mat(1)
thf(fact_211_index__transpose__mat_I1_J,axiom,
! [I: nat,A2: mat_int,J: nat] :
( ( ord_less_nat @ I @ ( dim_col_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_row_int @ A2 ) )
=> ( ( index_mat_int @ ( transpose_mat_int @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ J @ I ) ) ) ) ) ).
% index_transpose_mat(1)
thf(fact_212_index__minus__mat_I2_J,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( dim_row_nat @ ( minus_minus_mat_nat @ A2 @ B2 ) )
= ( dim_row_nat @ B2 ) ) ).
% index_minus_mat(2)
thf(fact_213_index__minus__mat_I2_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_row_int @ ( minus_minus_mat_int @ A2 @ B2 ) )
= ( dim_row_int @ B2 ) ) ).
% index_minus_mat(2)
thf(fact_214_index__minus__mat_I3_J,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( dim_col_nat @ ( minus_minus_mat_nat @ A2 @ B2 ) )
= ( dim_col_nat @ B2 ) ) ).
% index_minus_mat(3)
thf(fact_215_index__minus__mat_I3_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_col_int @ ( minus_minus_mat_int @ A2 @ B2 ) )
= ( dim_col_int @ B2 ) ) ).
% index_minus_mat(3)
thf(fact_216_Matrix_Otranspose__transpose,axiom,
! [A2: mat_int] :
( ( transpose_mat_int @ ( transpose_mat_int @ A2 ) )
= A2 ) ).
% Matrix.transpose_transpose
thf(fact_217_transpose__mat__eq,axiom,
! [A2: mat_int,B2: mat_int] :
( ( ( transpose_mat_int @ A2 )
= ( transpose_mat_int @ B2 ) )
= ( A2 = B2 ) ) ).
% transpose_mat_eq
thf(fact_218_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_219_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_220_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_221_index__mult__mat_I2_J,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( dim_row_nat @ ( times_times_mat_nat @ A2 @ B2 ) )
= ( dim_row_nat @ A2 ) ) ).
% index_mult_mat(2)
thf(fact_222_index__mult__mat_I2_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_row_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_mult_mat(2)
thf(fact_223_index__mult__mat_I3_J,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( dim_col_nat @ ( times_times_mat_nat @ A2 @ B2 ) )
= ( dim_col_nat @ B2 ) ) ).
% index_mult_mat(3)
thf(fact_224_index__mult__mat_I3_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_col_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( dim_col_int @ B2 ) ) ).
% index_mult_mat(3)
thf(fact_225_add__row__to__multiple__dim_I1_J,axiom,
! [A: nat,Ks: list_nat,L: nat,A2: mat_nat] :
( ( dim_row_nat @ ( rank_A6933685734760956328le_nat @ A @ Ks @ L @ A2 ) )
= ( dim_row_nat @ A2 ) ) ).
% add_row_to_multiple_dim(1)
thf(fact_226_add__row__to__multiple__dim_I1_J,axiom,
! [A: int,Ks: list_nat,L: nat,A2: mat_int] :
( ( dim_row_int @ ( rank_A6931195264251906052le_int @ A @ Ks @ L @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% add_row_to_multiple_dim(1)
thf(fact_227_add__row__to__multiple__dim_I2_J,axiom,
! [A: nat,Ks: list_nat,L: nat,A2: mat_nat] :
( ( dim_col_nat @ ( rank_A6933685734760956328le_nat @ A @ Ks @ L @ A2 ) )
= ( dim_col_nat @ A2 ) ) ).
% add_row_to_multiple_dim(2)
thf(fact_228_add__row__to__multiple__dim_I2_J,axiom,
! [A: int,Ks: list_nat,L: nat,A2: mat_int] :
( ( dim_col_int @ ( rank_A6931195264251906052le_int @ A @ Ks @ L @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% add_row_to_multiple_dim(2)
thf(fact_229_add__multiple__cols__dim_I1_J,axiom,
! [A: nat,K: nat,Ls: list_nat,A2: mat_nat] :
( ( dim_row_nat @ ( rank_A5094960629828624546ls_nat @ A @ K @ Ls @ A2 ) )
= ( dim_row_nat @ A2 ) ) ).
% add_multiple_cols_dim(1)
thf(fact_230_add__multiple__cols__dim_I1_J,axiom,
! [A: int,K: nat,Ls: list_nat,A2: mat_int] :
( ( dim_row_int @ ( rank_A5092470159319574270ls_int @ A @ K @ Ls @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% add_multiple_cols_dim(1)
thf(fact_231_add__multiple__cols__dim_I2_J,axiom,
! [A: nat,K: nat,Ls: list_nat,A2: mat_nat] :
( ( dim_col_nat @ ( rank_A5094960629828624546ls_nat @ A @ K @ Ls @ A2 ) )
= ( dim_col_nat @ A2 ) ) ).
% add_multiple_cols_dim(2)
thf(fact_232_add__multiple__cols__dim_I2_J,axiom,
! [A: int,K: nat,Ls: list_nat,A2: mat_int] :
( ( dim_col_int @ ( rank_A5092470159319574270ls_int @ A @ K @ Ls @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% add_multiple_cols_dim(2)
thf(fact_233_M__not__one__simp,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
!= one_one_int )
=> ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ).
% M_not_one_simp
thf(fact_234_M__not__zero__simp,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
!= zero_zero_int )
=> ( ( index_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ) ).
% M_not_zero_simp
thf(fact_235_index__transpose__mat_I2_J,axiom,
! [A2: mat_nat] :
( ( dim_row_nat @ ( transpose_mat_nat @ A2 ) )
= ( dim_col_nat @ A2 ) ) ).
% index_transpose_mat(2)
thf(fact_236_index__transpose__mat_I2_J,axiom,
! [A2: mat_int] :
( ( dim_row_int @ ( transpose_mat_int @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% index_transpose_mat(2)
thf(fact_237_index__transpose__mat_I3_J,axiom,
! [A2: mat_nat] :
( ( dim_col_nat @ ( transpose_mat_nat @ A2 ) )
= ( dim_row_nat @ A2 ) ) ).
% index_transpose_mat(3)
thf(fact_238_index__transpose__mat_I3_J,axiom,
! [A2: mat_int] :
( ( dim_col_int @ ( transpose_mat_int @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_transpose_mat(3)
thf(fact_239_transform__N__step1__vals_I5_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( I != J )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ) ).
% transform_N_step1_vals(5)
thf(fact_240_eq__matI,axiom,
! [B2: mat_nat,A2: mat_nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_nat @ B2 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ B2 ) )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= ( index_mat_nat @ B2 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) )
=> ( ( ( dim_row_nat @ A2 )
= ( dim_row_nat @ B2 ) )
=> ( ( ( dim_col_nat @ A2 )
= ( dim_col_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% eq_matI
thf(fact_241_eq__matI,axiom,
! [B2: mat_int,A2: mat_int] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_int @ B2 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_int @ B2 ) )
=> ( ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= ( index_mat_int @ B2 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) )
=> ( ( ( dim_row_int @ A2 )
= ( dim_row_int @ B2 ) )
=> ( ( ( dim_col_int @ A2 )
= ( dim_col_int @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% eq_matI
thf(fact_242_transform__N__step2__vals_I4_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( I != J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ).
% transform_N_step2_vals(4)
thf(fact_243_index__minus__mat_I1_J,axiom,
! [I: nat,B2: mat_mat_int,J: nat,A2: mat_mat_int] :
( ( ord_less_nat @ I @ ( dim_row_mat_int @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_mat_int @ B2 ) )
=> ( ( index_mat_mat_int @ ( minus_5217014392749413595at_int @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_mat_int @ ( index_mat_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_mat_int @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_244_index__minus__mat_I1_J,axiom,
! [I: nat,B2: mat_mat_nat,J: nat,A2: mat_mat_nat] :
( ( ord_less_nat @ I @ ( dim_row_mat_nat @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_mat_nat @ B2 ) )
=> ( ( index_mat_mat_nat @ ( minus_7707097537681140351at_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_mat_nat @ ( index_mat_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_mat_nat @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_245_index__minus__mat_I1_J,axiom,
! [I: nat,B2: mat_set_nat,J: nat,A2: mat_set_nat] :
( ( ord_less_nat @ I @ ( dim_row_set_nat @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_set_nat @ B2 ) )
=> ( ( index_mat_set_nat @ ( minus_6025431102612251238et_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_set_nat @ ( index_mat_set_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_set_nat @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_246_index__minus__mat_I1_J,axiom,
! [I: nat,B2: mat_int,J: nat,A2: mat_int] :
( ( ord_less_nat @ I @ ( dim_row_int @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ B2 ) )
=> ( ( index_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_int @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_247_index__minus__mat_I1_J,axiom,
! [I: nat,B2: mat_nat,J: nat,A2: mat_nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ B2 ) )
=> ( ( index_mat_nat @ ( minus_minus_mat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_nat @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_nat @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_248_index__uminus__mat_I1_J,axiom,
! [I: nat,A2: mat_mat_int,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_mat_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_mat_int @ A2 ) )
=> ( ( index_mat_mat_int @ ( uminus1241013589067180107at_int @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( uminus5233555848319219356at_int @ ( index_mat_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_uminus_mat(1)
thf(fact_249_index__uminus__mat_I1_J,axiom,
! [I: nat,A2: mat_int,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( uminus5233555848319219356at_int @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( uminus_uminus_int @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_uminus_mat(1)
thf(fact_250_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_251_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_252_square__eq__1__iff,axiom,
! [X3: int] :
( ( ( times_times_int @ X3 @ X3 )
= one_one_int )
= ( ( X3 = one_one_int )
| ( X3
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_253_ordered__bibd_Otransform__N__step1__vals_I5_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( I != J )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(5)
thf(fact_254_ordered__bibd_Otransform__N__step1__vals_I5_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( I != J )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(5)
thf(fact_255_ordered__bibd_Otransform__N__step2__vals_I4_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( I != J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(4)
thf(fact_256_ordered__bibd_Otransform__N__step2__vals_I4_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( I != J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(4)
thf(fact_257_add__first__row__to__multiple__index_I1_J,axiom,
! [I: nat,M3: mat_nat,J: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ M3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ M3 ) )
=> ( ( I = zero_zero_nat )
=> ( ( index_mat_nat @ ( rank_A6933685734760956328le_nat @ A @ ( upt @ one_one_nat @ ( dim_row_nat @ M3 ) ) @ zero_zero_nat @ M3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ M3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_first_row_to_multiple_index(1)
thf(fact_258_add__first__row__to__multiple__index_I1_J,axiom,
! [I: nat,M3: mat_int,J: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ M3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ M3 ) )
=> ( ( I = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ A @ ( upt @ one_one_nat @ ( dim_row_int @ M3 ) ) @ zero_zero_nat @ M3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ M3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_first_row_to_multiple_index(1)
thf(fact_259_mat__eq__iff,axiom,
( ( ^ [Y2: mat_nat,Z: mat_nat] : ( Y2 = Z ) )
= ( ^ [X: mat_nat,Y3: mat_nat] :
( ( ( dim_row_nat @ X )
= ( dim_row_nat @ Y3 ) )
& ( ( dim_col_nat @ X )
= ( dim_col_nat @ Y3 ) )
& ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ Y3 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ Y3 ) )
=> ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ Y3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% mat_eq_iff
thf(fact_260_mat__eq__iff,axiom,
( ( ^ [Y2: mat_int,Z: mat_int] : ( Y2 = Z ) )
= ( ^ [X: mat_int,Y3: mat_int] :
( ( ( dim_row_int @ X )
= ( dim_row_int @ Y3 ) )
& ( ( dim_col_int @ X )
= ( dim_col_int @ Y3 ) )
& ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_int @ Y3 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_int @ Y3 ) )
=> ( ( index_mat_int @ X @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_int @ Y3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% mat_eq_iff
thf(fact_261_ordered__bibd_Otransform__N__step1__vals_I3_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(3)
thf(fact_262_ordered__bibd_Otransform__N__step1__vals_I3_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(3)
thf(fact_263_ordered__bibd_Otransform__N__step2__vals_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(2)
thf(fact_264_ordered__bibd_Otransform__N__step2__vals_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(2)
thf(fact_265_add__all__cols__to__first_I1_J,axiom,
! [I: nat,M3: mat_nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ M3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ M3 ) )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_nat @ ( rank_A5094960629828624546ls_nat @ one_one_nat @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_nat @ M3 ) ) @ M3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ M3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_all_cols_to_first(1)
thf(fact_266_add__all__cols__to__first_I1_J,axiom,
! [I: nat,M3: mat_int,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ M3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ M3 ) )
=> ( ( J != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ M3 ) ) @ M3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ M3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_all_cols_to_first(1)
thf(fact_267_ordered__bibd_Otransform__N__step1__vals_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(1)
thf(fact_268_ordered__bibd_Otransform__N__step1__vals_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(1)
thf(fact_269_ordered__bibd_Otransform__N__step1__vals_I4_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( I = J )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(4)
thf(fact_270_ordered__bibd_Otransform__N__step1__vals_I4_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J != zero_zero_nat )
=> ( ( I = J )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(4)
thf(fact_271_ordered__bibd_Otransform__N__step1__vals_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ Lambda ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(2)
thf(fact_272_ordered__bibd_Otransform__N__step1__vals_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ Lambda ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step1_vals(2)
thf(fact_273_ordered__bibd_Otransform__N__step2__vals_I3_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( I = J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(3)
thf(fact_274_ordered__bibd_Otransform__N__step2__vals_I3_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I != zero_zero_nat )
=> ( ( I = J )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(3)
thf(fact_275_transform__upper__triangular,axiom,
upper_triangular_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% transform_upper_triangular
thf(fact_276_transform__N__step2__vals_I1_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ lambda ) @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% transform_N_step2_vals(1)
thf(fact_277_d00,axiom,
( ( index_mat_int @ d @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ lambda ) @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ) ) ).
% d00
thf(fact_278_ordered__constant__rep_Otranspose__N__mult__diag,axiom,
! [V_s: list_nat,B_s: list_set_nat,R: nat,I: nat,J: nat] :
( ( incide6328174194974408527ep_nat @ V_s @ B_s @ R )
=> ( ( I = J )
=> ( ( ord_less_nat @ I @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) )
=> ( ( ord_less_nat @ J @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) )
=> ( ( index_mat_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ R ) ) ) ) ) ) ).
% ordered_constant_rep.transpose_N_mult_diag
thf(fact_279_ordered__constant__rep_Otranspose__N__mult__diag,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat,I: nat,J: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( ( I = J )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ V_s ) ) )
=> ( ( ord_less_nat @ J @ ( finite_card_a @ ( set_a2 @ V_s ) ) )
=> ( ( index_mat_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ R ) ) ) ) ) ) ).
% ordered_constant_rep.transpose_N_mult_diag
thf(fact_280_ordered__pairwise__balance_Otranspose__N__mult__off__diag,axiom,
! [V_s: list_nat,B_s: list_set_nat,Lambda: nat,I: nat,J: nat] :
( ( incide3388802471754236788ce_nat @ V_s @ B_s @ Lambda )
=> ( ( I != J )
=> ( ( ord_less_nat @ I @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) )
=> ( ( ord_less_nat @ J @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) )
=> ( ( index_mat_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ).
% ordered_pairwise_balance.transpose_N_mult_off_diag
thf(fact_281_ordered__pairwise__balance_Otranspose__N__mult__off__diag,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat,I: nat,J: nat] :
( ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda )
=> ( ( I != J )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ V_s ) ) )
=> ( ( ord_less_nat @ J @ ( finite_card_a @ ( set_a2 @ V_s ) ) )
=> ( ( index_mat_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( semiri1314217659103216013at_int @ Lambda ) ) ) ) ) ) ).
% ordered_pairwise_balance.transpose_N_mult_off_diag
thf(fact_282_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_283_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_284_uminus__uminus__mat,axiom,
! [A2: mat_int] :
( ( uminus5233555848319219356at_int @ ( uminus5233555848319219356at_int @ A2 ) )
= A2 ) ).
% uminus_uminus_mat
thf(fact_285_uminus__eq__mat,axiom,
! [A2: mat_int,B2: mat_int] :
( ( ( uminus5233555848319219356at_int @ A2 )
= ( uminus5233555848319219356at_int @ B2 ) )
= ( A2 = B2 ) ) ).
% uminus_eq_mat
thf(fact_286_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_287_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_288_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_289_index__uminus__mat_I2_J,axiom,
! [A2: mat_int] :
( ( dim_row_int @ ( uminus5233555848319219356at_int @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_uminus_mat(2)
thf(fact_290_index__uminus__mat_I3_J,axiom,
! [A2: mat_int] :
( ( dim_col_int @ ( uminus5233555848319219356at_int @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% index_uminus_mat(3)
thf(fact_291__092_060open_0620_A_092_060notin_062_Aset_A_0911_O_O_060dim__row_A_IN_A_K_AN_092_060_094sup_062T_J_093_092_060close_062,axiom,
~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ) ) ).
% \<open>0 \<notin> set [1..<dim_row (N * N\<^sup>T)]\<close>
thf(fact_292_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_293_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_294_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_295_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_296_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_297_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_298_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_299_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_300_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_301_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_302_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_303_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_304_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_305_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_306_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_307_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_308_uminus__mult__left__mat,axiom,
! [A2: mat_int,B2: mat_int] :
( ( ( dim_col_int @ A2 )
= ( dim_row_int @ B2 ) )
=> ( ( times_times_mat_int @ ( uminus5233555848319219356at_int @ A2 ) @ B2 )
= ( uminus5233555848319219356at_int @ ( times_times_mat_int @ A2 @ B2 ) ) ) ) ).
% uminus_mult_left_mat
thf(fact_309_uminus__mult__right__mat,axiom,
! [A2: mat_int,B2: mat_int] :
( ( ( dim_col_int @ A2 )
= ( dim_row_int @ B2 ) )
=> ( ( times_times_mat_int @ A2 @ ( uminus5233555848319219356at_int @ B2 ) )
= ( uminus5233555848319219356at_int @ ( times_times_mat_int @ A2 @ B2 ) ) ) ) ).
% uminus_mult_right_mat
thf(fact_310_index__add__mat_I1_J,axiom,
! [I: nat,B2: mat_mat_nat,J: nat,A2: mat_mat_nat] :
( ( ord_less_nat @ I @ ( dim_row_mat_nat @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_mat_nat @ B2 ) )
=> ( ( index_mat_mat_nat @ ( plus_p2963301051743621423at_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_mat_nat @ ( index_mat_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_mat_nat @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_311_index__add__mat_I1_J,axiom,
! [I: nat,B2: mat_mat_int,J: nat,A2: mat_mat_int] :
( ( ord_less_nat @ I @ ( dim_row_mat_int @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_mat_int @ B2 ) )
=> ( ( index_mat_mat_int @ ( plus_p473217906811894667at_int @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_mat_int @ ( index_mat_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_mat_int @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_312_index__add__mat_I1_J,axiom,
! [I: nat,B2: mat_int,J: nat,A2: mat_int] :
( ( ord_less_nat @ I @ ( dim_row_int @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ B2 ) )
=> ( ( index_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_int @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_313_index__add__mat_I1_J,axiom,
! [I: nat,B2: mat_nat,J: nat,A2: mat_nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ B2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ B2 ) )
=> ( ( index_mat_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_nat @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_314_add__row__to__multiple__index__unchanged,axiom,
! [I: nat,A2: mat_nat,J: nat,Ks: list_nat,A: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ~ ( member_nat @ I @ ( set_nat2 @ Ks ) )
=> ( ( index_mat_nat @ ( rank_A6933685734760956328le_nat @ A @ Ks @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_row_to_multiple_index_unchanged
thf(fact_315_add__row__to__multiple__index__unchanged,axiom,
! [I: nat,A2: mat_int,J: nat,Ks: list_nat,A: int,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ~ ( member_nat @ I @ ( set_nat2 @ Ks ) )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ A @ Ks @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_row_to_multiple_index_unchanged
thf(fact_316_upper__triangularI,axiom,
! [A2: mat_nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_nat @ A2 ) )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_nat ) ) )
=> ( upper_triangular_nat @ A2 ) ) ).
% upper_triangularI
thf(fact_317_upper__triangularI,axiom,
! [A2: mat_int] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_int @ A2 ) )
=> ( ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_int ) ) )
=> ( upper_triangular_int @ A2 ) ) ).
% upper_triangularI
thf(fact_318_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_319_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_320_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_321_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_322_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_323_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_324_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_325_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_326_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_327_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_328_transpose__uminus,axiom,
! [A2: mat_int] :
( ( transpose_mat_int @ ( uminus5233555848319219356at_int @ A2 ) )
= ( uminus5233555848319219356at_int @ ( transpose_mat_int @ A2 ) ) ) ).
% transpose_uminus
thf(fact_329_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_330_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_331_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_332_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_333_add__less__zeroD,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X3 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X3 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_334_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_335_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_336_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_337_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_338_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_339_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_340_square__diff__square__factored,axiom,
! [X3: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X3 @ Y ) @ ( minus_minus_int @ X3 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_341_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_342_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_343_not__sum__squares__lt__zero,axiom,
! [X3: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_344_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_345_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_346_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_347_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_348_square__diff__one__factored,axiom,
! [X3: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_349_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_350_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_351_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_352_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_353_int__div__less__self,axiom,
! [X3: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_354_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_355_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_356_inc__mat__of__01__mat,axiom,
! [Vs2: list_nat,Bs2: list_set_nat] : ( incide6080938071136783841_1_int @ ( incide1177982898701834729at_int @ Vs2 @ Bs2 ) ) ).
% inc_mat_of_01_mat
thf(fact_357_inc__mat__of__01__mat,axiom,
! [Vs2: list_a,Bs2: list_set_a] : ( incide6080938071136783841_1_int @ ( incide7016682120514301311_a_int @ Vs2 @ Bs2 ) ) ).
% inc_mat_of_01_mat
thf(fact_358_upper__triangular__def,axiom,
( upper_triangular_nat
= ( ^ [A3: mat_nat] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ( ( index_mat_nat @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ).
% upper_triangular_def
thf(fact_359_upper__triangular__def,axiom,
( upper_triangular_int
= ( ^ [A3: mat_int] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_int @ A3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ( ( index_mat_int @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_int ) ) ) ) ) ).
% upper_triangular_def
thf(fact_360_upper__triangularD,axiom,
! [A2: mat_nat,J: nat,I: nat] :
( ( upper_triangular_nat @ A2 )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ).
% upper_triangularD
thf(fact_361_upper__triangularD,axiom,
! [A2: mat_int,J: nat,I: nat] :
( ( upper_triangular_int @ A2 )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ).
% upper_triangularD
thf(fact_362_ordered__incomplete__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat] :
( ( incide1377962018248667206sign_a @ V_s @ B_s @ K2 )
=> ( incide5219153079875704461sign_a @ V_s @ B_s @ K2 ) ) ).
% ordered_incomplete_design.axioms(1)
thf(fact_363_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_364_add__first__row__to__multiple__index_I2_J,axiom,
! [I: nat,M3: mat_nat,J: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ M3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ M3 ) )
=> ( ( I != zero_zero_nat )
=> ( ( index_mat_nat @ ( rank_A6933685734760956328le_nat @ A @ ( upt @ one_one_nat @ ( dim_row_nat @ M3 ) ) @ zero_zero_nat @ M3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ M3 @ ( product_Pair_nat_nat @ zero_zero_nat @ J ) ) ) @ ( index_mat_nat @ M3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% add_first_row_to_multiple_index(2)
thf(fact_365_add__first__row__to__multiple__index_I2_J,axiom,
! [I: nat,M3: mat_int,J: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ M3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ M3 ) )
=> ( ( I != zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ A @ ( upt @ one_one_nat @ ( dim_row_int @ M3 ) ) @ zero_zero_nat @ M3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ M3 @ ( product_Pair_nat_nat @ zero_zero_nat @ J ) ) ) @ ( index_mat_int @ M3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% add_first_row_to_multiple_index(2)
thf(fact_366_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_367_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_368_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_369_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_370_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_371_proper__inc__mat__def,axiom,
( incide296956673391143413at_nat
= ( ^ [M4: mat_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_nat @ M4 ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_nat @ M4 ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_372_proper__inc__mat__def,axiom,
( incide294466202882093137at_int
= ( ^ [M4: mat_int] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_int @ M4 ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_int @ M4 ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_373_ordered__design_Oall__cols__non__empty,axiom,
! [V_s: list_nat,B_s: list_set_nat,J: nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) )
=> ( incide6851923868969248411ol_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ J ) ) ) ).
% ordered_design.all_cols_non_empty
thf(fact_374_ordered__design_Oall__cols__non__empty,axiom,
! [V_s: list_a,B_s: list_set_a,J: nat] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) )
=> ( incide6851923868969248411ol_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ J ) ) ) ).
% ordered_design.all_cols_non_empty
thf(fact_375_ordered__bibd_Otransform__upper__triangular,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( upper_triangular_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) ) ).
% ordered_bibd.transform_upper_triangular
thf(fact_376_ordered__bibd_Otransform__upper__triangular,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( upper_triangular_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) ) ).
% ordered_bibd.transform_upper_triangular
thf(fact_377_ordered__constant__rep_Oincidence__mat__rep__num,axiom,
! [V_s: list_nat,B_s: list_set_nat,R: nat,I: nat] :
( ( incide6328174194974408527ep_nat @ V_s @ B_s @ R )
=> ( ( ord_less_nat @ I @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) )
=> ( ( incide7000514267430604580um_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ I )
= R ) ) ) ).
% ordered_constant_rep.incidence_mat_rep_num
thf(fact_378_ordered__constant__rep_Oincidence__mat__rep__num,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat,I: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ V_s ) ) )
=> ( ( incide7000514267430604580um_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ I )
= R ) ) ) ).
% ordered_constant_rep.incidence_mat_rep_num
thf(fact_379_ordered__bibd_Otransform__N__step2__vals_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ Lambda ) @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(1)
thf(fact_380_ordered__bibd_Otransform__N__step2__vals_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat,I: nat,J: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) )
=> ( ( I = zero_zero_nat )
=> ( ( J = zero_zero_nat )
=> ( ( index_mat_int @ ( rank_A5092470159319574270ls_int @ one_one_int @ zero_zero_nat @ ( upt @ one_one_nat @ ( dim_col_int @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) @ ( rank_A6931195264251906052le_int @ ( uminus_uminus_int @ one_one_int ) @ ( upt @ one_one_nat @ ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ zero_zero_nat @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ Lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ Lambda ) @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% ordered_bibd.transform_N_step2_vals(1)
thf(fact_381_non__empty__col__alt__def,axiom,
! [J: nat,M3: mat_nat] :
( ( ord_less_nat @ J @ ( dim_col_nat @ M3 ) )
=> ( ( incide6854414339478298687ol_nat @ M3 @ J )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ M3 ) )
& ( ( index_mat_nat @ M3 @ ( product_Pair_nat_nat @ I3 @ J ) )
!= zero_zero_nat ) ) ) ) ) ).
% non_empty_col_alt_def
thf(fact_382_non__empty__col__alt__def,axiom,
! [J: nat,M3: mat_int] :
( ( ord_less_nat @ J @ ( dim_col_int @ M3 ) )
=> ( ( incide6851923868969248411ol_int @ M3 @ J )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_int @ M3 ) )
& ( ( index_mat_int @ M3 @ ( product_Pair_nat_nat @ I3 @ J ) )
!= zero_zero_int ) ) ) ) ) ).
% non_empty_col_alt_def
thf(fact_383_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_384_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_385_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_386_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_387_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_388_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_389_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_390_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_391_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_392_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_393_index__add__mat_I2_J,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( dim_row_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) )
= ( dim_row_nat @ B2 ) ) ).
% index_add_mat(2)
thf(fact_394_index__add__mat_I2_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_row_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( dim_row_int @ B2 ) ) ).
% index_add_mat(2)
thf(fact_395_index__add__mat_I3_J,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( dim_col_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) )
= ( dim_col_nat @ B2 ) ) ).
% index_add_mat(3)
thf(fact_396_index__add__mat_I3_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_col_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( dim_col_int @ B2 ) ) ).
% index_add_mat(3)
thf(fact_397_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_398_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_399_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_400_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_401_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_402_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_403_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_404_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_405_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_406_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_407_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_408_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_409_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_410_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_411_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_412_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_413_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_414_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_415_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_416_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_417_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_418_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_419_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_420_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_421_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_422_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_423_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P2 @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_424_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P2 @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_425_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_426_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_427_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_428_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_429_split__div,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% split_div
thf(fact_430_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_431_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_432_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_433_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_434_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_435_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_436_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_437_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_438_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_439_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_440_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_441_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_442_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_443_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_444_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_445_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_446_int__gr__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_447_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_448_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_449_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_450_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_451_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_452_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_453_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M6: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_454_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_455_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_456_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_457_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_458_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_459_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_460_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_461_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_462_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_463_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_464_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_465_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_466_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_467_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_468_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_469_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_470_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_471_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_472_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_473_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_474_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_475_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_476_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_477_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_478_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_479_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_480_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_481_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_482_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_483_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_484_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_485_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_486_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_487_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_488_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_489_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_490_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_491_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_492_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_493_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_494_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_495_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_496_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_497_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_498_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_499_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_500_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_501_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_502_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_503_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_504_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_505_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_506_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_507_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_508_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_509_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_510_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_511_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_512_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_513_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_514_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_515_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_516_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_517_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_518_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_519_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_520_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_521_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_522_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_523_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_524_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_525_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_526_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_527_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_528_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_529_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_530_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_531_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_532_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_533_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_534_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_535_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_536_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_537_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_538_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_539_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_540_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_541_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_542_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_543_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_544_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_545_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_546_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_547_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_548_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_549_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_550_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_551_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_552_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_553_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_554_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_555_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_556_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_557_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_558_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_559_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_560_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_561_verit__comp__simplify1_I1_J,axiom,
! [A: mat_int] :
~ ( ord_less_mat_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_562_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_563_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_564_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_565_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_566_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_567_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_568_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_569_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_570_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_571_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_572_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_573_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_574_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_575_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_576_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_577_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_578_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_579_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_580_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_581_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_582_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_583_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_584_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_585_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_586_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_587_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_588_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_589_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_590_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_591_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_592_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_593_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_594_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_595_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_596_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_597_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_598_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_599_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_600_int__if,axiom,
! [P2: $o,A: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_601_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_602_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_603_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_604_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_605_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_606_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_607_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_608_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_609_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_610_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_611_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_612_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_613_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_614_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_615_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_616_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_617_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_618_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_619_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_620_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_621_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_622_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_623_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_624_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_625_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_626_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_627_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_628_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_629_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_630_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_631_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_632_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_633_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_634_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_635_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_636_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_637_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_638_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_639_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_640_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_641_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_642_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_643_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_644_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_645_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_646_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_647_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_648_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_649_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_650_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_651_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_652_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_653_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_654_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_655_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_656_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_657_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_658_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_659_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_660_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_661_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_662_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_663_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_664_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_665_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_666_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_667_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_668_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_669_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_670_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_671_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_672_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_673_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_674_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_675_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_676_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_677_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_678_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_679_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_680_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_681_sum__squares__eq__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_682_add__delete__point__inv,axiom,
! [P: a] :
( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ P )
= ( set_a2 @ v_s ) ) ) ).
% add_delete_point_inv
thf(fact_683_ordered__proper__design__axioms,axiom,
incide3676903341588786676sign_a @ v_s @ b_s ).
% ordered_proper_design_axioms
thf(fact_684_cm,axiom,
member_mat_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( carrier_mat_int @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% cm
thf(fact_685_carrier__matD_I1_J,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( dim_row_nat @ A2 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_686_carrier__matD_I1_J,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( dim_row_int @ A2 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_687_carrier__matD_I2_J,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( dim_col_nat @ A2 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_688_carrier__matD_I2_J,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( dim_col_int @ A2 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_689_assoc__mult__mat,axiom,
! [A2: mat_int,N_1: nat,N_2: nat,B2: mat_int,N_3: nat,C3: mat_int,N_4: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ N_1 @ N_2 ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N_2 @ N_3 ) )
=> ( ( member_mat_int @ C3 @ ( carrier_mat_int @ N_3 @ N_4 ) )
=> ( ( times_times_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ C3 )
= ( times_times_mat_int @ A2 @ ( times_times_mat_int @ B2 @ C3 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_690_transpose__carrier__mat,axiom,
! [A2: mat_int,Nc: nat,Nr: nat] :
( ( member_mat_int @ ( transpose_mat_int @ A2 ) @ ( carrier_mat_int @ Nc @ Nr ) )
= ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_691_assoc__add__mat,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B2: mat_nat,C3: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ C3 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) @ C3 )
= ( plus_plus_mat_nat @ A2 @ ( plus_plus_mat_nat @ B2 @ C3 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_692_assoc__add__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int,C3: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ C3 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( plus_plus_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ C3 )
= ( plus_plus_mat_int @ A2 @ ( plus_plus_mat_int @ B2 @ C3 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_693_uminus__carrier__iff__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ ( uminus5233555848319219356at_int @ A2 ) @ ( carrier_mat_int @ Nr @ Nc ) )
= ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_694_carrier__matI,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat] :
( ( ( dim_row_nat @ A2 )
= Nr )
=> ( ( ( dim_col_nat @ A2 )
= Nc )
=> ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_695_carrier__matI,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( ( dim_row_int @ A2 )
= Nr )
=> ( ( ( dim_col_int @ A2 )
= Nc )
=> ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_696_add__existing__point,axiom,
! [P: a] :
( ( member_a @ P @ ( set_a2 @ v_s ) )
=> ( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P )
= ( set_a2 @ v_s ) ) ) ).
% add_existing_point
thf(fact_697_mult__carrier__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( member_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_698_comm__add__mat,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B2: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ A2 @ B2 )
= ( plus_plus_mat_nat @ B2 @ A2 ) ) ) ) ).
% comm_add_mat
thf(fact_699_comm__add__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( plus_plus_mat_int @ A2 @ B2 )
= ( plus_plus_mat_int @ B2 @ A2 ) ) ) ) ).
% comm_add_mat
thf(fact_700_add__carrier__mat,axiom,
! [B2: mat_nat,Nr: nat,Nc: nat,A2: mat_nat] :
( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_701_add__carrier__mat,axiom,
! [B2: mat_int,Nr: nat,Nc: nat,A2: mat_int] :
( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( member_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_702_add__row__to__multiple__carrier,axiom,
! [A2: mat_nat,N: nat,A: nat,Ks: list_nat,L: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ N @ N ) )
=> ( member_mat_nat @ ( rank_A6933685734760956328le_nat @ A @ Ks @ L @ A2 ) @ ( carrier_mat_nat @ N @ N ) ) ) ).
% add_row_to_multiple_carrier
thf(fact_703_add__row__to__multiple__carrier,axiom,
! [A2: mat_int,N: nat,A: int,Ks: list_nat,L: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ N @ N ) )
=> ( member_mat_int @ ( rank_A6931195264251906052le_int @ A @ Ks @ L @ A2 ) @ ( carrier_mat_int @ N @ N ) ) ) ).
% add_row_to_multiple_carrier
thf(fact_704_uminus__carrier__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( member_mat_int @ ( uminus5233555848319219356at_int @ A2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_705_add__multiple__cols__carrier,axiom,
! [A2: mat_nat,N: nat,A: nat,K: nat,Ls: list_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ N @ N ) )
=> ( member_mat_nat @ ( rank_A5094960629828624546ls_nat @ A @ K @ Ls @ A2 ) @ ( carrier_mat_nat @ N @ N ) ) ) ).
% add_multiple_cols_carrier
thf(fact_706_add__multiple__cols__carrier,axiom,
! [A2: mat_int,N: nat,A: int,K: nat,Ls: list_nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ N @ N ) )
=> ( member_mat_int @ ( rank_A5092470159319574270ls_int @ A @ K @ Ls @ A2 ) @ ( carrier_mat_int @ N @ N ) ) ) ).
% add_multiple_cols_carrier
thf(fact_707_minus__carrier__mat,axiom,
! [B2: mat_nat,Nr: nat,Nc: nat,A2: mat_nat] :
( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( member_mat_nat @ ( minus_minus_mat_nat @ A2 @ B2 ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_708_minus__carrier__mat,axiom,
! [B2: mat_int,Nr: nat,Nc: nat,A2: mat_int] :
( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( member_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_709_ordered__bibd_Oaxioms_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K2: nat,Lambda: nat] :
( ( incide4338384322530710165bd_nat @ V_s @ B_s @ K2 @ Lambda )
=> ( incide1001368407746664282gn_nat @ V_s @ B_s ) ) ).
% ordered_bibd.axioms(1)
thf(fact_710_ordered__bibd_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat,Lambda: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K2 @ Lambda )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_bibd.axioms(1)
thf(fact_711_ordered__proper__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( incide2848671379600480836sign_a @ V_s @ B_s ) ) ).
% ordered_proper_design.axioms(1)
thf(fact_712_ordered__block__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K2: nat] :
( ( incide5219153079875704461sign_a @ V_s @ B_s @ K2 )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_block_design.axioms(1)
thf(fact_713_carrier__mat__triv,axiom,
! [M: mat_nat] : ( member_mat_nat @ M @ ( carrier_mat_nat @ ( dim_row_nat @ M ) @ ( dim_col_nat @ M ) ) ) ).
% carrier_mat_triv
thf(fact_714_carrier__mat__triv,axiom,
! [M: mat_int] : ( member_mat_int @ M @ ( carrier_mat_int @ ( dim_row_int @ M ) @ ( dim_col_int @ M ) ) ) ).
% carrier_mat_triv
thf(fact_715_ordered__constant__rep_Oaxioms_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,R: nat] :
( ( incide6328174194974408527ep_nat @ V_s @ B_s @ R )
=> ( incide1001368407746664282gn_nat @ V_s @ B_s ) ) ).
% ordered_constant_rep.axioms(1)
thf(fact_716_ordered__constant__rep_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_constant_rep.axioms(1)
thf(fact_717_transpose__mult,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( transpose_mat_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( times_times_mat_int @ ( transpose_mat_int @ B2 ) @ ( transpose_mat_int @ A2 ) ) ) ) ) ).
% transpose_mult
thf(fact_718_add__mult__distrib__mat,axiom,
! [A2: mat_nat,Nr: nat,N: nat,B2: mat_nat,C3: mat_nat,Nc: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ C3 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) @ C3 )
= ( plus_plus_mat_nat @ ( times_times_mat_nat @ A2 @ C3 ) @ ( times_times_mat_nat @ B2 @ C3 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_719_add__mult__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,C3: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ C3 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ C3 )
= ( plus_plus_mat_int @ ( times_times_mat_int @ A2 @ C3 ) @ ( times_times_mat_int @ B2 @ C3 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_720_mult__add__distrib__mat,axiom,
! [A2: mat_nat,Nr: nat,N: nat,B2: mat_nat,Nc: nat,C3: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( member_mat_nat @ C3 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( times_times_mat_nat @ A2 @ ( plus_plus_mat_nat @ B2 @ C3 ) )
= ( plus_plus_mat_nat @ ( times_times_mat_nat @ A2 @ B2 ) @ ( times_times_mat_nat @ A2 @ C3 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_721_mult__add__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat,C3: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( member_mat_int @ C3 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ A2 @ ( plus_plus_mat_int @ B2 @ C3 ) )
= ( plus_plus_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ ( times_times_mat_int @ A2 @ C3 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_722_transpose__add,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B2: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( transpose_mat_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) )
= ( plus_plus_mat_nat @ ( transpose_mat_nat @ A2 ) @ ( transpose_mat_nat @ B2 ) ) ) ) ) ).
% transpose_add
thf(fact_723_transpose__add,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( transpose_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( plus_plus_mat_int @ ( transpose_mat_int @ A2 ) @ ( transpose_mat_int @ B2 ) ) ) ) ) ).
% transpose_add
thf(fact_724_minus__mult__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,C3: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ C3 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) @ C3 )
= ( minus_minus_mat_int @ ( times_times_mat_int @ A2 @ C3 ) @ ( times_times_mat_int @ B2 @ C3 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_725_mult__minus__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat,C3: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( member_mat_int @ C3 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ A2 @ ( minus_minus_mat_int @ B2 @ C3 ) )
= ( minus_minus_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ ( times_times_mat_int @ A2 @ C3 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_726_transpose__minus,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B2: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( transpose_mat_nat @ ( minus_minus_mat_nat @ A2 @ B2 ) )
= ( minus_minus_mat_nat @ ( transpose_mat_nat @ A2 ) @ ( transpose_mat_nat @ B2 ) ) ) ) ) ).
% transpose_minus
thf(fact_727_transpose__minus,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( transpose_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) )
= ( minus_minus_mat_int @ ( transpose_mat_int @ A2 ) @ ( transpose_mat_int @ B2 ) ) ) ) ) ).
% transpose_minus
thf(fact_728_uminus__add__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( uminus5233555848319219356at_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( plus_plus_mat_int @ ( uminus5233555848319219356at_int @ B2 ) @ ( uminus5233555848319219356at_int @ A2 ) ) ) ) ) ).
% uminus_add_mat
thf(fact_729_minus__add__minus__mat,axiom,
! [U: mat_int,Nr: nat,Nc: nat,V: mat_int,W: mat_int] :
( ( member_mat_int @ U @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ V @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ W @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( minus_minus_mat_int @ U @ ( plus_plus_mat_int @ V @ W ) )
= ( minus_minus_mat_int @ ( minus_minus_mat_int @ U @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_730_add__uminus__minus__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( plus_plus_mat_int @ A2 @ ( uminus5233555848319219356at_int @ B2 ) )
= ( minus_minus_mat_int @ A2 @ B2 ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_731_minus__add__uminus__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( minus_minus_mat_int @ A2 @ B2 )
= ( plus_plus_mat_int @ A2 @ ( uminus5233555848319219356at_int @ B2 ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_732_uminus__add__minus__mat,axiom,
! [L: mat_int,Nr: nat,Nc: nat,R2: mat_int] :
( ( member_mat_int @ L @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ R2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( uminus5233555848319219356at_int @ ( plus_plus_mat_int @ L @ R2 ) )
= ( minus_minus_mat_int @ ( uminus5233555848319219356at_int @ L ) @ R2 ) ) ) ) ).
% uminus_add_minus_mat
thf(fact_733_ordered__proper__design_Omat__is__proper,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide1001368407746664282gn_nat @ V_s @ B_s )
=> ( incide294466202882093137at_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ).
% ordered_proper_design.mat_is_proper
thf(fact_734_ordered__proper__design_Omat__is__proper,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( incide294466202882093137at_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ).
% ordered_proper_design.mat_is_proper
thf(fact_735_sum__squares__gt__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X3 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_736_index__update__mat1,axiom,
! [I: nat,A2: mat_nat,J: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( index_mat_nat @ ( update_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) ) ) ).
% index_update_mat1
thf(fact_737_index__update__mat1,axiom,
! [I: nat,A2: mat_int,J: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( update_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) ) ) ).
% index_update_mat1
thf(fact_738_index__update__mat2,axiom,
! [I4: nat,A2: mat_nat,J4: nat,Ij: product_prod_nat_nat,A: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J4 @ ( dim_col_nat @ A2 ) )
=> ( ( ( product_Pair_nat_nat @ I4 @ J4 )
!= Ij )
=> ( ( index_mat_nat @ ( update_mat_nat @ A2 @ Ij @ A ) @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I4 @ J4 ) ) ) ) ) ) ).
% index_update_mat2
thf(fact_739_index__update__mat2,axiom,
! [I4: nat,A2: mat_int,J4: nat,Ij: product_prod_nat_nat,A: int] :
( ( ord_less_nat @ I4 @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J4 @ ( dim_col_int @ A2 ) )
=> ( ( ( product_Pair_nat_nat @ I4 @ J4 )
!= Ij )
=> ( ( index_mat_int @ ( update_mat_int @ A2 @ Ij @ A ) @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I4 @ J4 ) ) ) ) ) ) ).
% index_update_mat2
thf(fact_740_lift__01__mat__simp_I2_J,axiom,
! [M3: mat_int] :
( ( dim_col_nat @ ( matrix326359094246023743nt_nat @ M3 ) )
= ( dim_col_int @ M3 ) ) ).
% lift_01_mat_simp(2)
thf(fact_741_lift__01__mat__simp_I2_J,axiom,
! [M3: mat_nat] :
( ( dim_col_int @ ( matrix8547886948690694719at_int @ M3 ) )
= ( dim_col_nat @ M3 ) ) ).
% lift_01_mat_simp(2)
thf(fact_742_lift__01__mat__simp_I2_J,axiom,
! [M3: mat_nat] :
( ( dim_col_nat @ ( matrix8550377419199744995at_nat @ M3 ) )
= ( dim_col_nat @ M3 ) ) ).
% lift_01_mat_simp(2)
thf(fact_743_lift__01__mat__simp_I2_J,axiom,
! [M3: mat_int] :
( ( dim_col_int @ ( matrix323868623736973467nt_int @ M3 ) )
= ( dim_col_int @ M3 ) ) ).
% lift_01_mat_simp(2)
thf(fact_744_lift__01__mat__simp_I1_J,axiom,
! [M3: mat_int] :
( ( dim_row_nat @ ( matrix326359094246023743nt_nat @ M3 ) )
= ( dim_row_int @ M3 ) ) ).
% lift_01_mat_simp(1)
thf(fact_745_lift__01__mat__simp_I1_J,axiom,
! [M3: mat_nat] :
( ( dim_row_int @ ( matrix8547886948690694719at_int @ M3 ) )
= ( dim_row_nat @ M3 ) ) ).
% lift_01_mat_simp(1)
thf(fact_746_lift__01__mat__simp_I1_J,axiom,
! [M3: mat_nat] :
( ( dim_row_nat @ ( matrix8550377419199744995at_nat @ M3 ) )
= ( dim_row_nat @ M3 ) ) ).
% lift_01_mat_simp(1)
thf(fact_747_lift__01__mat__simp_I1_J,axiom,
! [M3: mat_int] :
( ( dim_row_int @ ( matrix323868623736973467nt_int @ M3 ) )
= ( dim_row_int @ M3 ) ) ).
% lift_01_mat_simp(1)
thf(fact_748_dim__update__mat_I1_J,axiom,
! [A2: mat_nat,Ij: product_prod_nat_nat,A: nat] :
( ( dim_row_nat @ ( update_mat_nat @ A2 @ Ij @ A ) )
= ( dim_row_nat @ A2 ) ) ).
% dim_update_mat(1)
thf(fact_749_dim__update__mat_I1_J,axiom,
! [A2: mat_int,Ij: product_prod_nat_nat,A: int] :
( ( dim_row_int @ ( update_mat_int @ A2 @ Ij @ A ) )
= ( dim_row_int @ A2 ) ) ).
% dim_update_mat(1)
thf(fact_750_dim__update__mat_I2_J,axiom,
! [A2: mat_nat,Ij: product_prod_nat_nat,A: nat] :
( ( dim_col_nat @ ( update_mat_nat @ A2 @ Ij @ A ) )
= ( dim_col_nat @ A2 ) ) ).
% dim_update_mat(2)
thf(fact_751_dim__update__mat_I2_J,axiom,
! [A2: mat_int,Ij: product_prod_nat_nat,A: int] :
( ( dim_col_int @ ( update_mat_int @ A2 @ Ij @ A ) )
= ( dim_col_int @ A2 ) ) ).
% dim_update_mat(2)
thf(fact_752_lift__01__mat__carrier,axiom,
! [M3: mat_int] : ( member_mat_nat @ ( matrix326359094246023743nt_nat @ M3 ) @ ( carrier_mat_nat @ ( dim_row_int @ M3 ) @ ( dim_col_int @ M3 ) ) ) ).
% lift_01_mat_carrier
thf(fact_753_lift__01__mat__carrier,axiom,
! [M3: mat_nat] : ( member_mat_int @ ( matrix8547886948690694719at_int @ M3 ) @ ( carrier_mat_int @ ( dim_row_nat @ M3 ) @ ( dim_col_nat @ M3 ) ) ) ).
% lift_01_mat_carrier
thf(fact_754_lift__01__mat__carrier,axiom,
! [M3: mat_nat] : ( member_mat_nat @ ( matrix8550377419199744995at_nat @ M3 ) @ ( carrier_mat_nat @ ( dim_row_nat @ M3 ) @ ( dim_col_nat @ M3 ) ) ) ).
% lift_01_mat_carrier
thf(fact_755_lift__01__mat__carrier,axiom,
! [M3: mat_int] : ( member_mat_int @ ( matrix323868623736973467nt_int @ M3 ) @ ( carrier_mat_int @ ( dim_row_int @ M3 ) @ ( dim_col_int @ M3 ) ) ) ).
% lift_01_mat_carrier
thf(fact_756_diagonal__mat__def,axiom,
( diagonal_mat_nat
= ( ^ [A3: mat_nat] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_nat @ A3 ) )
=> ( ( I3 != J3 )
=> ( ( index_mat_nat @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_757_diagonal__mat__def,axiom,
( diagonal_mat_int
= ( ^ [A3: mat_int] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_int @ A3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_int @ A3 ) )
=> ( ( I3 != J3 )
=> ( ( index_mat_int @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_int ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_758_detbc,axiom,
( ( det_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( det_int @ c ) ) ).
% detbc
thf(fact_759_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_760_add__row__to__multiple__det,axiom,
! [L: nat,Ks: list_nat,N: nat,A2: mat_int,A: int] :
( ~ ( member_nat @ L @ ( set_nat2 @ Ks ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( member_mat_int @ A2 @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( rank_A6931195264251906052le_int @ A @ Ks @ L @ A2 ) )
= ( det_int @ A2 ) ) ) ) ) ).
% add_row_to_multiple_det
thf(fact_761_add__multiple__cols__det,axiom,
! [K: nat,Ls: list_nat,N: nat,A2: mat_int,A: int] :
( ~ ( member_nat @ K @ ( set_nat2 @ Ls ) )
=> ( ! [L2: nat] :
( ( member_nat @ L2 @ ( set_nat2 @ Ls ) )
=> ( ord_less_nat @ L2 @ N ) )
=> ( ( member_mat_int @ A2 @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( rank_A5092470159319574270ls_int @ A @ K @ Ls @ A2 ) )
= ( det_int @ A2 ) ) ) ) ) ).
% add_multiple_cols_det
thf(fact_762_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_763_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_764_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_765_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_766_Ico__eq__Ico,axiom,
! [L: int,H: int,L3: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L3 @ H2 ) )
= ( ( ( L = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L3 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_767_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L3: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L3 @ H2 ) )
= ( ( ( L = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L3 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_768_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_769_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_770_det__dim__zero,axiom,
! [A2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( det_int @ A2 )
= one_one_int ) ) ).
% det_dim_zero
thf(fact_771_det__single,axiom,
! [A2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ one_one_nat @ one_one_nat ) )
=> ( ( det_int @ A2 )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% det_single
thf(fact_772_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( index_mat_int @ ( matrix8485685120660989714at_int @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ).
% all_ones_mat_index
thf(fact_773_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( index_mat_nat @ ( matrix8488175591170039990at_nat @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ).
% all_ones_mat_index
thf(fact_774_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( index_mat_int @ ( matrix8485685120660989714at_int @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ).
% all_ones_mat_index
thf(fact_775_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( index_mat_nat @ ( matrix8488175591170039990at_nat @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ).
% all_ones_mat_index
thf(fact_776_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( index_mat_int @ ( matrix8485685120660989714at_int @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ).
% all_ones_mat_index
thf(fact_777_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ ( matrix8488175591170039990at_nat @ N ) ) )
=> ( ( index_mat_nat @ ( matrix8488175591170039990at_nat @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ).
% all_ones_mat_index
thf(fact_778_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( index_mat_nat @ ( matrix8488175591170039990at_nat @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ).
% all_ones_mat_index
thf(fact_779_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) ) )
=> ( ( index_mat_int @ ( matrix8485685120660989714at_int @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ).
% all_ones_mat_index
thf(fact_780_all__ones__mat__dim__row,axiom,
! [N: nat] :
( ( dim_row_nat @ ( matrix8488175591170039990at_nat @ N ) )
= N ) ).
% all_ones_mat_dim_row
thf(fact_781_all__ones__mat__dim__row,axiom,
! [N: nat] :
( ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) )
= N ) ).
% all_ones_mat_dim_row
thf(fact_782_all__ones__mat__dim__col,axiom,
! [N: nat] :
( ( dim_col_nat @ ( matrix8488175591170039990at_nat @ N ) )
= N ) ).
% all_ones_mat_dim_col
thf(fact_783_all__ones__mat__dim__col,axiom,
! [N: nat] :
( ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) )
= N ) ).
% all_ones_mat_dim_col
thf(fact_784_det__transpose,axiom,
! [A2: mat_int,N: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( transpose_mat_int @ A2 ) )
= ( det_int @ A2 ) ) ) ).
% det_transpose
thf(fact_785_det__mult,axiom,
! [A2: mat_int,N: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ N @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ N ) )
=> ( ( det_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( times_times_int @ ( det_int @ A2 ) @ ( det_int @ B2 ) ) ) ) ) ).
% det_mult
thf(fact_786_ideal_Oscale__minus__right,axiom,
! [A: int,X3: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ X3 ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ X3 ) ) ) ).
% ideal.scale_minus_right
thf(fact_787_ideal_Oscale__minus__left,axiom,
! [A: int,X3: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ X3 )
= ( uminus_uminus_int @ ( times_times_int @ A @ X3 ) ) ) ).
% ideal.scale_minus_left
thf(fact_788_ideal_Oscale__zero__left,axiom,
! [X3: int] :
( ( times_times_int @ zero_zero_int @ X3 )
= zero_zero_int ) ).
% ideal.scale_zero_left
thf(fact_789_ideal_Oscale__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% ideal.scale_zero_right
thf(fact_790_ideal_Oscale__one,axiom,
! [X3: int] :
( ( times_times_int @ one_one_int @ X3 )
= X3 ) ).
% ideal.scale_one
thf(fact_791_ideal_Oscale__minus__both,axiom,
! [A: int,X3: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ X3 ) )
= ( times_times_int @ A @ X3 ) ) ).
% ideal.scale_minus_both
thf(fact_792_ideal_Oscale__scale,axiom,
! [A: int,B: int,X3: int] :
( ( times_times_int @ A @ ( times_times_int @ B @ X3 ) )
= ( times_times_int @ ( times_times_int @ A @ B ) @ X3 ) ) ).
% ideal.scale_scale
thf(fact_793_ideal_Oscale__left__commute,axiom,
! [A: int,B: int,X3: int] :
( ( times_times_int @ A @ ( times_times_int @ B @ X3 ) )
= ( times_times_int @ B @ ( times_times_int @ A @ X3 ) ) ) ).
% ideal.scale_left_commute
thf(fact_794_ideal_Oscale__right__distrib,axiom,
! [A: int,X3: int,Y: int] :
( ( times_times_int @ A @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ A @ X3 ) @ ( times_times_int @ A @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_795_ideal_Oscale__left__distrib,axiom,
! [A: int,B: int,X3: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ X3 )
= ( plus_plus_int @ ( times_times_int @ A @ X3 ) @ ( times_times_int @ B @ X3 ) ) ) ).
% ideal.scale_left_distrib
thf(fact_796_ideal_Oscale__left__diff__distrib,axiom,
! [A: int,B: int,X3: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ X3 )
= ( minus_minus_int @ ( times_times_int @ A @ X3 ) @ ( times_times_int @ B @ X3 ) ) ) ).
% ideal.scale_left_diff_distrib
thf(fact_797_ideal_Oscale__right__diff__distrib,axiom,
! [A: int,X3: int,Y: int] :
( ( times_times_int @ A @ ( minus_minus_int @ X3 @ Y ) )
= ( minus_minus_int @ ( times_times_int @ A @ X3 ) @ ( times_times_int @ A @ Y ) ) ) ).
% ideal.scale_right_diff_distrib
thf(fact_798_class__ring_Ominus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% class_ring.minus_zero
thf(fact_799_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_800_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_801_class__cring_Ofactors__equal,axiom,
! [A: int,B: int,C: int,D: int] :
( ( A = B )
=> ( ( C = D )
=> ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_802_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: int,B: int,C: int,D: int] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_int @ A @ C )
= ( plus_plus_int @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_803_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_804_mult__hom_Ohom__add,axiom,
! [C: nat,X3: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X3 @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X3 ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_805_mult__hom_Ohom__add,axiom,
! [C: int,X3: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X3 ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_806_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% class_cring.cring_simprules(22)
thf(fact_807_mult__hom_Ohom__add__eq__zero,axiom,
! [X3: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X3 ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_808_mult__hom_Ohom__add__eq__zero,axiom,
! [X3: int,Y: int,C: int] :
( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C @ X3 ) @ ( times_times_int @ C @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_809_class__ring_Ominus__eq,axiom,
( minus_minus_int
= ( ^ [X: int,Y3: int] : ( plus_plus_int @ X @ ( uminus_uminus_int @ Y3 ) ) ) ) ).
% class_ring.minus_eq
thf(fact_810_is__dual__def,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs )
= ( ( incide6998539924841383625em_nat @ Vs @ Bs )
& ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% is_dual_def
thf(fact_811_is__dual__def,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs )
= ( ( incide1624170830610365509stem_a @ Vs @ Bs )
& ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% is_dual_def
thf(fact_812_is__dualI,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ Vs @ Bs )
=> ( ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs ) ) ) ).
% is_dualI
thf(fact_813_is__dualI,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ Vs @ Bs )
=> ( ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs ) ) ) ).
% is_dualI
thf(fact_814_lift__mat__is__0__1,axiom,
incide4966654671090901726ix_nat @ ( matrix326359094246023743nt_nat @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ).
% lift_mat_is_0_1
thf(fact_815_lift__mat__is__0__1,axiom,
incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ).
% lift_mat_is_0_1
thf(fact_816_ordered__incidence__system__axioms,axiom,
incide1624170830610365509stem_a @ v_s @ b_s ).
% ordered_incidence_system_axioms
thf(fact_817_zero__one__matrix__axioms,axiom,
incide4964164200581851450ix_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% zero_one_matrix_axioms
thf(fact_818_is__dualD2,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs )
=> ( incide6998539924841383625em_nat @ Vs @ Bs ) ) ).
% is_dualD2
thf(fact_819_is__dualD2,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs )
=> ( incide1624170830610365509stem_a @ Vs @ Bs ) ) ).
% is_dualD2
thf(fact_820_zero__one__matrix_Olift__mat__is__0__1,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( incide4966654671090901726ix_nat @ ( matrix326359094246023743nt_nat @ Matrix ) ) ) ).
% zero_one_matrix.lift_mat_is_0_1
thf(fact_821_zero__one__matrix_Olift__mat__is__0__1,axiom,
! [Matrix: mat_nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( incide4964164200581851450ix_int @ ( matrix8547886948690694719at_int @ Matrix ) ) ) ).
% zero_one_matrix.lift_mat_is_0_1
thf(fact_822_zero__one__matrix_Olift__mat__is__0__1,axiom,
! [Matrix: mat_nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( incide4966654671090901726ix_nat @ ( matrix8550377419199744995at_nat @ Matrix ) ) ) ).
% zero_one_matrix.lift_mat_is_0_1
thf(fact_823_zero__one__matrix_Olift__mat__is__0__1,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ Matrix ) ) ) ).
% zero_one_matrix.lift_mat_is_0_1
thf(fact_824_zero__one__matrix__ring__1_Ointro,axiom,
! [M3: mat_int] :
( ( incide4964164200581851450ix_int @ M3 )
=> ( incide6080938071136783841_1_int @ M3 ) ) ).
% zero_one_matrix_ring_1.intro
thf(fact_825_zero__one__matrix__ring__1_Oaxioms,axiom,
! [M3: mat_int] :
( ( incide6080938071136783841_1_int @ M3 )
=> ( incide4964164200581851450ix_int @ M3 ) ) ).
% zero_one_matrix_ring_1.axioms
thf(fact_826_zero__one__matrix__ring__1__def,axiom,
incide6080938071136783841_1_int = incide4964164200581851450ix_int ).
% zero_one_matrix_ring_1_def
thf(fact_827_ordered__design_Oaxioms_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( incide6998539924841383625em_nat @ V_s @ B_s ) ) ).
% ordered_design.axioms(1)
thf(fact_828_ordered__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( incide1624170830610365509stem_a @ V_s @ B_s ) ) ).
% ordered_design.axioms(1)
thf(fact_829_ordered__incidence__system_Odim__row__is__v,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_row_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dim_row_is_v
thf(fact_830_ordered__incidence__system_Odim__row__is__v,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_row_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dim_row_is_v
thf(fact_831_ordered__incidence__system_Otranspose__N__mult__dim_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(1)
thf(fact_832_ordered__incidence__system_Otranspose__N__mult__dim_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(1)
thf(fact_833_ordered__incidence__system_Otranspose__N__mult__dim_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(2)
thf(fact_834_ordered__incidence__system_Otranspose__N__mult__dim_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(2)
thf(fact_835_zero__one__matrix_OM__not__one__simp,axiom,
! [Matrix: mat_nat,J: nat,I: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
!= one_one_nat )
=> ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ).
% zero_one_matrix.M_not_one_simp
thf(fact_836_zero__one__matrix_OM__not__one__simp,axiom,
! [Matrix: mat_int,J: nat,I: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
!= one_one_int )
=> ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ).
% zero_one_matrix.M_not_one_simp
thf(fact_837_zero__one__matrix_OM__not__zero__simp,axiom,
! [Matrix: mat_nat,J: nat,I: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
!= zero_zero_nat )
=> ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.M_not_zero_simp
thf(fact_838_zero__one__matrix_OM__not__zero__simp,axiom,
! [Matrix: mat_int,J: nat,I: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
!= zero_zero_int )
=> ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ) ) ).
% zero_one_matrix.M_not_zero_simp
thf(fact_839_zero__one__matrix_Olift__mat__01__index__iff_I1_J,axiom,
! [Matrix: mat_nat,I: nat,J: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Matrix ) )
=> ( ( ( index_mat_nat @ ( matrix8550377419199744995at_nat @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat )
= ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(1)
thf(fact_840_zero__one__matrix_Olift__mat__01__index__iff_I1_J,axiom,
! [Matrix: mat_nat,I: nat,J: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Matrix ) )
=> ( ( ( index_mat_int @ ( matrix8547886948690694719at_int @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int )
= ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(1)
thf(fact_841_zero__one__matrix_Olift__mat__01__index__iff_I1_J,axiom,
! [Matrix: mat_int,I: nat,J: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ Matrix ) )
=> ( ( ( index_mat_nat @ ( matrix326359094246023743nt_nat @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat )
= ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(1)
thf(fact_842_zero__one__matrix_Olift__mat__01__index__iff_I1_J,axiom,
! [Matrix: mat_int,I: nat,J: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ Matrix ) )
=> ( ( ( index_mat_int @ ( matrix323868623736973467nt_int @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int )
= ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(1)
thf(fact_843_zero__one__matrix_Olift__mat__01__index__iff_I2_J,axiom,
! [Matrix: mat_nat,I: nat,J: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Matrix ) )
=> ( ( ( index_mat_nat @ ( matrix8550377419199744995at_nat @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat )
= ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(2)
thf(fact_844_zero__one__matrix_Olift__mat__01__index__iff_I2_J,axiom,
! [Matrix: mat_nat,I: nat,J: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Matrix ) )
=> ( ( ( index_mat_int @ ( matrix8547886948690694719at_int @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int )
= ( ( index_mat_nat @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(2)
thf(fact_845_zero__one__matrix_Olift__mat__01__index__iff_I2_J,axiom,
! [Matrix: mat_int,I: nat,J: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ Matrix ) )
=> ( ( ( index_mat_nat @ ( matrix326359094246023743nt_nat @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat )
= ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(2)
thf(fact_846_zero__one__matrix_Olift__mat__01__index__iff_I2_J,axiom,
! [Matrix: mat_int,I: nat,J: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ Matrix ) )
=> ( ( ( index_mat_int @ ( matrix323868623736973467nt_int @ Matrix ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int )
= ( ( index_mat_int @ Matrix @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ) ) ).
% zero_one_matrix.lift_mat_01_index_iff(2)
thf(fact_847_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs )
= ( ( incide6998539924841383625em_nat @ Vs @ Bs )
& ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_848_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs )
= ( ( incide1624170830610365509stem_a @ Vs @ Bs )
& ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_849_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs )
= ( ( incide6998539924841383625em_nat @ Vs @ Bs )
& ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_850_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs )
= ( ( incide1624170830610365509stem_a @ Vs @ Bs )
& ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_851_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_852_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_853_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_854_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_855_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( incide6998539924841383625em_nat @ Vs @ Bs )
=> ( ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) )
=> ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_856_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( incide1624170830610365509stem_a @ Vs @ Bs )
=> ( ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) )
=> ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_857_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( incide6998539924841383625em_nat @ Vs @ Bs )
=> ( ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) )
=> ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_858_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( incide1624170830610365509stem_a @ Vs @ Bs )
=> ( ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) )
=> ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_859_ordered__incidence__system_Ois__dual_Ocong,axiom,
dual_o5859382014055506072al_a_a = dual_o5859382014055506072al_a_a ).
% ordered_incidence_system.is_dual.cong
thf(fact_860_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs )
=> ( incide6998539924841383625em_nat @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_861_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs )
=> ( incide1624170830610365509stem_a @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_862_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs )
=> ( incide6998539924841383625em_nat @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_863_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs )
=> ( incide1624170830610365509stem_a @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_864_block__size__lt__v,axiom,
ord_less_eq_nat @ k @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% block_size_lt_v
thf(fact_865_zero__one__matrix__int_Otranspose__cond__diag__r,axiom,
! [M3: mat_int,I: nat,R2: nat] :
( ( incide8301514189696901506ix_int @ M3 )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ ( times_times_mat_int @ M3 @ ( transpose_mat_int @ M3 ) ) ) )
=> ( ! [J2: nat] :
( ( I = J2 )
=> ( ( index_mat_int @ ( times_times_mat_int @ M3 @ ( transpose_mat_int @ M3 ) ) @ ( product_Pair_nat_nat @ I @ J2 ) )
= ( semiri1314217659103216013at_int @ R2 ) ) )
=> ( ( incide7000514267430604580um_int @ M3 @ I )
= R2 ) ) ) ) ).
% zero_one_matrix_int.transpose_cond_diag_r
thf(fact_866_minusinfinity,axiom,
! [D: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P2 @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_867_index__not__zero,axiom,
ord_less_eq_nat @ one_one_nat @ lambda ).
% index_not_zero
thf(fact_868_k__non__zero,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% k_non_zero
thf(fact_869_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_870_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_871_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_872_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_873_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_874_compl__le__compl__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X3 ) ) ).
% compl_le_compl_iff
thf(fact_875_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_876_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_877_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_878_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_879_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_880_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_881_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_882_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_883_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_884_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_885_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_886_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_887_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_888_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_889_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_890_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_891_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_892_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_893_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_894_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_895_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_896_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_897_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_898_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_899_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_900_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_901_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_902_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_903_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_904_atLeastLessThan__iff,axiom,
! [I: mat_nat,L: mat_nat,U: mat_nat] :
( ( member_mat_nat @ I @ ( set_or6965765401998554842at_nat @ L @ U ) )
= ( ( ord_less_eq_mat_nat @ L @ I )
& ( ord_less_mat_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_905_atLeastLessThan__iff,axiom,
! [I: mat_int,L: mat_int,U: mat_int] :
( ( member_mat_int @ I @ ( set_or2787914382489358134at_int @ L @ U ) )
= ( ( ord_less_eq_mat_int @ L @ I )
& ( ord_less_mat_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_906_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_907_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_908_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_909_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_910_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_911_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_912_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_913_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_914_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_915_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_916_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_917_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_918_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_919_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_920_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_921_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_922_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_923_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_924_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_925_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_926_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_927_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_928_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_929_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_930_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_931_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_932_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_933_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_934_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_935_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_936_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_937_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_938_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_939_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_940_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_941_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_942_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_943_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_944_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_945_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_946_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_947_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_948_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_949_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_950_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_951_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_952_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_953_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_954_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_955_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_956_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_957_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_958_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_959_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_960_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_961_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_962_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_963_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_964_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_965_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_966_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_967_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_968_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_969_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_970_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_971_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_972_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_973_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_974_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_975_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_976_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_977_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_978_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_979_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_980_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_981_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_982_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_983_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_984_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_985_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_986_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_987_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_988_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_989_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_990_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_991_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_992_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_993_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_994_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_995_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_996_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_997_compl__mono,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X3 ) ) ) ).
% compl_mono
thf(fact_998_compl__le__swap1,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X3 ) )
=> ( ord_less_eq_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_999_compl__le__swap2,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1000_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1001_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1002_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1003_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1004_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1005_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1006_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1007_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1008_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1009_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1010_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1011_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1012_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1013_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1014_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1015_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_1016_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1017_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1018_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1019_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1020_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1021_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1022_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1023_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1024_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1025_verit__comp__simplify1_I2_J,axiom,
! [A: mat_int] : ( ord_less_eq_mat_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1026_verit__comp__simplify1_I2_J,axiom,
! [A: mat_nat] : ( ord_less_eq_mat_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1027_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1028_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1029_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1030_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1031_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1032_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1033_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1034_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_1035_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_1036_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_1037_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1038_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1039_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1040_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1041_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1042_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_1043_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1044_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_1045_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_1046_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1047_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1048_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1049_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1050_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1051_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_1052_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1053_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1054_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_1055_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1056_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1057_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1058_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1059_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1060_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1061_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1062_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1063_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1064_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1065_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1066_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1067_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1068_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K4: nat] :
( N3
= ( plus_plus_nat @ M5 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1069_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1070_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1071_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1072_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1073_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1074_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1075_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1076_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1077_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1078_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1079_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_1080_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_1081_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1082_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1083_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_1084_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1085_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1086_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1087_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1088_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_1089_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_1090_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_1091_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_1092_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1093_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1094_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1095_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1096_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1097_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1098_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1099_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1100_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1101_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1102_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1103_less__eq__mat__def,axiom,
( ord_le4000661125347319327et_nat
= ( ^ [A3: mat_set_nat,B5: mat_set_nat] :
( ( ( dim_row_set_nat @ A3 )
= ( dim_row_set_nat @ B5 ) )
& ( ( dim_col_set_nat @ A3 )
= ( dim_col_set_nat @ B5 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_set_nat @ B5 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_set_nat @ B5 ) )
=> ( ord_less_eq_set_nat @ ( index_mat_set_nat @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_set_nat @ B5 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_1104_less__eq__mat__def,axiom,
( ord_le6155007575904973794at_int
= ( ^ [A3: mat_mat_int,B5: mat_mat_int] :
( ( ( dim_row_mat_int @ A3 )
= ( dim_row_mat_int @ B5 ) )
& ( ( dim_col_mat_int @ A3 )
= ( dim_col_mat_int @ B5 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_mat_int @ B5 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_mat_int @ B5 ) )
=> ( ord_less_eq_mat_int @ ( index_mat_mat_int @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_mat_int @ B5 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_1105_less__eq__mat__def,axiom,
( ord_le8645090720836700550at_nat
= ( ^ [A3: mat_mat_nat,B5: mat_mat_nat] :
( ( ( dim_row_mat_nat @ A3 )
= ( dim_row_mat_nat @ B5 ) )
& ( ( dim_col_mat_nat @ A3 )
= ( dim_col_mat_nat @ B5 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_mat_nat @ B5 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_mat_nat @ B5 ) )
=> ( ord_less_eq_mat_nat @ ( index_mat_mat_nat @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_mat_nat @ B5 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_1106_less__eq__mat__def,axiom,
( ord_less_eq_mat_int
= ( ^ [A3: mat_int,B5: mat_int] :
( ( ( dim_row_int @ A3 )
= ( dim_row_int @ B5 ) )
& ( ( dim_col_int @ A3 )
= ( dim_col_int @ B5 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_int @ B5 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_int @ B5 ) )
=> ( ord_less_eq_int @ ( index_mat_int @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_int @ B5 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_1107_less__eq__mat__def,axiom,
( ord_less_eq_mat_nat
= ( ^ [A3: mat_nat,B5: mat_nat] :
( ( ( dim_row_nat @ A3 )
= ( dim_row_nat @ B5 ) )
& ( ( dim_col_nat @ A3 )
= ( dim_col_nat @ B5 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ B5 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_nat @ B5 ) )
=> ( ord_less_eq_nat @ ( index_mat_nat @ A3 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_nat @ B5 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_1108_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1109_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1110_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1111_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1112_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1113_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1114_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1115_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1116_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1117_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1118_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1119_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1120_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1121_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1122_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1123_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1124_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1125_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1126_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1127_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1128_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1129_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1130_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1131_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1132_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1133_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1134_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1135_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1136_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1137_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1138_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1139_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1140_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1141_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1142_sum__squares__le__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1143_sum__squares__ge__zero,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1144_mult__left__le__one__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1145_mult__right__le__one__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X3 @ Y ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1146_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1147_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1148_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1149_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1150_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1151_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1152_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1153_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1154_pinf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1155_pinf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1156_pinf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1157_pinf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1158_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1159_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1160_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1161_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1162_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1163_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1164_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1165_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1166_minf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1167_minf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1168_minf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1169_minf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1170_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1171_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1172_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1173_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1174_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_1175_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_1176_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_1177_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_1178_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1179_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1180_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1181_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1182_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1183_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1184_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1185_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1186_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1187_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1188_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1189_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1190_zero__one__matrix__int__def,axiom,
incide8301514189696901506ix_int = incide6080938071136783841_1_int ).
% zero_one_matrix_int_def
thf(fact_1191_zero__one__matrix__int_Oaxioms,axiom,
! [M3: mat_int] :
( ( incide8301514189696901506ix_int @ M3 )
=> ( incide6080938071136783841_1_int @ M3 ) ) ).
% zero_one_matrix_int.axioms
thf(fact_1192_zero__one__matrix__int_Ointro,axiom,
! [M3: mat_int] :
( ( incide6080938071136783841_1_int @ M3 )
=> ( incide8301514189696901506ix_int @ M3 ) ) ).
% zero_one_matrix_int.intro
thf(fact_1193_zdiff__int__split,axiom,
! [P2: int > $o,X3: nat,Y: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X3 @ Y )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1194_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1195_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1196_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1197_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1198_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1199_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1200_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1201_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1202_convex__bound__le,axiom,
! [X3: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X3 @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1203_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1204_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1205_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1206_convex__bound__lt,axiom,
! [X3: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X3 @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1207_inf__period_I2_J,axiom,
! [P2: int > $o,D3: int,Q: int > $o] :
( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X4: int,K5: int] :
( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D3 ) ) )
| ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1208_inf__period_I1_J,axiom,
! [P2: int > $o,D3: int,Q: int > $o] :
( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X4: int,K5: int] :
( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D3 ) ) )
& ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
member_nat @ i @ ( set_or4665077453230672383an_nat @ one_one_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
thf(conj_1,conjecture,
( ( index_mat_int @ d @ ( product_Pair_nat_nat @ i @ i ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ lambda ) ) ) ).
%------------------------------------------------------------------------------