TPTP Problem File: SLH0065^1.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 : LP_Duality/0001_LP_Duality/prob_00122_004943__28757138_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1493 ( 486 unt; 216 typ; 0 def)
% Number of atoms : 3719 (1095 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 13041 ( 208 ~; 104 |; 145 &;10853 @)
% ( 0 <=>;1731 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 453 ( 453 >; 0 *; 0 +; 0 <<)
% Number of symbols : 192 ( 189 usr; 22 con; 0-6 aty)
% Number of variables : 3751 ( 166 ^;3538 !; 47 ?;3751 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:56:38.226
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Matrix__Ovec_Itf__a_J_J_J,type,
set_vec_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J_J,type,
set_vec_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Matrix__Ovec_Itf__a_J_J_J,type,
set_mat_vec_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Set__Oset_It__Matrix__Ovec_Itf__a_J_J_J,type,
vec_set_vec_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
vec_set_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Ovec_Itf__a_J_J_J,type,
set_set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
set_set_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
set_vec_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Set__Oset_It__Nat__Onat_J_J,type,
vec_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Matrix__Ovec_Itf__a_J_J,type,
vec_vec_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
vec_mat_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Ovec_Itf__a_J_J,type,
mat_vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Omat_Itf__a_J_J,type,
mat_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J,type,
poly_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (189)
thf(sy_c_Column__Operations_Oswap__col__to__front_001tf__a,type,
column2924081423933032910ront_a: mat_a > nat > mat_a ).
thf(sy_c_Column__Operations_Oswap__row__to__front_001tf__a,type,
column973622294476583016ront_a: mat_a > nat > mat_a ).
thf(sy_c_Determinant_Omat__delete_001tf__a,type,
mat_delete_a: mat_a > nat > nat > mat_a ).
thf(sy_c_Determinant_Opermutation__insert_001t__Nat__Onat,type,
permut3695043542826343943rt_nat: nat > nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Gauss__Jordan__Elimination_Oaddrow__mat_001t__Nat__Onat,type,
gauss_6496870380031412486at_nat: nat > nat > nat > nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Oaddrow__mat_001tf__a,type,
gauss_8159914756388622152_mat_a: nat > a > nat > nat > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Ofind__base__vector_001tf__a,type,
gauss_6280258074615264798ctor_a: mat_a > vec_a ).
thf(sy_c_Gauss__Jordan__Elimination_Ogauss__jordan__single_001tf__a,type,
gauss_4684855476144371464ngle_a: mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Nat__Onat,type,
gauss_8885043348566651034en_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat > mat_nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001tf__a,type,
gauss_3441994962245461172_gen_a: ( a > a > a ) > ( a > a > a ) > a > nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Orow__echelon__form_001tf__a,type,
gauss_5855338539171749649form_a: mat_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_Itf__a_J,type,
minus_minus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
minus_minus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_Itf__a_J,type,
minus_minus_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_Itf__a_J,type,
plus_plus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
plus_plus_vec_mat_a: vec_mat_a > vec_mat_a > vec_mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Matrix__Ovec_Itf__a_J_J,type,
plus_plus_vec_vec_a: vec_vec_a > vec_vec_a > vec_vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
plus_plus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
plus_p8204160854631895605_mat_a: vec_set_mat_a > vec_set_mat_a > vec_set_mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Set__Oset_It__Matrix__Ovec_Itf__a_J_J_J,type,
plus_p5241491716512693953_vec_a: vec_set_vec_a > vec_set_vec_a > vec_set_vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Set__Oset_It__Nat__Onat_J_J,type,
plus_p6405046879299855402et_nat: vec_set_nat > vec_set_nat > vec_set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_Itf__a_J,type,
plus_plus_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
plus_p2215855510709889632at_nat: set_mat_nat > set_mat_nat > set_mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
plus_plus_set_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
plus_p1963516127331757268ec_nat: set_vec_nat > set_vec_nat > set_vec_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
plus_plus_set_vec_a: set_vec_a > set_vec_a > set_vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
plus_p8188135320652551888_mat_a: set_set_mat_a > set_set_mat_a > set_set_mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Matrix__Ovec_Itf__a_J_J_J,type,
plus_p5225466182533350236_vec_a: set_set_vec_a > set_set_vec_a > set_set_vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
plus_p4817606893110106565et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
plus_p5801569732252685179et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_Itf__a_J,type,
plus_plus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
plus_plus_a: a > a > a ).
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__Matrix__Omat_Itf__a_J,type,
times_times_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
times_1230744552615602198_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Matrix__Omat_Itf__a_J_J,type,
uminus2462596480751992523_mat_a: mat_mat_a > mat_mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_Itf__a_J,type,
uminus_uminus_mat_a: mat_a > mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
uminus6789456888195538751_mat_a: vec_mat_a > vec_mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_It__Matrix__Ovec_Itf__a_J_J,type,
uminus8262787361227035083_vec_a: vec_vec_a > vec_vec_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_Itf__a_J,type,
uminus_uminus_vec_a: vec_a > vec_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
uminus1296375033039821146_mat_a: set_mat_a > set_mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
uminus2769705506071317478_vec_a: set_vec_a > set_vec_a ).
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_Ouminus__class_Ouminus_001tf__a,type,
uminus_uminus_a: a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Integral__Bounded__Vectors_OBounded__vec_001tf__a,type,
integr5272830646797813307_vec_a: a > set_vec_a ).
thf(sy_c_Integral__Bounded__Vectors_Ois__det__bound_001tf__a,type,
integr1439407231847824701ound_a: ( nat > a > a ) > $o ).
thf(sy_c_Matrix_Oappend__rows_001tf__a,type,
append_rows_a: mat_a > mat_a > mat_a ).
thf(sy_c_Matrix_Oappend__vec_001t__Matrix__Ovec_Itf__a_J,type,
append_vec_vec_a: vec_vec_a > vec_vec_a > vec_vec_a ).
thf(sy_c_Matrix_Oappend__vec_001t__Nat__Onat,type,
append_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Oappend__vec_001tf__a,type,
append_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Matrix__Ovec_Itf__a_J,type,
carrier_mat_vec_a: nat > nat > set_mat_vec_a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Matrix__Omat_Itf__a_J,type,
carrier_vec_mat_a: nat > set_vec_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Matrix__Ovec_Itf__a_J,type,
carrier_vec_vec_a: nat > set_vec_vec_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Nat__Onat,type,
carrier_vec_nat: nat > set_vec_nat ).
thf(sy_c_Matrix_Ocarrier__vec_001tf__a,type,
carrier_vec_a: nat > set_vec_a ).
thf(sy_c_Matrix_Ocomponent__mult_001t__Nat__Onat,type,
component_mult_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Ocomponent__mult_001tf__a,type,
component_mult_a: vec_a > vec_a > vec_a ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Matrix__Omat_Itf__a_J,type,
dim_vec_mat_a: vec_mat_a > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Matrix__Ovec_Itf__a_J,type,
dim_vec_vec_a: vec_vec_a > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Nat__Onat,type,
dim_vec_nat: vec_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
dim_vec_set_mat_a: vec_set_mat_a > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
dim_vec_set_vec_a: vec_set_vec_a > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Set__Oset_It__Nat__Onat_J,type,
dim_vec_set_nat: vec_set_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001tf__a,type,
dim_vec_a: vec_a > nat ).
thf(sy_c_Matrix_Ofour__block__mat_001t__Nat__Onat,type,
four_block_mat_nat: mat_nat > mat_nat > mat_nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Ofour__block__mat_001tf__a,type,
four_block_mat_a: mat_a > mat_a > mat_a > mat_a > mat_a ).
thf(sy_c_Matrix_Oinverts__mat_001tf__a,type,
inverts_mat_a: mat_a > mat_a > $o ).
thf(sy_c_Matrix_Omat__of__row_001t__Matrix__Omat_Itf__a_J,type,
mat_of_row_mat_a: vec_mat_a > mat_mat_a ).
thf(sy_c_Matrix_Omat__of__row_001t__Matrix__Ovec_Itf__a_J,type,
mat_of_row_vec_a: vec_vec_a > mat_vec_a ).
thf(sy_c_Matrix_Omat__of__row_001t__Nat__Onat,type,
mat_of_row_nat: vec_nat > mat_nat ).
thf(sy_c_Matrix_Omat__of__row_001tf__a,type,
mat_of_row_a: vec_a > mat_a ).
thf(sy_c_Matrix_Omult__mat__vec_001t__Nat__Onat,type,
mult_mat_vec_nat: mat_nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Omult__mat__vec_001tf__a,type,
mult_mat_vec_a: mat_a > vec_a > vec_a ).
thf(sy_c_Matrix_Oone__mat_001tf__a,type,
one_mat_a: nat > mat_a ).
thf(sy_c_Matrix_Orow_001tf__a,type,
row_a: mat_a > nat > vec_a ).
thf(sy_c_Matrix_Oscalar__prod_001t__Nat__Onat,type,
scalar_prod_nat: vec_nat > vec_nat > nat ).
thf(sy_c_Matrix_Oscalar__prod_001tf__a,type,
scalar_prod_a: vec_a > vec_a > a ).
thf(sy_c_Matrix_Otranspose__mat_001t__Nat__Onat,type,
transpose_mat_nat: mat_nat > mat_nat ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Oupdate__vec_001tf__a,type,
update_vec_a: vec_a > nat > a > vec_a ).
thf(sy_c_Matrix_Ovec__first_001t__Matrix__Ovec_Itf__a_J,type,
vec_first_vec_a: vec_vec_a > nat > vec_vec_a ).
thf(sy_c_Matrix_Ovec__first_001t__Nat__Onat,type,
vec_first_nat: vec_nat > nat > vec_nat ).
thf(sy_c_Matrix_Ovec__first_001tf__a,type,
vec_first_a: vec_a > nat > vec_a ).
thf(sy_c_Matrix_Ovec__index_001t__Matrix__Omat_Itf__a_J,type,
vec_index_mat_a: vec_mat_a > nat > mat_a ).
thf(sy_c_Matrix_Ovec__index_001t__Matrix__Ovec_Itf__a_J,type,
vec_index_vec_a: vec_vec_a > nat > vec_a ).
thf(sy_c_Matrix_Ovec__index_001t__Nat__Onat,type,
vec_index_nat: vec_nat > nat > nat ).
thf(sy_c_Matrix_Ovec__index_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
vec_index_set_mat_a: vec_set_mat_a > nat > set_mat_a ).
thf(sy_c_Matrix_Ovec__index_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
vec_index_set_vec_a: vec_set_vec_a > nat > set_vec_a ).
thf(sy_c_Matrix_Ovec__index_001t__Set__Oset_It__Nat__Onat_J,type,
vec_index_set_nat: vec_set_nat > nat > set_nat ).
thf(sy_c_Matrix_Ovec__index_001tf__a,type,
vec_index_a: vec_a > nat > a ).
thf(sy_c_Matrix_Ovec__last_001t__Matrix__Ovec_Itf__a_J,type,
vec_last_vec_a: vec_vec_a > nat > vec_vec_a ).
thf(sy_c_Matrix_Ovec__last_001t__Nat__Onat,type,
vec_last_nat: vec_nat > nat > vec_nat ).
thf(sy_c_Matrix_Ovec__last_001tf__a,type,
vec_last_a: vec_a > nat > vec_a ).
thf(sy_c_Matrix_Ozero__mat_001t__Nat__Onat,type,
zero_mat_nat: nat > nat > mat_nat ).
thf(sy_c_Matrix_Ozero__mat_001tf__a,type,
zero_mat_a: nat > nat > mat_a ).
thf(sy_c_Matrix_Ozero__vec_001t__Nat__Onat,type,
zero_vec_nat: nat > vec_nat ).
thf(sy_c_Matrix_Ozero__vec_001tf__a,type,
zero_vec_a: nat > vec_a ).
thf(sy_c_Matrix__Kernel_Omat__kernel_001tf__a,type,
matrix_mat_kernel_a: mat_a > set_vec_a ).
thf(sy_c_Matrix__Kernel_Ovardim_Ounpadl_001tf__a,type,
matrix_unpadl_a: nat > vec_a > vec_a ).
thf(sy_c_Matrix__Kernel_Ovardim_Ounpadr_001tf__a,type,
matrix_unpadr_a: nat > vec_a > vec_a ).
thf(sy_c_Missing__Matrix_Oappend__cols_001t__Nat__Onat,type,
missin2716118257506319449ls_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Missing__Matrix_Oappend__cols_001tf__a,type,
missin386308114684349109cols_a: mat_a > mat_a > mat_a ).
thf(sy_c_Missing__Matrix_Omat__of__col_001tf__a,type,
missing_mat_of_col_a: vec_a > mat_a ).
thf(sy_c_Missing__Matrix_Omat__row__first_001tf__a,type,
missin3040492613037353666irst_a: mat_a > nat > mat_a ).
thf(sy_c_Missing__Matrix_Omat__row__last_001tf__a,type,
missin5577565584678110354last_a: mat_a > nat > mat_a ).
thf(sy_c_Missing__Matrix_Ovec__of__scal_001tf__a,type,
missin5951511974119752530scal_a: a > vec_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Norms_Olinf__norm__vec_001tf__a,type,
linf_norm_vec_a: vec_a > a ).
thf(sy_c_Norms_Onorm1_001tf__a,type,
norm1_a: poly_a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001tf__a,type,
neg_numeral_dbl_a: a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001tf__a,type,
neg_nu6917059380386235053_inc_a: a > a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Ovec_Itf__a_J,type,
ord_less_vec_a: vec_a > vec_a > $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_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4012615358376148468_vec_a: vec_vec_a > vec_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Nat__Onat_J,type,
ord_less_eq_vec_nat: vec_nat > vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_Itf__a_J,type,
ord_less_eq_vec_a: vec_a > vec_a > $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__Matrix__Omat_Itf__a_J_J,type,
ord_le3318621148231462513_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4791951621262958845_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Schur__Decomposition_Ovec__inv_001tf__a,type,
schur_vec_inv_a: vec_a > vec_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Nat__Onat_J,type,
collect_mat_nat: ( mat_nat > $o ) > set_mat_nat ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_It__Nat__Onat_J,type,
collect_vec_nat: ( vec_nat > $o ) > set_vec_nat ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__a_J,type,
collect_vec_a: ( vec_a > $o ) > set_vec_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
collect_set_mat_a: ( set_mat_a > $o ) > set_set_mat_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
collect_set_vec_a: ( set_vec_a > $o ) > set_set_vec_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Matrix__Omat_It__Nat__Onat_J,type,
set_el538880749040126264at_nat: mat_nat > set_mat_nat > set_mat_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Matrix__Omat_Itf__a_J,type,
set_el3843030600875166508_mat_a: mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
set_el4025340090195379628ec_nat: vec_nat > set_vec_nat > set_vec_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Matrix__Ovec_Itf__a_J,type,
set_el4823081622967894968_vec_a: vec_a > set_vec_a > set_vec_a ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Nat__Onat,type,
set_elt_set_plus_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_el1154695046483745890_mat_a: set_mat_a > set_set_mat_a > set_set_mat_a ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_el2628025519515242222_vec_a: set_vec_a > set_set_vec_a > set_set_vec_a ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Nat__Onat_J,type,
set_el2470847132782436567et_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_el1246813096228348301et_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001tf__a,type,
set_elt_set_plus_a: a > set_a > set_a ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Matrix__Omat_Itf__a_J,type,
set_el1062546952344711308_mat_a: mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Nat__Onat,type,
set_el2933305810450955905es_nat: nat > set_nat > set_nat ).
thf(sy_c_member_001t__Matrix__Omat_It__Matrix__Ovec_Itf__a_J_J,type,
member_mat_vec_a: mat_vec_a > set_mat_vec_a > $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__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
member_vec_mat_a: vec_mat_a > set_vec_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Matrix__Ovec_Itf__a_J_J,type,
member_vec_vec_a: vec_vec_a > set_vec_vec_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Nat__Onat_J,type,
member_vec_nat: vec_nat > set_vec_nat > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__a_J,type,
member_vec_a: vec_a > set_vec_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
member_set_mat_a: set_mat_a > set_set_mat_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
member_set_vec_a: set_vec_a > set_set_vec_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_M____,type,
m: mat_a ).
thf(sy_v_M__last____,type,
m_last: mat_a ).
thf(sy_v_M__low____,type,
m_low: mat_a ).
thf(sy_v_M__up____,type,
m_up: mat_a ).
thf(sy_v_b,type,
b: vec_a ).
thf(sy_v_bc____,type,
bc: vec_a ).
thf(sy_v_c,type,
c: vec_a ).
thf(sy_v_nc,type,
nc: nat ).
thf(sy_v_nr,type,
nr: nat ).
thf(sy_v_t____,type,
t: vec_a ).
thf(sy_v_u1____,type,
u1: vec_a ).
thf(sy_v_u2____,type,
u2: vec_a ).
thf(sy_v_u3____,type,
u3: vec_a ).
thf(sy_v_ulv____,type,
ulv: vec_a ).
% Relevant facts (1273)
thf(fact_0_c,axiom,
member_vec_a @ c @ ( carrier_vec_a @ nc ) ).
% c
thf(fact_1_u3__def,axiom,
( u3
= ( vec_last_a @ u1 @ ( plus_plus_nat @ nc @ nc ) ) ) ).
% u3_def
thf(fact_2_add__left__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_3_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_4_add__right__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_5_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_6_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_7_set__plus__intro,axiom,
! [A: vec_a,C2: set_vec_a,B: vec_a,D: set_vec_a] :
( ( member_vec_a @ A @ C2 )
=> ( ( member_vec_a @ B @ D )
=> ( member_vec_a @ ( plus_plus_vec_a @ A @ B ) @ ( plus_plus_set_vec_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_8_set__plus__intro,axiom,
! [A: mat_a,C2: set_mat_a,B: mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( member_mat_a @ B @ D )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( plus_plus_set_mat_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_9_set__plus__intro,axiom,
! [A: set_nat,C2: set_set_nat,B: set_nat,D: set_set_nat] :
( ( member_set_nat @ A @ C2 )
=> ( ( member_set_nat @ B @ D )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_10_set__plus__intro,axiom,
! [A: set_mat_a,C2: set_set_mat_a,B: set_mat_a,D: set_set_mat_a] :
( ( member_set_mat_a @ A @ C2 )
=> ( ( member_set_mat_a @ B @ D )
=> ( member_set_mat_a @ ( plus_plus_set_mat_a @ A @ B ) @ ( plus_p8188135320652551888_mat_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_11_set__plus__intro,axiom,
! [A: set_vec_a,C2: set_set_vec_a,B: set_vec_a,D: set_set_vec_a] :
( ( member_set_vec_a @ A @ C2 )
=> ( ( member_set_vec_a @ B @ D )
=> ( member_set_vec_a @ ( plus_plus_set_vec_a @ A @ B ) @ ( plus_p5225466182533350236_vec_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_12_set__plus__intro,axiom,
! [A: a,C2: set_a,B: a,D: set_a] :
( ( member_a @ A @ C2 )
=> ( ( member_a @ B @ D )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_13_set__plus__intro,axiom,
! [A: mat_nat,C2: set_mat_nat,B: mat_nat,D: set_mat_nat] :
( ( member_mat_nat @ A @ C2 )
=> ( ( member_mat_nat @ B @ D )
=> ( member_mat_nat @ ( plus_plus_mat_nat @ A @ B ) @ ( plus_p2215855510709889632at_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_14_set__plus__intro,axiom,
! [A: vec_nat,C2: set_vec_nat,B: vec_nat,D: set_vec_nat] :
( ( member_vec_nat @ A @ C2 )
=> ( ( member_vec_nat @ B @ D )
=> ( member_vec_nat @ ( plus_plus_vec_nat @ A @ B ) @ ( plus_p1963516127331757268ec_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_15_set__plus__intro,axiom,
! [A: set_set_nat,C2: set_set_set_nat,B: set_set_nat,D: set_set_set_nat] :
( ( member_set_set_nat @ A @ C2 )
=> ( ( member_set_set_nat @ B @ D )
=> ( member_set_set_nat @ ( plus_p4817606893110106565et_nat @ A @ B ) @ ( plus_p5801569732252685179et_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_16_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ ( plus_p4817606893110106565et_nat @ A @ B ) @ C )
= ( plus_p4817606893110106565et_nat @ A @ ( plus_p4817606893110106565et_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_17_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_18_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_19_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_20_is__num__normalize_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_21_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_a @ I @ K )
= ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_22_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_23_group__cancel_Oadd1,axiom,
! [A2: set_set_nat,K: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( A2
= ( plus_p4817606893110106565et_nat @ K @ A ) )
=> ( ( plus_p4817606893110106565et_nat @ A2 @ B )
= ( plus_p4817606893110106565et_nat @ K @ ( plus_p4817606893110106565et_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_24_group__cancel_Oadd1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_25_group__cancel_Oadd1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( plus_plus_set_nat @ K @ A ) )
=> ( ( plus_plus_set_nat @ A2 @ B )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_26_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_27_group__cancel_Oadd2,axiom,
! [B2: set_set_nat,K: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( B2
= ( plus_p4817606893110106565et_nat @ K @ B ) )
=> ( ( plus_p4817606893110106565et_nat @ A @ B2 )
= ( plus_p4817606893110106565et_nat @ K @ ( plus_p4817606893110106565et_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_28_group__cancel_Oadd2,axiom,
! [B2: a,K: a,B: a,A: a] :
( ( B2
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B2 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_29_group__cancel_Oadd2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( plus_plus_set_nat @ K @ B ) )
=> ( ( plus_plus_set_nat @ A @ B2 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_30_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_31_add_Oassoc,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ ( plus_p4817606893110106565et_nat @ A @ B ) @ C )
= ( plus_p4817606893110106565et_nat @ A @ ( plus_p4817606893110106565et_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_32_add_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_33_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_34_add_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_35_add_Oleft__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_36_add_Oright__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_37_set__plus__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( plus_plus_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_38_set__plus__elim,axiom,
! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ X @ ( plus_p4817606893110106565et_nat @ A2 @ B2 ) )
=> ~ ! [A3: set_nat,B3: set_nat] :
( ( X
= ( plus_plus_set_nat @ A3 @ B3 ) )
=> ( ( member_set_nat @ A3 @ A2 )
=> ~ ( member_set_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_39_set__plus__elim,axiom,
! [X: mat_a,A2: set_mat_a,B2: set_mat_a] :
( ( member_mat_a @ X @ ( plus_plus_set_mat_a @ A2 @ B2 ) )
=> ~ ! [A3: mat_a,B3: mat_a] :
( ( X
= ( plus_plus_mat_a @ A3 @ B3 ) )
=> ( ( member_mat_a @ A3 @ A2 )
=> ~ ( member_mat_a @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_40_set__plus__elim,axiom,
! [X: vec_a,A2: set_vec_a,B2: set_vec_a] :
( ( member_vec_a @ X @ ( plus_plus_set_vec_a @ A2 @ B2 ) )
=> ~ ! [A3: vec_a,B3: vec_a] :
( ( X
= ( plus_plus_vec_a @ A3 @ B3 ) )
=> ( ( member_vec_a @ A3 @ A2 )
=> ~ ( member_vec_a @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_41_set__plus__elim,axiom,
! [X: set_mat_a,A2: set_set_mat_a,B2: set_set_mat_a] :
( ( member_set_mat_a @ X @ ( plus_p8188135320652551888_mat_a @ A2 @ B2 ) )
=> ~ ! [A3: set_mat_a,B3: set_mat_a] :
( ( X
= ( plus_plus_set_mat_a @ A3 @ B3 ) )
=> ( ( member_set_mat_a @ A3 @ A2 )
=> ~ ( member_set_mat_a @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_42_set__plus__elim,axiom,
! [X: set_vec_a,A2: set_set_vec_a,B2: set_set_vec_a] :
( ( member_set_vec_a @ X @ ( plus_p5225466182533350236_vec_a @ A2 @ B2 ) )
=> ~ ! [A3: set_vec_a,B3: set_vec_a] :
( ( X
= ( plus_plus_set_vec_a @ A3 @ B3 ) )
=> ( ( member_set_vec_a @ A3 @ A2 )
=> ~ ( member_set_vec_a @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_43_set__plus__elim,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( member_a @ X @ ( plus_plus_set_a @ A2 @ B2 ) )
=> ~ ! [A3: a,B3: a] :
( ( X
= ( plus_plus_a @ A3 @ B3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ~ ( member_a @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_44_set__plus__elim,axiom,
! [X: mat_nat,A2: set_mat_nat,B2: set_mat_nat] :
( ( member_mat_nat @ X @ ( plus_p2215855510709889632at_nat @ A2 @ B2 ) )
=> ~ ! [A3: mat_nat,B3: mat_nat] :
( ( X
= ( plus_plus_mat_nat @ A3 @ B3 ) )
=> ( ( member_mat_nat @ A3 @ A2 )
=> ~ ( member_mat_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_45_set__plus__elim,axiom,
! [X: vec_nat,A2: set_vec_nat,B2: set_vec_nat] :
( ( member_vec_nat @ X @ ( plus_p1963516127331757268ec_nat @ A2 @ B2 ) )
=> ~ ! [A3: vec_nat,B3: vec_nat] :
( ( X
= ( plus_plus_vec_nat @ A3 @ B3 ) )
=> ( ( member_vec_nat @ A3 @ A2 )
=> ~ ( member_vec_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_46_set__plus__elim,axiom,
! [X: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ X @ ( plus_p5801569732252685179et_nat @ A2 @ B2 ) )
=> ~ ! [A3: set_set_nat,B3: set_set_nat] :
( ( X
= ( plus_p4817606893110106565et_nat @ A3 @ B3 ) )
=> ( ( member_set_set_nat @ A3 @ A2 )
=> ~ ( member_set_set_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_47_add__right__imp__eq,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_48_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_49_add__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_50_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_51_add_Oleft__commute,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ B @ ( plus_p4817606893110106565et_nat @ A @ C ) )
= ( plus_p4817606893110106565et_nat @ A @ ( plus_p4817606893110106565et_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_52_add_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_53_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_54_add_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ B @ ( plus_plus_set_nat @ A @ C ) )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_55_add_Ocommute,axiom,
( plus_p4817606893110106565et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( plus_p4817606893110106565et_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_56_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A4: a,B4: a] : ( plus_plus_a @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_57_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_58_add_Ocommute,axiom,
( plus_plus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( plus_plus_set_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_59_assoc__add__vec,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat,V_3: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_3 @ ( carrier_vec_nat @ N ) )
=> ( ( plus_plus_vec_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_vec_nat @ V_1 @ ( plus_plus_vec_nat @ V_2 @ V_3 ) ) ) ) ) ) ).
% assoc_add_vec
thf(fact_60_assoc__add__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_vec_a @ V_1 @ ( plus_plus_vec_a @ V_2 @ V_3 ) ) ) ) ) ) ).
% assoc_add_vec
thf(fact_61_vec__last__carrier,axiom,
! [V: vec_nat,N: nat] : ( member_vec_nat @ ( vec_last_nat @ V @ N ) @ ( carrier_vec_nat @ N ) ) ).
% vec_last_carrier
thf(fact_62_vec__last__carrier,axiom,
! [V: vec_vec_a,N: nat] : ( member_vec_vec_a @ ( vec_last_vec_a @ V @ N ) @ ( carrier_vec_vec_a @ N ) ) ).
% vec_last_carrier
thf(fact_63_vec__last__carrier,axiom,
! [V: vec_a,N: nat] : ( member_vec_a @ ( vec_last_a @ V @ N ) @ ( carrier_vec_a @ N ) ) ).
% vec_last_carrier
thf(fact_64_u1id,axiom,
( u1
= ( append_vec_a @ u2 @ u3 ) ) ).
% u1id
thf(fact_65_u1,axiom,
member_vec_a @ u1 @ ( carrier_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) ) ).
% u1
thf(fact_66_add__carrier__vec,axiom,
! [V_1: vec_vec_a,N: nat,V_2: vec_vec_a] :
( ( member_vec_vec_a @ V_1 @ ( carrier_vec_vec_a @ N ) )
=> ( ( member_vec_vec_a @ V_2 @ ( carrier_vec_vec_a @ N ) )
=> ( member_vec_vec_a @ ( plus_plus_vec_vec_a @ V_1 @ V_2 ) @ ( carrier_vec_vec_a @ N ) ) ) ) ).
% add_carrier_vec
thf(fact_67_add__carrier__vec,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( member_vec_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ ( carrier_vec_nat @ N ) ) ) ) ).
% add_carrier_vec
thf(fact_68_add__carrier__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% add_carrier_vec
thf(fact_69_comm__add__vec,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( plus_plus_vec_nat @ V_1 @ V_2 )
= ( plus_plus_vec_nat @ V_2 @ V_1 ) ) ) ) ).
% comm_add_vec
thf(fact_70_comm__add__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V_1 @ V_2 )
= ( plus_plus_vec_a @ V_2 @ V_1 ) ) ) ) ).
% comm_add_vec
thf(fact_71_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( A = B )
=> ( ( C = D2 )
=> ( ( plus_plus_a @ A @ C )
= ( plus_plus_a @ B @ D2 ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_72_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( A = B )
=> ( ( C = D2 )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_73_set__plus__rearrange,axiom,
! [A: set_set_nat,C2: set_set_set_nat,B: set_set_nat,D: set_set_set_nat] :
( ( plus_p5801569732252685179et_nat @ ( set_el1246813096228348301et_nat @ A @ C2 ) @ ( set_el1246813096228348301et_nat @ B @ D ) )
= ( set_el1246813096228348301et_nat @ ( plus_p4817606893110106565et_nat @ A @ B ) @ ( plus_p5801569732252685179et_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_74_set__plus__rearrange,axiom,
! [A: a,C2: set_a,B: a,D: set_a] :
( ( plus_plus_set_a @ ( set_elt_set_plus_a @ A @ C2 ) @ ( set_elt_set_plus_a @ B @ D ) )
= ( set_elt_set_plus_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_75_set__plus__rearrange,axiom,
! [A: set_nat,C2: set_set_nat,B: set_nat,D: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ ( set_el2470847132782436567et_nat @ A @ C2 ) @ ( set_el2470847132782436567et_nat @ B @ D ) )
= ( set_el2470847132782436567et_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_76_set__plus__rearrange,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( plus_plus_set_nat @ ( set_elt_set_plus_nat @ A @ C2 ) @ ( set_elt_set_plus_nat @ B @ D ) )
= ( set_elt_set_plus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange
thf(fact_77_b,axiom,
member_vec_a @ b @ ( carrier_vec_a @ nr ) ).
% b
thf(fact_78_bc,axiom,
member_vec_a @ bc @ ( carrier_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) ).
% bc
thf(fact_79_dbl__def,axiom,
( neg_numeral_dbl_a
= ( ^ [X2: a] : ( plus_plus_a @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_80_sum__in__Bounded__vecI,axiom,
! [X: vec_a,B1: a,Y: vec_a,B22: a,N: nat] :
( ( member_vec_a @ X @ ( integr5272830646797813307_vec_a @ B1 ) )
=> ( ( member_vec_a @ Y @ ( integr5272830646797813307_vec_a @ B22 ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ Y @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( plus_plus_vec_a @ X @ Y ) @ ( integr5272830646797813307_vec_a @ ( plus_plus_a @ B1 @ B22 ) ) ) ) ) ) ) ).
% sum_in_Bounded_vecI
thf(fact_81_u2,axiom,
member_vec_a @ u2 @ ( carrier_vec_a @ ( plus_plus_nat @ nr @ one_one_nat ) ) ).
% u2
thf(fact_82_append__vec__eq,axiom,
! [V: vec_nat,N: nat,V2: vec_nat,W: vec_nat,W2: vec_nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V2 @ ( carrier_vec_nat @ N ) )
=> ( ( ( append_vec_nat @ V @ W )
= ( append_vec_nat @ V2 @ W2 ) )
= ( ( V = V2 )
& ( W = W2 ) ) ) ) ) ).
% append_vec_eq
thf(fact_83_append__vec__eq,axiom,
! [V: vec_vec_a,N: nat,V2: vec_vec_a,W: vec_vec_a,W2: vec_vec_a] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N ) )
=> ( ( member_vec_vec_a @ V2 @ ( carrier_vec_vec_a @ N ) )
=> ( ( ( append_vec_vec_a @ V @ W )
= ( append_vec_vec_a @ V2 @ W2 ) )
= ( ( V = V2 )
& ( W = W2 ) ) ) ) ) ).
% append_vec_eq
thf(fact_84_append__vec__eq,axiom,
! [V: vec_a,N: nat,V2: vec_a,W: vec_a,W2: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( ( append_vec_a @ V @ W )
= ( append_vec_a @ V2 @ W2 ) )
= ( ( V = V2 )
& ( W = W2 ) ) ) ) ) ).
% append_vec_eq
thf(fact_85_set__plus__intro2,axiom,
! [B: nat,C2: set_nat,A: nat] :
( ( member_nat @ B @ C2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( set_elt_set_plus_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_86_set__plus__intro2,axiom,
! [B: set_nat,C2: set_set_nat,A: set_nat] :
( ( member_set_nat @ B @ C2 )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( set_el2470847132782436567et_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_87_set__plus__intro2,axiom,
! [B: mat_a,C2: set_mat_a,A: mat_a] :
( ( member_mat_a @ B @ C2 )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( set_el3843030600875166508_mat_a @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_88_set__plus__intro2,axiom,
! [B: vec_a,C2: set_vec_a,A: vec_a] :
( ( member_vec_a @ B @ C2 )
=> ( member_vec_a @ ( plus_plus_vec_a @ A @ B ) @ ( set_el4823081622967894968_vec_a @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_89_set__plus__intro2,axiom,
! [B: set_mat_a,C2: set_set_mat_a,A: set_mat_a] :
( ( member_set_mat_a @ B @ C2 )
=> ( member_set_mat_a @ ( plus_plus_set_mat_a @ A @ B ) @ ( set_el1154695046483745890_mat_a @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_90_set__plus__intro2,axiom,
! [B: set_vec_a,C2: set_set_vec_a,A: set_vec_a] :
( ( member_set_vec_a @ B @ C2 )
=> ( member_set_vec_a @ ( plus_plus_set_vec_a @ A @ B ) @ ( set_el2628025519515242222_vec_a @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_91_set__plus__intro2,axiom,
! [B: a,C2: set_a,A: a] :
( ( member_a @ B @ C2 )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( set_elt_set_plus_a @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_92_set__plus__intro2,axiom,
! [B: mat_nat,C2: set_mat_nat,A: mat_nat] :
( ( member_mat_nat @ B @ C2 )
=> ( member_mat_nat @ ( plus_plus_mat_nat @ A @ B ) @ ( set_el538880749040126264at_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_93_set__plus__intro2,axiom,
! [B: vec_nat,C2: set_vec_nat,A: vec_nat] :
( ( member_vec_nat @ B @ C2 )
=> ( member_vec_nat @ ( plus_plus_vec_nat @ A @ B ) @ ( set_el4025340090195379628ec_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_94_set__plus__intro2,axiom,
! [B: set_set_nat,C2: set_set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B @ C2 )
=> ( member_set_set_nat @ ( plus_p4817606893110106565et_nat @ A @ B ) @ ( set_el1246813096228348301et_nat @ A @ C2 ) ) ) ).
% set_plus_intro2
thf(fact_95_append__carrier__vec,axiom,
! [V: vec_nat,N1: nat,W: vec_nat,N2: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N1 ) )
=> ( ( member_vec_nat @ W @ ( carrier_vec_nat @ N2 ) )
=> ( member_vec_nat @ ( append_vec_nat @ V @ W ) @ ( carrier_vec_nat @ ( plus_plus_nat @ N1 @ N2 ) ) ) ) ) ).
% append_carrier_vec
thf(fact_96_append__carrier__vec,axiom,
! [V: vec_vec_a,N1: nat,W: vec_vec_a,N2: nat] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N1 ) )
=> ( ( member_vec_vec_a @ W @ ( carrier_vec_vec_a @ N2 ) )
=> ( member_vec_vec_a @ ( append_vec_vec_a @ V @ W ) @ ( carrier_vec_vec_a @ ( plus_plus_nat @ N1 @ N2 ) ) ) ) ) ).
% append_carrier_vec
thf(fact_97_append__carrier__vec,axiom,
! [V: vec_a,N1: nat,W: vec_a,N2: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N1 ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N2 ) )
=> ( member_vec_a @ ( append_vec_a @ V @ W ) @ ( carrier_vec_a @ ( plus_plus_nat @ N1 @ N2 ) ) ) ) ) ).
% append_carrier_vec
thf(fact_98_t,axiom,
member_vec_a @ t @ ( carrier_vec_a @ nr ) ).
% t
thf(fact_99_one__reorient,axiom,
! [X: a] :
( ( one_one_a = X )
= ( X = one_one_a ) ) ).
% one_reorient
thf(fact_100_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_101_append__vec__add,axiom,
! [V: vec_vec_a,N: nat,V2: vec_vec_a,W: vec_vec_a,M: nat,W2: vec_vec_a] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N ) )
=> ( ( member_vec_vec_a @ V2 @ ( carrier_vec_vec_a @ N ) )
=> ( ( member_vec_vec_a @ W @ ( carrier_vec_vec_a @ M ) )
=> ( ( member_vec_vec_a @ W2 @ ( carrier_vec_vec_a @ M ) )
=> ( ( plus_plus_vec_vec_a @ ( append_vec_vec_a @ V @ W ) @ ( append_vec_vec_a @ V2 @ W2 ) )
= ( append_vec_vec_a @ ( plus_plus_vec_vec_a @ V @ V2 ) @ ( plus_plus_vec_vec_a @ W @ W2 ) ) ) ) ) ) ) ).
% append_vec_add
thf(fact_102_append__vec__add,axiom,
! [V: vec_nat,N: nat,V2: vec_nat,W: vec_nat,M: nat,W2: vec_nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V2 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ W @ ( carrier_vec_nat @ M ) )
=> ( ( member_vec_nat @ W2 @ ( carrier_vec_nat @ M ) )
=> ( ( plus_plus_vec_nat @ ( append_vec_nat @ V @ W ) @ ( append_vec_nat @ V2 @ W2 ) )
= ( append_vec_nat @ ( plus_plus_vec_nat @ V @ V2 ) @ ( plus_plus_vec_nat @ W @ W2 ) ) ) ) ) ) ) ).
% append_vec_add
thf(fact_103_append__vec__add,axiom,
! [V: vec_a,N: nat,V2: vec_a,W: vec_a,M: nat,W2: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ M ) )
=> ( ( member_vec_a @ W2 @ ( carrier_vec_a @ M ) )
=> ( ( plus_plus_vec_a @ ( append_vec_a @ V @ W ) @ ( append_vec_a @ V2 @ W2 ) )
= ( append_vec_a @ ( plus_plus_vec_a @ V @ V2 ) @ ( plus_plus_vec_a @ W @ W2 ) ) ) ) ) ) ) ).
% append_vec_add
thf(fact_104_all__vec__append,axiom,
! [N: nat,M: nat,P: vec_nat > $o] :
( ( ! [X2: vec_nat] :
( ( member_vec_nat @ X2 @ ( carrier_vec_nat @ ( plus_plus_nat @ N @ M ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: vec_nat] :
( ( member_vec_nat @ X2 @ ( carrier_vec_nat @ N ) )
=> ! [Y2: vec_nat] :
( ( member_vec_nat @ Y2 @ ( carrier_vec_nat @ M ) )
=> ( P @ ( append_vec_nat @ X2 @ Y2 ) ) ) ) ) ) ).
% all_vec_append
thf(fact_105_all__vec__append,axiom,
! [N: nat,M: nat,P: vec_vec_a > $o] :
( ( ! [X2: vec_vec_a] :
( ( member_vec_vec_a @ X2 @ ( carrier_vec_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: vec_vec_a] :
( ( member_vec_vec_a @ X2 @ ( carrier_vec_vec_a @ N ) )
=> ! [Y2: vec_vec_a] :
( ( member_vec_vec_a @ Y2 @ ( carrier_vec_vec_a @ M ) )
=> ( P @ ( append_vec_vec_a @ X2 @ Y2 ) ) ) ) ) ) ).
% all_vec_append
thf(fact_106_all__vec__append,axiom,
! [N: nat,M: nat,P: vec_a > $o] :
( ( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ ( carrier_vec_a @ N ) )
=> ! [Y2: vec_a] :
( ( member_vec_a @ Y2 @ ( carrier_vec_a @ M ) )
=> ( P @ ( append_vec_a @ X2 @ Y2 ) ) ) ) ) ) ).
% all_vec_append
thf(fact_107_set__plus__rearrange2,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_set_nat] :
( ( set_el1246813096228348301et_nat @ A @ ( set_el1246813096228348301et_nat @ B @ C2 ) )
= ( set_el1246813096228348301et_nat @ ( plus_p4817606893110106565et_nat @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_108_set__plus__rearrange2,axiom,
! [A: a,B: a,C2: set_a] :
( ( set_elt_set_plus_a @ A @ ( set_elt_set_plus_a @ B @ C2 ) )
= ( set_elt_set_plus_a @ ( plus_plus_a @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_109_set__plus__rearrange2,axiom,
! [A: nat,B: nat,C2: set_nat] :
( ( set_elt_set_plus_nat @ A @ ( set_elt_set_plus_nat @ B @ C2 ) )
= ( set_elt_set_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_110_set__plus__rearrange2,axiom,
! [A: set_nat,B: set_nat,C2: set_set_nat] :
( ( set_el2470847132782436567et_nat @ A @ ( set_el2470847132782436567et_nat @ B @ C2 ) )
= ( set_el2470847132782436567et_nat @ ( plus_plus_set_nat @ A @ B ) @ C2 ) ) ).
% set_plus_rearrange2
thf(fact_111_set__plus__rearrange4,axiom,
! [C2: set_set_nat,A: set_nat,D: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ C2 @ ( set_el2470847132782436567et_nat @ A @ D ) )
= ( set_el2470847132782436567et_nat @ A @ ( plus_p4817606893110106565et_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange4
thf(fact_112_set__plus__rearrange4,axiom,
! [C2: set_nat,A: nat,D: set_nat] :
( ( plus_plus_set_nat @ C2 @ ( set_elt_set_plus_nat @ A @ D ) )
= ( set_elt_set_plus_nat @ A @ ( plus_plus_set_nat @ C2 @ D ) ) ) ).
% set_plus_rearrange4
thf(fact_113_set__plus__rearrange3,axiom,
! [A: set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( plus_p4817606893110106565et_nat @ ( set_el2470847132782436567et_nat @ A @ B2 ) @ C2 )
= ( set_el2470847132782436567et_nat @ A @ ( plus_p4817606893110106565et_nat @ B2 @ C2 ) ) ) ).
% set_plus_rearrange3
thf(fact_114_set__plus__rearrange3,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( plus_plus_set_nat @ ( set_elt_set_plus_nat @ A @ B2 ) @ C2 )
= ( set_elt_set_plus_nat @ A @ ( plus_plus_set_nat @ B2 @ C2 ) ) ) ).
% set_plus_rearrange3
thf(fact_115_u2__def,axiom,
( u2
= ( vec_first_a @ u1 @ ( plus_plus_nat @ nr @ one_one_nat ) ) ) ).
% u2_def
thf(fact_116_ulv,axiom,
member_vec_a @ ulv @ ( carrier_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) ).
% ulv
thf(fact_117_exists__vec__append,axiom,
! [N: nat,M: nat,P: vec_nat > $o] :
( ( ? [X2: vec_nat] :
( ( member_vec_nat @ X2 @ ( carrier_vec_nat @ ( plus_plus_nat @ N @ M ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: vec_nat] :
( ( member_vec_nat @ X2 @ ( carrier_vec_nat @ N ) )
& ? [Y2: vec_nat] :
( ( member_vec_nat @ Y2 @ ( carrier_vec_nat @ M ) )
& ( P @ ( append_vec_nat @ X2 @ Y2 ) ) ) ) ) ) ).
% exists_vec_append
thf(fact_118_exists__vec__append,axiom,
! [N: nat,M: nat,P: vec_vec_a > $o] :
( ( ? [X2: vec_vec_a] :
( ( member_vec_vec_a @ X2 @ ( carrier_vec_vec_a @ ( plus_plus_nat @ N @ M ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: vec_vec_a] :
( ( member_vec_vec_a @ X2 @ ( carrier_vec_vec_a @ N ) )
& ? [Y2: vec_vec_a] :
( ( member_vec_vec_a @ Y2 @ ( carrier_vec_vec_a @ M ) )
& ( P @ ( append_vec_vec_a @ X2 @ Y2 ) ) ) ) ) ) ).
% exists_vec_append
thf(fact_119_exists__vec__append,axiom,
! [N: nat,M: nat,P: vec_a > $o] :
( ( ? [X2: vec_a] :
( ( member_vec_a @ X2 @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: vec_a] :
( ( member_vec_a @ X2 @ ( carrier_vec_a @ N ) )
& ? [Y2: vec_a] :
( ( member_vec_a @ Y2 @ ( carrier_vec_a @ M ) )
& ( P @ ( append_vec_a @ X2 @ Y2 ) ) ) ) ) ) ).
% exists_vec_append
thf(fact_120_mem__Collect__eq,axiom,
! [A: vec_nat,P: vec_nat > $o] :
( ( member_vec_nat @ A @ ( collect_vec_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
! [A: mat_nat,P: mat_nat > $o] :
( ( member_mat_nat @ A @ ( collect_mat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
! [A: set_vec_a,P: set_vec_a > $o] :
( ( member_set_vec_a @ A @ ( collect_set_vec_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
! [A: set_mat_a,P: set_mat_a > $o] :
( ( member_set_mat_a @ A @ ( collect_set_mat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A: vec_a,P: vec_a > $o] :
( ( member_vec_a @ A @ ( collect_vec_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A @ ( collect_mat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A2: set_vec_nat] :
( ( collect_vec_nat
@ ^ [X2: vec_nat] : ( member_vec_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A2: set_mat_nat] :
( ( collect_mat_nat
@ ^ [X2: mat_nat] : ( member_mat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A2: set_set_vec_a] :
( ( collect_set_vec_a
@ ^ [X2: set_vec_a] : ( member_set_vec_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A2: set_set_mat_a] :
( ( collect_set_mat_a
@ ^ [X2: set_mat_a] : ( member_set_mat_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A2: set_vec_a] :
( ( collect_vec_a
@ ^ [X2: vec_a] : ( member_vec_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A2: set_mat_a] :
( ( collect_mat_a
@ ^ [X2: mat_a] : ( member_mat_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_136_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_137_Collect__cong,axiom,
! [P: mat_a > $o,Q: mat_a > $o] :
( ! [X3: mat_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_mat_a @ P )
= ( collect_mat_a @ Q ) ) ) ).
% Collect_cong
thf(fact_138_Collect__cong,axiom,
! [P: vec_a > $o,Q: vec_a > $o] :
( ! [X3: vec_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_vec_a @ P )
= ( collect_vec_a @ Q ) ) ) ).
% Collect_cong
thf(fact_139_vec__first__last__append,axiom,
! [V: vec_nat,N: nat,M: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( append_vec_nat @ ( vec_first_nat @ V @ N ) @ ( vec_last_nat @ V @ M ) )
= V ) ) ).
% vec_first_last_append
thf(fact_140_vec__first__last__append,axiom,
! [V: vec_vec_a,N: nat,M: nat] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( append_vec_vec_a @ ( vec_first_vec_a @ V @ N ) @ ( vec_last_vec_a @ V @ M ) )
= V ) ) ).
% vec_first_last_append
thf(fact_141_vec__first__last__append,axiom,
! [V: vec_a,N: nat,M: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( append_vec_a @ ( vec_first_a @ V @ N ) @ ( vec_last_a @ V @ M ) )
= V ) ) ).
% vec_first_last_append
thf(fact_142_u1__def,axiom,
( u1
= ( vec_first_a @ ulv @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) ) ) ).
% u1_def
thf(fact_143_bc__def,axiom,
( bc
= ( append_vec_a @ ( append_vec_a @ ( append_vec_a @ b @ ( zero_vec_a @ one_one_nat ) ) @ ( append_vec_a @ c @ ( uminus_uminus_vec_a @ c ) ) ) @ ( zero_vec_a @ nr ) ) ) ).
% bc_def
thf(fact_144_A,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ nr @ nc ) ).
% A
thf(fact_145_M,axiom,
member_mat_a @ m @ ( carrier_mat_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) @ ( plus_plus_nat @ nc @ nr ) ) ).
% M
thf(fact_146_M__up,axiom,
member_mat_a @ m_up @ ( carrier_mat_a @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_up
thf(fact_147_t__def,axiom,
( t
= ( vec_last_a @ ulv @ nr ) ) ).
% t_def
thf(fact_148_ulvid,axiom,
( ulv
= ( append_vec_a @ u1 @ t ) ) ).
% ulvid
thf(fact_149_neg__equal__iff__equal,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_150_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_151_assoc__add__mat,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B2: mat_nat,C2: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ ( plus_plus_mat_nat @ A2 @ B2 ) @ C2 )
= ( plus_plus_mat_nat @ A2 @ ( plus_plus_mat_nat @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_152_assoc__add__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ C2 )
= ( plus_plus_mat_a @ A2 @ ( plus_plus_mat_a @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_153_uminus__uminus__vec,axiom,
! [V: vec_a] :
( ( uminus_uminus_vec_a @ ( uminus_uminus_vec_a @ V ) )
= V ) ).
% uminus_uminus_vec
thf(fact_154_uminus__eq__vec,axiom,
! [V: vec_a,W: vec_a] :
( ( ( uminus_uminus_vec_a @ V )
= ( uminus_uminus_vec_a @ W ) )
= ( V = W ) ) ).
% uminus_eq_vec
thf(fact_155_minus__add__distrib,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) ) ) ).
% minus_add_distrib
thf(fact_156_minus__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( plus_plus_a @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_157_add__minus__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ A @ ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_158_uminus__carrier__vec,axiom,
! [V: vec_vec_a,N: nat] :
( ( member_vec_vec_a @ ( uminus8262787361227035083_vec_a @ V ) @ ( carrier_vec_vec_a @ N ) )
= ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N ) ) ) ).
% uminus_carrier_vec
thf(fact_159_uminus__carrier__vec,axiom,
! [V: vec_mat_a,N: nat] :
( ( member_vec_mat_a @ ( uminus6789456888195538751_mat_a @ V ) @ ( carrier_vec_mat_a @ N ) )
= ( member_vec_mat_a @ V @ ( carrier_vec_mat_a @ N ) ) ) ).
% uminus_carrier_vec
thf(fact_160_uminus__carrier__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ ( uminus_uminus_vec_a @ V ) @ ( carrier_vec_a @ N ) )
= ( member_vec_a @ V @ ( carrier_vec_a @ N ) ) ) ).
% uminus_carrier_vec
thf(fact_161_uminus__zero__vec,axiom,
! [N: nat] :
( ( uminus_uminus_vec_a @ ( zero_vec_a @ N ) )
= ( zero_vec_a @ N ) ) ).
% uminus_zero_vec
thf(fact_162_minus__in__Bounded__vec,axiom,
! [X: vec_a,B: a] :
( ( member_vec_a @ ( uminus_uminus_vec_a @ X ) @ ( integr5272830646797813307_vec_a @ B ) )
= ( member_vec_a @ X @ ( integr5272830646797813307_vec_a @ B ) ) ) ).
% minus_in_Bounded_vec
thf(fact_163_left__zero__vec,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( plus_plus_vec_nat @ ( zero_vec_nat @ N ) @ V )
= V ) ) ).
% left_zero_vec
thf(fact_164_left__zero__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( zero_vec_a @ N ) @ V )
= V ) ) ).
% left_zero_vec
thf(fact_165_right__zero__vec,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( plus_plus_vec_nat @ V @ ( zero_vec_nat @ N ) )
= V ) ) ).
% right_zero_vec
thf(fact_166_right__zero__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V @ ( zero_vec_a @ N ) )
= V ) ) ).
% right_zero_vec
thf(fact_167_uminus__r__inv__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V @ ( uminus_uminus_vec_a @ V ) )
= ( zero_vec_a @ N ) ) ) ).
% uminus_r_inv_vec
thf(fact_168_uminus__l__inv__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( uminus_uminus_vec_a @ V ) @ V )
= ( zero_vec_a @ N ) ) ) ).
% uminus_l_inv_vec
thf(fact_169_ulv0,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) @ ulv ).
% ulv0
thf(fact_170_M__low,axiom,
member_mat_a @ m_low @ ( carrier_mat_a @ ( plus_plus_nat @ nc @ nc ) @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_low
thf(fact_171_M__last,axiom,
member_mat_a @ m_last @ ( carrier_mat_a @ nr @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_last
thf(fact_172_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_173_comm__add__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A2 @ B2 )
= ( plus_plus_mat_a @ B2 @ A2 ) ) ) ) ).
% comm_add_mat
thf(fact_174_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_175_add__carrier__mat,axiom,
! [B2: mat_a,Nr: nat,Nc: nat,A2: mat_a] :
( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_176_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_177_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_178_uminus__zero__vec__eq,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ( uminus_uminus_vec_a @ V )
= ( zero_vec_a @ N ) )
= ( V
= ( zero_vec_a @ N ) ) ) ) ).
% uminus_zero_vec_eq
thf(fact_179_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_nat @ ( zero_vec_nat @ N ) @ ( carrier_vec_nat @ N ) ) ).
% zero_carrier_vec
thf(fact_180_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_a @ ( zero_vec_a @ N ) @ ( carrier_vec_a @ N ) ) ).
% zero_carrier_vec
thf(fact_181_one__neq__neg__one,axiom,
( one_one_a
!= ( uminus_uminus_a @ one_one_a ) ) ).
% one_neq_neg_one
thf(fact_182_add_Oinverse__distrib__swap,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_183_group__cancel_Oneg1,axiom,
! [A2: a,K: a,A: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( uminus_uminus_a @ A2 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( uminus_uminus_a @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_184_is__num__normalize_I8_J,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_185_vec__first__carrier,axiom,
! [V: vec_nat,N: nat] : ( member_vec_nat @ ( vec_first_nat @ V @ N ) @ ( carrier_vec_nat @ N ) ) ).
% vec_first_carrier
thf(fact_186_vec__first__carrier,axiom,
! [V: vec_vec_a,N: nat] : ( member_vec_vec_a @ ( vec_first_vec_a @ V @ N ) @ ( carrier_vec_vec_a @ N ) ) ).
% vec_first_carrier
thf(fact_187_vec__first__carrier,axiom,
! [V: vec_a,N: nat] : ( member_vec_a @ ( vec_first_a @ V @ N ) @ ( carrier_vec_a @ N ) ) ).
% vec_first_carrier
thf(fact_188_vec__first__append,axiom,
! [V: vec_nat,N: nat,W: vec_nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( vec_first_nat @ ( append_vec_nat @ V @ W ) @ N )
= V ) ) ).
% vec_first_append
thf(fact_189_vec__first__append,axiom,
! [V: vec_vec_a,N: nat,W: vec_vec_a] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N ) )
=> ( ( vec_first_vec_a @ ( append_vec_vec_a @ V @ W ) @ N )
= V ) ) ).
% vec_first_append
thf(fact_190_vec__first__append,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( vec_first_a @ ( append_vec_a @ V @ W ) @ N )
= V ) ) ).
% vec_first_append
thf(fact_191_add__inv__exists__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ N ) )
& ( ( plus_plus_vec_a @ X3 @ V )
= ( zero_vec_a @ N ) )
& ( ( plus_plus_vec_a @ V @ X3 )
= ( zero_vec_a @ N ) ) ) ) ).
% add_inv_exists_vec
thf(fact_192_Mulv,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ m ) @ ulv )
= ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) ) ) ).
% Mulv
thf(fact_193_vardim_Opadr__padl__eq,axiom,
! [V: vec_nat,N: nat,M: nat,U: vec_nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( ( append_vec_nat @ V @ ( zero_vec_nat @ M ) )
= ( append_vec_nat @ ( zero_vec_nat @ N ) @ U ) )
= ( ( V
= ( zero_vec_nat @ N ) )
& ( U
= ( zero_vec_nat @ M ) ) ) ) ) ).
% vardim.padr_padl_eq
thf(fact_194_vardim_Opadr__padl__eq,axiom,
! [V: vec_a,N: nat,M: nat,U: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ( append_vec_a @ V @ ( zero_vec_a @ M ) )
= ( append_vec_a @ ( zero_vec_a @ N ) @ U ) )
= ( ( V
= ( zero_vec_a @ N ) )
& ( U
= ( zero_vec_a @ M ) ) ) ) ) ).
% vardim.padr_padl_eq
thf(fact_195_primal,axiom,
? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ X3 ) @ b ) ) ).
% primal
thf(fact_196_verit__minus__simplify_I4_J,axiom,
! [B: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_197_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_vec_a] :
( ( uminus2769705506071317478_vec_a @ ( uminus2769705506071317478_vec_a @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_198_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_mat_a] :
( ( uminus1296375033039821146_mat_a @ ( uminus1296375033039821146_mat_a @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_199_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_200_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_vec_a,Y: set_vec_a] :
( ( ( uminus2769705506071317478_vec_a @ X )
= ( uminus2769705506071317478_vec_a @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_201_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_mat_a,Y: set_mat_a] :
( ( ( uminus1296375033039821146_mat_a @ X )
= ( uminus1296375033039821146_mat_a @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_202_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( uminus5710092332889474511et_nat @ X )
= ( uminus5710092332889474511et_nat @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_203_carrier__append__cols,axiom,
! [A2: mat_nat,Nr: nat,Nc1: nat,B2: mat_nat,Nc2: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc1 ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr @ Nc2 ) )
=> ( member_mat_nat @ ( missin2716118257506319449ls_nat @ A2 @ B2 ) @ ( carrier_mat_nat @ Nr @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% carrier_append_cols
thf(fact_204_carrier__append__cols,axiom,
! [A2: mat_a,Nr: nat,Nc1: nat,B2: mat_a,Nc2: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( member_mat_a @ ( missin386308114684349109cols_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% carrier_append_cols
thf(fact_205_mat__of__row__carrier_I1_J,axiom,
! [Y: vec_nat,N: nat] :
( ( member_vec_nat @ Y @ ( carrier_vec_nat @ N ) )
=> ( member_mat_nat @ ( mat_of_row_nat @ Y ) @ ( carrier_mat_nat @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_206_mat__of__row__carrier_I1_J,axiom,
! [Y: vec_vec_a,N: nat] :
( ( member_vec_vec_a @ Y @ ( carrier_vec_vec_a @ N ) )
=> ( member_mat_vec_a @ ( mat_of_row_vec_a @ Y ) @ ( carrier_mat_vec_a @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_207_mat__of__row__carrier_I1_J,axiom,
! [Y: vec_a,N: nat] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ N ) )
=> ( member_mat_a @ ( mat_of_row_a @ Y ) @ ( carrier_mat_a @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_208_M__up__def,axiom,
( m_up
= ( four_block_mat_a @ a2 @ ( zero_mat_a @ nr @ nr ) @ ( mat_of_row_a @ ( uminus_uminus_vec_a @ c ) ) @ ( mat_of_row_a @ b ) ) ) ).
% M_up_def
thf(fact_209_dbl__inc__simps_I4_J,axiom,
( ( neg_nu6917059380386235053_inc_a @ ( uminus_uminus_a @ one_one_a ) )
= ( uminus_uminus_a @ one_one_a ) ) ).
% dbl_inc_simps(4)
thf(fact_210_uminus__carrier__iff__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ ( carrier_mat_a @ Nr @ Nc ) )
= ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_211_transpose__mat__eq,axiom,
! [A2: mat_nat,B2: mat_nat] :
( ( ( transpose_mat_nat @ A2 )
= ( transpose_mat_nat @ B2 ) )
= ( A2 = B2 ) ) ).
% transpose_mat_eq
thf(fact_212_transpose__mat__eq,axiom,
! [A2: mat_a,B2: mat_a] :
( ( ( transpose_mat_a @ A2 )
= ( transpose_mat_a @ B2 ) )
= ( A2 = B2 ) ) ).
% transpose_mat_eq
thf(fact_213_Matrix_Otranspose__transpose,axiom,
! [A2: mat_nat] :
( ( transpose_mat_nat @ ( transpose_mat_nat @ A2 ) )
= A2 ) ).
% Matrix.transpose_transpose
thf(fact_214_Matrix_Otranspose__transpose,axiom,
! [A2: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A2 ) )
= A2 ) ).
% Matrix.transpose_transpose
thf(fact_215_dual,axiom,
? [Y3: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ Y3 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ Y3 )
= c ) ) ).
% dual
thf(fact_216_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_217_add__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_218_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_219_add__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_220_compl__le__compl__iff,axiom,
! [X: set_vec_a,Y: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ X ) @ ( uminus2769705506071317478_vec_a @ Y ) )
= ( ord_le4791951621262958845_vec_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_221_compl__le__compl__iff,axiom,
! [X: set_mat_a,Y: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( uminus1296375033039821146_mat_a @ X ) @ ( uminus1296375033039821146_mat_a @ Y ) )
= ( ord_le3318621148231462513_mat_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_222_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_223_neg__le__iff__le,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_224_transpose__carrier__mat,axiom,
! [A2: mat_nat,Nc: nat,Nr: nat] :
( ( member_mat_nat @ ( transpose_mat_nat @ A2 ) @ ( carrier_mat_nat @ Nc @ Nr ) )
= ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_225_transpose__carrier__mat,axiom,
! [A2: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A2 ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_226_zero__transpose__mat,axiom,
! [N: nat,M: nat] :
( ( transpose_mat_nat @ ( zero_mat_nat @ N @ M ) )
= ( zero_mat_nat @ M @ N ) ) ).
% zero_transpose_mat
thf(fact_227_zero__transpose__mat,axiom,
! [N: nat,M: nat] :
( ( transpose_mat_a @ ( zero_mat_a @ N @ M ) )
= ( zero_mat_a @ M @ N ) ) ).
% zero_transpose_mat
thf(fact_228_M__low__def,axiom,
( m_low
= ( four_block_mat_a @ ( zero_mat_a @ nc @ nc ) @ ( transpose_mat_a @ a2 ) @ ( zero_mat_a @ nc @ nc ) @ ( uminus_uminus_mat_a @ ( transpose_mat_a @ a2 ) ) ) ) ).
% M_low_def
thf(fact_229_four__block__zero__mat,axiom,
! [Nr1: nat,Nc1: nat,Nc2: nat,Nr2: nat] :
( ( four_block_mat_a @ ( zero_mat_a @ Nr1 @ Nc1 ) @ ( zero_mat_a @ Nr1 @ Nc2 ) @ ( zero_mat_a @ Nr2 @ Nc1 ) @ ( zero_mat_a @ Nr2 @ Nc2 ) )
= ( zero_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ).
% four_block_zero_mat
thf(fact_230_uminus__l__inv__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ A2 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% uminus_l_inv_mat
thf(fact_231_right__add__zero__mat,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ A2 @ ( zero_mat_nat @ Nr @ Nc ) )
= A2 ) ) ).
% right_add_zero_mat
thf(fact_232_right__add__zero__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A2 @ ( zero_mat_a @ Nr @ Nc ) )
= A2 ) ) ).
% right_add_zero_mat
thf(fact_233_left__add__zero__mat,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( plus_plus_mat_nat @ ( zero_mat_nat @ Nr @ Nc ) @ A2 )
= A2 ) ) ).
% left_add_zero_mat
thf(fact_234_left__add__zero__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ A2 )
= A2 ) ) ).
% left_add_zero_mat
thf(fact_235_zero__mat__mult__vector,axiom,
! [X: vec_nat,Nc: nat,Nr: nat] :
( ( member_vec_nat @ X @ ( carrier_vec_nat @ Nc ) )
=> ( ( mult_mat_vec_nat @ ( zero_mat_nat @ Nr @ Nc ) @ X )
= ( zero_vec_nat @ Nr ) ) ) ).
% zero_mat_mult_vector
thf(fact_236_zero__mat__mult__vector,axiom,
! [X: vec_a,Nc: nat,Nr: nat] :
( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( zero_mat_a @ Nr @ Nc ) @ X )
= ( zero_vec_a @ Nr ) ) ) ).
% zero_mat_mult_vector
thf(fact_237_transpose__uminus,axiom,
! [A2: mat_a] :
( ( transpose_mat_a @ ( uminus_uminus_mat_a @ A2 ) )
= ( uminus_uminus_mat_a @ ( transpose_mat_a @ A2 ) ) ) ).
% transpose_uminus
thf(fact_238_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_239_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_240_verit__comp__simplify1_I2_J,axiom,
! [A: set_vec_a] : ( ord_le4791951621262958845_vec_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_241_verit__comp__simplify1_I2_J,axiom,
! [A: vec_nat] : ( ord_less_eq_vec_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_242_verit__comp__simplify1_I2_J,axiom,
! [A: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_243_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_244_verit__comp__simplify1_I2_J,axiom,
! [A: vec_vec_a] : ( ord_le4012615358376148468_vec_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_245_verit__comp__simplify1_I2_J,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_246_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_247_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_248_cong__four__block__mat,axiom,
! [A1: mat_a,B1: mat_a,A22: mat_a,B22: mat_a,A32: mat_a,B32: mat_a,A42: mat_a,B42: mat_a] :
( ( A1 = B1 )
=> ( ( A22 = B22 )
=> ( ( A32 = B32 )
=> ( ( A42 = B42 )
=> ( ( four_block_mat_a @ A1 @ A22 @ A32 @ A42 )
= ( four_block_mat_a @ B1 @ B22 @ B32 @ B42 ) ) ) ) ) ) ).
% cong_four_block_mat
thf(fact_249_transpose__four__block__mat,axiom,
! [A2: mat_nat,Nr1: nat,Nc1: nat,B2: mat_nat,Nc2: nat,C2: mat_nat,Nr2: nat,D: mat_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr1 @ Nc1 ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr1 @ Nc2 ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ Nr2 @ Nc1 ) )
=> ( ( member_mat_nat @ D @ ( carrier_mat_nat @ Nr2 @ Nc2 ) )
=> ( ( transpose_mat_nat @ ( four_block_mat_nat @ A2 @ B2 @ C2 @ D ) )
= ( four_block_mat_nat @ ( transpose_mat_nat @ A2 ) @ ( transpose_mat_nat @ C2 ) @ ( transpose_mat_nat @ B2 ) @ ( transpose_mat_nat @ D ) ) ) ) ) ) ) ).
% transpose_four_block_mat
thf(fact_250_transpose__four__block__mat,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( transpose_mat_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) )
= ( four_block_mat_a @ ( transpose_mat_a @ A2 ) @ ( transpose_mat_a @ C2 ) @ ( transpose_mat_a @ B2 ) @ ( transpose_mat_a @ D ) ) ) ) ) ) ) ).
% transpose_four_block_mat
thf(fact_251_compl__le__swap2,axiom,
! [Y: set_vec_a,X: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ Y ) @ X )
=> ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_252_compl__le__swap2,axiom,
! [Y: set_mat_a,X: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( uminus1296375033039821146_mat_a @ Y ) @ X )
=> ( ord_le3318621148231462513_mat_a @ ( uminus1296375033039821146_mat_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_253_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_254_compl__le__swap1,axiom,
! [Y: set_vec_a,X: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ Y @ ( uminus2769705506071317478_vec_a @ X ) )
=> ( ord_le4791951621262958845_vec_a @ X @ ( uminus2769705506071317478_vec_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_255_compl__le__swap1,axiom,
! [Y: set_mat_a,X: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ Y @ ( uminus1296375033039821146_mat_a @ X ) )
=> ( ord_le3318621148231462513_mat_a @ X @ ( uminus1296375033039821146_mat_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_256_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_257_compl__mono,axiom,
! [X: set_vec_a,Y: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y )
=> ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ Y ) @ ( uminus2769705506071317478_vec_a @ X ) ) ) ).
% compl_mono
thf(fact_258_compl__mono,axiom,
! [X: set_mat_a,Y: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X @ Y )
=> ( ord_le3318621148231462513_mat_a @ ( uminus1296375033039821146_mat_a @ Y ) @ ( uminus1296375033039821146_mat_a @ X ) ) ) ).
% compl_mono
thf(fact_259_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_260_Bounded__vec__mono,axiom,
! [B: a,B2: a] :
( ( ord_less_eq_a @ B @ B2 )
=> ( ord_le4791951621262958845_vec_a @ ( integr5272830646797813307_vec_a @ B ) @ ( integr5272830646797813307_vec_a @ B2 ) ) ) ).
% Bounded_vec_mono
thf(fact_261_mat__of__row__uminus,axiom,
! [V: vec_mat_a] :
( ( mat_of_row_mat_a @ ( uminus6789456888195538751_mat_a @ V ) )
= ( uminus2462596480751992523_mat_a @ ( mat_of_row_mat_a @ V ) ) ) ).
% mat_of_row_uminus
thf(fact_262_mat__of__row__uminus,axiom,
! [V: vec_a] :
( ( mat_of_row_a @ ( uminus_uminus_vec_a @ V ) )
= ( uminus_uminus_mat_a @ ( mat_of_row_a @ V ) ) ) ).
% mat_of_row_uminus
thf(fact_263_mult__mat__vec__split,axiom,
! [A2: mat_nat,N: nat,D: mat_nat,M: nat,A: vec_nat,D2: vec_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ D @ ( carrier_mat_nat @ M @ M ) )
=> ( ( member_vec_nat @ A @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ D2 @ ( carrier_vec_nat @ M ) )
=> ( ( mult_mat_vec_nat @ ( four_block_mat_nat @ A2 @ ( zero_mat_nat @ N @ M ) @ ( zero_mat_nat @ M @ N ) @ D ) @ ( append_vec_nat @ A @ D2 ) )
= ( append_vec_nat @ ( mult_mat_vec_nat @ A2 @ A ) @ ( mult_mat_vec_nat @ D @ D2 ) ) ) ) ) ) ) ).
% mult_mat_vec_split
thf(fact_264_mult__mat__vec__split,axiom,
! [A2: mat_a,N: nat,D: mat_a,M: nat,A: vec_a,D2: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ D2 @ ( carrier_vec_a @ M ) )
=> ( ( mult_mat_vec_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ D ) @ ( append_vec_a @ A @ D2 ) )
= ( append_vec_a @ ( mult_mat_vec_a @ A2 @ A ) @ ( mult_mat_vec_a @ D @ D2 ) ) ) ) ) ) ) ).
% mult_mat_vec_split
thf(fact_265_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_nat @ ( zero_mat_nat @ Nr @ Nc ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_266_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ ( carrier_mat_a @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_267_add__inv__exists__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ? [X3: mat_a] :
( ( member_mat_a @ X3 @ ( carrier_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ X3 @ A2 )
= ( zero_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ A2 @ X3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ) ).
% add_inv_exists_mat
thf(fact_268_add__four__block__mat,axiom,
! [A1: mat_nat,Nr1: nat,Nc1: nat,B1: mat_nat,Nc2: nat,C1: mat_nat,Nr2: nat,D1: mat_nat,A22: mat_nat,B22: mat_nat,C22: mat_nat,D22: mat_nat] :
( ( member_mat_nat @ A1 @ ( carrier_mat_nat @ Nr1 @ Nc1 ) )
=> ( ( member_mat_nat @ B1 @ ( carrier_mat_nat @ Nr1 @ Nc2 ) )
=> ( ( member_mat_nat @ C1 @ ( carrier_mat_nat @ Nr2 @ Nc1 ) )
=> ( ( member_mat_nat @ D1 @ ( carrier_mat_nat @ Nr2 @ Nc2 ) )
=> ( ( member_mat_nat @ A22 @ ( carrier_mat_nat @ Nr1 @ Nc1 ) )
=> ( ( member_mat_nat @ B22 @ ( carrier_mat_nat @ Nr1 @ Nc2 ) )
=> ( ( member_mat_nat @ C22 @ ( carrier_mat_nat @ Nr2 @ Nc1 ) )
=> ( ( member_mat_nat @ D22 @ ( carrier_mat_nat @ Nr2 @ Nc2 ) )
=> ( ( plus_plus_mat_nat @ ( four_block_mat_nat @ A1 @ B1 @ C1 @ D1 ) @ ( four_block_mat_nat @ A22 @ B22 @ C22 @ D22 ) )
= ( four_block_mat_nat @ ( plus_plus_mat_nat @ A1 @ A22 ) @ ( plus_plus_mat_nat @ B1 @ B22 ) @ ( plus_plus_mat_nat @ C1 @ C22 ) @ ( plus_plus_mat_nat @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ).
% add_four_block_mat
thf(fact_269_add__four__block__mat,axiom,
! [A1: mat_a,Nr1: nat,Nc1: nat,B1: mat_a,Nc2: nat,C1: mat_a,Nr2: nat,D1: mat_a,A22: mat_a,B22: mat_a,C22: mat_a,D22: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B1 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C1 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D1 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B22 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C22 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D22 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( plus_plus_mat_a @ ( four_block_mat_a @ A1 @ B1 @ C1 @ D1 ) @ ( four_block_mat_a @ A22 @ B22 @ C22 @ D22 ) )
= ( four_block_mat_a @ ( plus_plus_mat_a @ A1 @ A22 ) @ ( plus_plus_mat_a @ B1 @ B22 ) @ ( plus_plus_mat_a @ C1 @ C22 ) @ ( plus_plus_mat_a @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ).
% add_four_block_mat
thf(fact_270_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_271_transpose__add,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) )
= ( plus_plus_mat_a @ ( transpose_mat_a @ A2 ) @ ( transpose_mat_a @ B2 ) ) ) ) ) ).
% transpose_add
thf(fact_272_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_273_le__numeral__extra_I4_J,axiom,
ord_less_eq_a @ one_one_a @ one_one_a ).
% le_numeral_extra(4)
thf(fact_274_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_275_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_276_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_277_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_278_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_279_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_280_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_281_add__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_282_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_283_add__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_284_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_285_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_286_add__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_287_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_288_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_289_add__le__imp__le__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_290_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_291_add__le__imp__le__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_292_le__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_eq_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% le_minus_iff
thf(fact_293_minus__le__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_294_le__imp__neg__le,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% le_imp_neg_le
thf(fact_295_uminus__carrier__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_296_uminus__add__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) )
= ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ B2 ) @ ( uminus_uminus_mat_a @ A2 ) ) ) ) ) ).
% uminus_add_mat
thf(fact_297_four__block__mat__mult__vec,axiom,
! [A2: mat_nat,Nr1: nat,Nc1: nat,B2: mat_nat,Nc2: nat,C2: mat_nat,Nr2: nat,D: mat_nat,A: vec_nat,D2: vec_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr1 @ Nc1 ) )
=> ( ( member_mat_nat @ B2 @ ( carrier_mat_nat @ Nr1 @ Nc2 ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ Nr2 @ Nc1 ) )
=> ( ( member_mat_nat @ D @ ( carrier_mat_nat @ Nr2 @ Nc2 ) )
=> ( ( member_vec_nat @ A @ ( carrier_vec_nat @ Nc1 ) )
=> ( ( member_vec_nat @ D2 @ ( carrier_vec_nat @ Nc2 ) )
=> ( ( mult_mat_vec_nat @ ( four_block_mat_nat @ A2 @ B2 @ C2 @ D ) @ ( append_vec_nat @ A @ D2 ) )
= ( append_vec_nat @ ( plus_plus_vec_nat @ ( mult_mat_vec_nat @ A2 @ A ) @ ( mult_mat_vec_nat @ B2 @ D2 ) ) @ ( plus_plus_vec_nat @ ( mult_mat_vec_nat @ C2 @ A ) @ ( mult_mat_vec_nat @ D @ D2 ) ) ) ) ) ) ) ) ) ) ).
% four_block_mat_mult_vec
thf(fact_298_four__block__mat__mult__vec,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D: mat_a,A: vec_a,D2: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ Nc1 ) )
=> ( ( member_vec_a @ D2 @ ( carrier_vec_a @ Nc2 ) )
=> ( ( mult_mat_vec_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) @ ( append_vec_a @ A @ D2 ) )
= ( append_vec_a @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ A ) @ ( mult_mat_vec_a @ B2 @ D2 ) ) @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ C2 @ A ) @ ( mult_mat_vec_a @ D @ D2 ) ) ) ) ) ) ) ) ) ) ).
% four_block_mat_mult_vec
thf(fact_299_mult__mat__vec__carrier,axiom,
! [A2: mat_nat,Nr: nat,N: nat,V: vec_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( member_vec_nat @ ( mult_mat_vec_nat @ A2 @ V ) @ ( carrier_vec_nat @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_300_mult__mat__vec__carrier,axiom,
! [A2: mat_a,Nr: nat,N: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_301_four__block__carrier__mat,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,D: mat_a,Nr2: nat,Nc2: nat,B2: mat_a,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( member_mat_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% four_block_carrier_mat
thf(fact_302_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% le_minus_one_simps(4)
thf(fact_303_le__minus__one__simps_I2_J,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a ).
% le_minus_one_simps(2)
thf(fact_304_append__vec__le,axiom,
! [V: vec_a,N: nat,W: vec_a,V2: vec_a,W2: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_eq_vec_a @ ( append_vec_a @ V @ V2 ) @ ( append_vec_a @ W @ W2 ) )
= ( ( ord_less_eq_vec_a @ V @ W )
& ( ord_less_eq_vec_a @ V2 @ W2 ) ) ) ) ) ).
% append_vec_le
thf(fact_305_mat__mult__append__cols,axiom,
! [A2: mat_a,Nr: nat,Nc1: nat,B2: mat_a,Nc2: nat,V1: vec_a,V22: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( ( member_vec_a @ V1 @ ( carrier_vec_a @ Nc1 ) )
=> ( ( member_vec_a @ V22 @ ( carrier_vec_a @ Nc2 ) )
=> ( ( mult_mat_vec_a @ ( missin386308114684349109cols_a @ A2 @ B2 ) @ ( append_vec_a @ V1 @ V22 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ V1 ) @ ( mult_mat_vec_a @ B2 @ V22 ) ) ) ) ) ) ) ).
% mat_mult_append_cols
thf(fact_306_add__mult__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ V )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% add_mult_distrib_mat_vec
thf(fact_307_mult__add__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V_1: vec_a,V_2: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V_1 @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ A2 @ ( plus_plus_vec_a @ V_1 @ V_2 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ V_1 ) @ ( mult_mat_vec_a @ A2 @ V_2 ) ) ) ) ) ) ).
% mult_add_distrib_mat_vec
thf(fact_308_M__def,axiom,
( m
= ( append_rows_a @ ( append_rows_a @ m_up @ m_low ) @ m_last ) ) ).
% M_def
thf(fact_309_M__last__def,axiom,
( m_last
= ( missin386308114684349109cols_a @ ( zero_mat_a @ nr @ nc ) @ ( uminus_uminus_mat_a @ ( one_mat_a @ nr ) ) ) ) ).
% M_last_def
thf(fact_310_append__rows__le,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat,A: vec_a,V: vec_a,B: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ Nr1 ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ ( append_rows_a @ A2 @ B2 ) @ V ) @ ( append_vec_a @ A @ B ) )
= ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ A )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ B2 @ V ) @ B ) ) ) ) ) ) ) ).
% append_rows_le
thf(fact_311_dual__order_Orefl,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% dual_order.refl
thf(fact_312_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_313_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_314_order__refl,axiom,
! [X: vec_a] : ( ord_less_eq_vec_a @ X @ X ) ).
% order_refl
thf(fact_315_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_316_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_317_mat__mult__append,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( append_rows_a @ A2 @ B2 ) @ V )
= ( append_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% mat_mult_append
thf(fact_318_weak__duality__theorem,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B: vec_nat,C: vec_nat,X: vec_nat,Y: vec_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_vec_nat @ B @ ( carrier_vec_nat @ Nr ) )
=> ( ( member_vec_nat @ C @ ( carrier_vec_nat @ Nc ) )
=> ( ( member_vec_nat @ X @ ( carrier_vec_nat @ Nc ) )
=> ( ( ord_less_eq_vec_nat @ ( mult_mat_vec_nat @ A2 @ X ) @ B )
=> ( ( ord_less_eq_vec_nat @ ( zero_vec_nat @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_nat @ ( transpose_mat_nat @ A2 ) @ Y )
= C )
=> ( ord_less_eq_nat @ ( scalar_prod_nat @ C @ X ) @ ( scalar_prod_nat @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_319_weak__duality__theorem,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a,X: vec_a,Y: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X ) @ B )
=> ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y )
= C )
=> ( ord_less_eq_a @ ( scalar_prod_a @ C @ X ) @ ( scalar_prod_a @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_320_unbounded__dual__solutions,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ! [V3: a] :
? [Y4: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y4 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y4 )
= C )
& ( ord_less_eq_a @ ( scalar_prod_a @ B @ Y4 ) @ V3 ) )
=> ~ ? [X4: vec_a] :
( ( member_vec_a @ X4 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X4 ) @ B ) ) ) ) ) ) ).
% unbounded_dual_solutions
thf(fact_321_unbounded__primal__solutions,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ! [V3: a] :
? [X4: vec_a] :
( ( member_vec_a @ X4 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X4 ) @ B )
& ( ord_less_eq_a @ V3 @ ( scalar_prod_a @ C @ X4 ) ) )
=> ~ ? [Y4: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y4 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y4 )
= C ) ) ) ) ) ) ).
% unbounded_primal_solutions
thf(fact_322_vardim_Ounpadr__padr,axiom,
! [M: nat,V: vec_a] :
( ( matrix_unpadr_a @ M @ ( append_vec_a @ V @ ( zero_vec_a @ M ) ) )
= V ) ).
% vardim.unpadr_padr
thf(fact_323_uminus__uminus__mat,axiom,
! [A2: mat_a] :
( ( uminus_uminus_mat_a @ ( uminus_uminus_mat_a @ A2 ) )
= A2 ) ).
% uminus_uminus_mat
thf(fact_324_uminus__eq__mat,axiom,
! [A2: mat_a,B2: mat_a] :
( ( ( uminus_uminus_mat_a @ A2 )
= ( uminus_uminus_mat_a @ B2 ) )
= ( A2 = B2 ) ) ).
% uminus_eq_mat
thf(fact_325_set__plus__mono2,axiom,
! [C2: set_nat,D: set_nat,E: set_nat,F: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ( ord_less_eq_set_nat @ E @ F )
=> ( ord_less_eq_set_nat @ ( plus_plus_set_nat @ C2 @ E ) @ ( plus_plus_set_nat @ D @ F ) ) ) ) ).
% set_plus_mono2
thf(fact_326_set__plus__mono2,axiom,
! [C2: set_mat_a,D: set_mat_a,E: set_mat_a,F: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ D )
=> ( ( ord_le3318621148231462513_mat_a @ E @ F )
=> ( ord_le3318621148231462513_mat_a @ ( plus_plus_set_mat_a @ C2 @ E ) @ ( plus_plus_set_mat_a @ D @ F ) ) ) ) ).
% set_plus_mono2
thf(fact_327_set__plus__mono2,axiom,
! [C2: set_vec_a,D: set_vec_a,E: set_vec_a,F: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ C2 @ D )
=> ( ( ord_le4791951621262958845_vec_a @ E @ F )
=> ( ord_le4791951621262958845_vec_a @ ( plus_plus_set_vec_a @ C2 @ E ) @ ( plus_plus_set_vec_a @ D @ F ) ) ) ) ).
% set_plus_mono2
thf(fact_328_transpose__one,axiom,
! [N: nat] :
( ( transpose_mat_a @ ( one_mat_a @ N ) )
= ( one_mat_a @ N ) ) ).
% transpose_one
thf(fact_329_vec__first__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( vec_first_a @ ( zero_vec_a @ N ) @ M )
= ( zero_vec_a @ M ) ) ) ).
% vec_first_zero
thf(fact_330_one__mult__mat__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( mult_mat_vec_a @ ( one_mat_a @ N ) @ V )
= V ) ) ).
% one_mult_mat_vec
thf(fact_331_carrier__append__rows,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( member_mat_a @ ( append_rows_a @ A2 @ B2 ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ Nc ) ) ) ) ).
% carrier_append_rows
thf(fact_332_set__plus__mono3,axiom,
! [A: nat,C2: set_nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ ( set_elt_set_plus_nat @ A @ D ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ).
% set_plus_mono3
thf(fact_333_set__plus__mono3,axiom,
! [A: mat_a,C2: set_mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ord_le3318621148231462513_mat_a @ ( set_el3843030600875166508_mat_a @ A @ D ) @ ( plus_plus_set_mat_a @ C2 @ D ) ) ) ).
% set_plus_mono3
thf(fact_334_set__plus__mono3,axiom,
! [A: vec_a,C2: set_vec_a,D: set_vec_a] :
( ( member_vec_a @ A @ C2 )
=> ( ord_le4791951621262958845_vec_a @ ( set_el4823081622967894968_vec_a @ A @ D ) @ ( plus_plus_set_vec_a @ C2 @ D ) ) ) ).
% set_plus_mono3
thf(fact_335_set__plus__mono4,axiom,
! [A: nat,C2: set_nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ ( set_elt_set_plus_nat @ A @ D ) @ ( plus_plus_set_nat @ D @ C2 ) ) ) ).
% set_plus_mono4
thf(fact_336_four__block__one__mat,axiom,
! [N1: nat,N2: nat] :
( ( four_block_mat_a @ ( one_mat_a @ N1 ) @ ( zero_mat_a @ N1 @ N2 ) @ ( zero_mat_a @ N2 @ N1 ) @ ( one_mat_a @ N2 ) )
= ( one_mat_a @ ( plus_plus_nat @ N1 @ N2 ) ) ) ).
% four_block_one_mat
thf(fact_337_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_a @ ( one_mat_a @ N ) @ ( carrier_mat_a @ N @ N ) ) ).
% one_carrier_mat
thf(fact_338_comm__scalar__prod,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ V_2 )
= ( scalar_prod_a @ V_2 @ V_1 ) ) ) ) ).
% comm_scalar_prod
thf(fact_339_set__plus__mono__b,axiom,
! [C2: set_vec_a,D: set_vec_a,X: vec_a,A: vec_a] :
( ( ord_le4791951621262958845_vec_a @ C2 @ D )
=> ( ( member_vec_a @ X @ ( set_el4823081622967894968_vec_a @ A @ C2 ) )
=> ( member_vec_a @ X @ ( set_el4823081622967894968_vec_a @ A @ D ) ) ) ) ).
% set_plus_mono_b
thf(fact_340_set__plus__mono__b,axiom,
! [C2: set_mat_a,D: set_mat_a,X: mat_a,A: mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ D )
=> ( ( member_mat_a @ X @ ( set_el3843030600875166508_mat_a @ A @ C2 ) )
=> ( member_mat_a @ X @ ( set_el3843030600875166508_mat_a @ A @ D ) ) ) ) ).
% set_plus_mono_b
thf(fact_341_set__plus__mono__b,axiom,
! [C2: set_nat,D: set_nat,X: nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ( member_nat @ X @ ( set_elt_set_plus_nat @ A @ C2 ) )
=> ( member_nat @ X @ ( set_elt_set_plus_nat @ A @ D ) ) ) ) ).
% set_plus_mono_b
thf(fact_342_set__plus__mono5,axiom,
! [A: nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( set_elt_set_plus_nat @ A @ B2 ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_mono5
thf(fact_343_set__plus__mono5,axiom,
! [A: mat_a,C2: set_mat_a,B2: set_mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( ord_le3318621148231462513_mat_a @ B2 @ D )
=> ( ord_le3318621148231462513_mat_a @ ( set_el3843030600875166508_mat_a @ A @ B2 ) @ ( plus_plus_set_mat_a @ C2 @ D ) ) ) ) ).
% set_plus_mono5
thf(fact_344_set__plus__mono5,axiom,
! [A: vec_a,C2: set_vec_a,B2: set_vec_a,D: set_vec_a] :
( ( member_vec_a @ A @ C2 )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ D )
=> ( ord_le4791951621262958845_vec_a @ ( set_el4823081622967894968_vec_a @ A @ B2 ) @ ( plus_plus_set_vec_a @ C2 @ D ) ) ) ) ).
% set_plus_mono5
thf(fact_345_scalar__prod__add__distrib,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat,V_3: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_3 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ V_1 @ ( plus_plus_vec_nat @ V_2 @ V_3 ) )
= ( plus_plus_nat @ ( scalar_prod_nat @ V_1 @ V_2 ) @ ( scalar_prod_nat @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_add_distrib
thf(fact_346_scalar__prod__add__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ ( plus_plus_vec_a @ V_2 @ V_3 ) )
= ( plus_plus_a @ ( scalar_prod_a @ V_1 @ V_2 ) @ ( scalar_prod_a @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_add_distrib
thf(fact_347_add__scalar__prod__distrib,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat,V_3: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_3 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_nat @ ( scalar_prod_nat @ V_1 @ V_3 ) @ ( scalar_prod_nat @ V_2 @ V_3 ) ) ) ) ) ) ).
% add_scalar_prod_distrib
thf(fact_348_add__scalar__prod__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_a @ ( scalar_prod_a @ V_1 @ V_3 ) @ ( scalar_prod_a @ V_2 @ V_3 ) ) ) ) ) ) ).
% add_scalar_prod_distrib
thf(fact_349_scalar__prod__append,axiom,
! [V1: vec_a,N1: nat,V22: vec_a,N2: nat,W1: vec_a,W22: vec_a] :
( ( member_vec_a @ V1 @ ( carrier_vec_a @ N1 ) )
=> ( ( member_vec_a @ V22 @ ( carrier_vec_a @ N2 ) )
=> ( ( member_vec_a @ W1 @ ( carrier_vec_a @ N1 ) )
=> ( ( member_vec_a @ W22 @ ( carrier_vec_a @ N2 ) )
=> ( ( scalar_prod_a @ ( append_vec_a @ V1 @ V22 ) @ ( append_vec_a @ W1 @ W22 ) )
= ( plus_plus_a @ ( scalar_prod_a @ V1 @ W1 ) @ ( scalar_prod_a @ V22 @ W22 ) ) ) ) ) ) ) ).
% scalar_prod_append
thf(fact_350_scalar__prod__append,axiom,
! [V1: vec_nat,N1: nat,V22: vec_nat,N2: nat,W1: vec_nat,W22: vec_nat] :
( ( member_vec_nat @ V1 @ ( carrier_vec_nat @ N1 ) )
=> ( ( member_vec_nat @ V22 @ ( carrier_vec_nat @ N2 ) )
=> ( ( member_vec_nat @ W1 @ ( carrier_vec_nat @ N1 ) )
=> ( ( member_vec_nat @ W22 @ ( carrier_vec_nat @ N2 ) )
=> ( ( scalar_prod_nat @ ( append_vec_nat @ V1 @ V22 ) @ ( append_vec_nat @ W1 @ W22 ) )
= ( plus_plus_nat @ ( scalar_prod_nat @ V1 @ W1 ) @ ( scalar_prod_nat @ V22 @ W22 ) ) ) ) ) ) ) ).
% scalar_prod_append
thf(fact_351_uminus__scalar__prod,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( uminus_uminus_a @ ( scalar_prod_a @ V @ W ) )
= ( scalar_prod_a @ ( uminus_uminus_vec_a @ V ) @ W ) ) ) ) ).
% uminus_scalar_prod
thf(fact_352_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_353_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_354_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_355_le__cases3,axiom,
! [X: a,Y: a,Z: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z ) )
=> ( ( ( ord_less_eq_a @ X @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y ) )
=> ( ( ( ord_less_eq_a @ Z @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z )
=> ~ ( ord_less_eq_a @ Z @ X ) )
=> ~ ( ( ord_less_eq_a @ Z @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_356_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_a,Z2: vec_a] : ( Y5 = Z2 ) )
= ( ^ [X2: vec_a,Y2: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
& ( ord_less_eq_vec_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_357_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_358_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( ord_less_eq_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_359_ord__eq__le__trans,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( A = B )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ord_less_eq_vec_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_360_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_361_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_362_ord__le__eq__trans,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_vec_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_363_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_364_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_365_order__antisym,axiom,
! [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_366_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_367_order__antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_368_order_Otrans,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ord_less_eq_vec_a @ A @ C ) ) ) ).
% order.trans
thf(fact_369_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_370_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_371_order__trans,axiom,
! [X: vec_a,Y: vec_a,Z: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ Y @ Z )
=> ( ord_less_eq_vec_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_372_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_373_order__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_eq_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_374_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_375_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_376_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: vec_a,Z2: vec_a] : ( Y5 = Z2 ) )
= ( ^ [A4: vec_a,B4: vec_a] :
( ( ord_less_eq_vec_a @ B4 @ A4 )
& ( ord_less_eq_vec_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_377_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_378_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_379_dual__order_Oantisym,axiom,
! [B: vec_a,A: vec_a] :
( ( ord_less_eq_vec_a @ B @ A )
=> ( ( ord_less_eq_vec_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_380_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_381_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_382_dual__order_Otrans,axiom,
! [B: vec_a,A: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ B @ A )
=> ( ( ord_less_eq_vec_a @ C @ B )
=> ( ord_less_eq_vec_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_383_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_384_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_385_antisym,axiom,
! [A: vec_a,B: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_386_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_387_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_388_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_a,Z2: vec_a] : ( Y5 = Z2 ) )
= ( ^ [A4: vec_a,B4: vec_a] :
( ( ord_less_eq_vec_a @ A4 @ B4 )
& ( ord_less_eq_vec_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_389_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_390_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_391_order__subst1,axiom,
! [A: vec_a,F2: vec_a > vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_392_order__subst1,axiom,
! [A: vec_a,F2: nat > vec_a,B: nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_393_order__subst1,axiom,
! [A: vec_a,F2: a > vec_a,B: a,C: a] :
( ( ord_less_eq_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_394_order__subst1,axiom,
! [A: nat,F2: vec_a > nat,B: vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_395_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_396_order__subst1,axiom,
! [A: nat,F2: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_397_order__subst1,axiom,
! [A: a,F2: vec_a > a,B: vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_398_order__subst1,axiom,
! [A: a,F2: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_399_order__subst1,axiom,
! [A: a,F2: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_400_order__subst2,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_401_order__subst2,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_402_order__subst2,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > a,C: a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F2 @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_403_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_404_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_405_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_406_order__subst2,axiom,
! [A: a,B: a,F2: a > vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_407_order__subst2,axiom,
! [A: a,B: a,F2: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_408_order__subst2,axiom,
! [A: a,B: a,F2: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F2 @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_409_order__eq__refl,axiom,
! [X: vec_a,Y: vec_a] :
( ( X = Y )
=> ( ord_less_eq_vec_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_410_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_411_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_412_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_413_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_414_ord__eq__le__subst,axiom,
! [A: vec_a,F2: vec_a > vec_a,B: vec_a,C: vec_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_415_ord__eq__le__subst,axiom,
! [A: nat,F2: vec_a > nat,B: vec_a,C: vec_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_416_ord__eq__le__subst,axiom,
! [A: a,F2: vec_a > a,B: vec_a,C: vec_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_417_ord__eq__le__subst,axiom,
! [A: vec_a,F2: nat > vec_a,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_418_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_419_ord__eq__le__subst,axiom,
! [A: a,F2: nat > a,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_420_ord__eq__le__subst,axiom,
! [A: vec_a,F2: a > vec_a,B: a,C: a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_421_ord__eq__le__subst,axiom,
! [A: nat,F2: a > nat,B: a,C: a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_422_ord__eq__le__subst,axiom,
! [A: a,F2: a > a,B: a,C: a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_423_ord__le__eq__subst,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_424_ord__le__eq__subst,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_425_ord__le__eq__subst,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > a,C: a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_426_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_427_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_428_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_429_ord__le__eq__subst,axiom,
! [A: a,B: a,F2: a > vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_430_ord__le__eq__subst,axiom,
! [A: a,B: a,F2: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_431_ord__le__eq__subst,axiom,
! [A: a,B: a,F2: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_432_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_433_linorder__le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_434_order__antisym__conv,axiom,
! [Y: vec_a,X: vec_a] :
( ( ord_less_eq_vec_a @ Y @ X )
=> ( ( ord_less_eq_vec_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_435_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_436_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_437_transpose__vec__mult__scalar,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,X: vec_a,Y: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ Y @ ( carrier_vec_a @ Nr ) )
=> ( ( scalar_prod_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y ) @ X )
= ( scalar_prod_a @ Y @ ( mult_mat_vec_a @ A2 @ X ) ) ) ) ) ) ).
% transpose_vec_mult_scalar
thf(fact_438_append__cols__def,axiom,
( missin386308114684349109cols_a
= ( ^ [A5: mat_a,B5: mat_a] : ( transpose_mat_a @ ( append_rows_a @ ( transpose_mat_a @ A5 ) @ ( transpose_mat_a @ B5 ) ) ) ) ) ).
% append_cols_def
thf(fact_439_sum__carrier__vec,axiom,
! [A2: set_vec_a,N: nat,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ ( carrier_vec_a @ N ) )
=> ( ord_le4791951621262958845_vec_a @ ( plus_plus_set_vec_a @ A2 @ B2 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% sum_carrier_vec
thf(fact_440_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_441_subsetI,axiom,
! [A2: set_vec_a,B2: set_vec_a] :
( ! [X3: vec_a] :
( ( member_vec_a @ X3 @ A2 )
=> ( member_vec_a @ X3 @ B2 ) )
=> ( ord_le4791951621262958845_vec_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_442_subsetI,axiom,
! [A2: set_mat_a,B2: set_mat_a] :
( ! [X3: mat_a] :
( ( member_mat_a @ X3 @ A2 )
=> ( member_mat_a @ X3 @ B2 ) )
=> ( ord_le3318621148231462513_mat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_443_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_444_mat__of__row__mult__append__rows,axiom,
! [Y1: vec_a,Nr1: nat,Y22: vec_a,Nr2: nat,A1: mat_a,Nc: nat,A22: mat_a] :
( ( member_vec_a @ Y1 @ ( carrier_vec_a @ Nr1 ) )
=> ( ( member_vec_a @ Y22 @ ( carrier_vec_a @ Nr2 ) )
=> ( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( times_times_mat_a @ ( mat_of_row_a @ ( append_vec_a @ Y1 @ Y22 ) ) @ ( append_rows_a @ A1 @ A22 ) )
= ( plus_plus_mat_a @ ( times_times_mat_a @ ( mat_of_row_a @ Y1 ) @ A1 ) @ ( times_times_mat_a @ ( mat_of_row_a @ Y22 ) @ A22 ) ) ) ) ) ) ) ).
% mat_of_row_mult_append_rows
thf(fact_445_comm__add__vecset,axiom,
! [A2: set_vec_a,N: nat,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A2 @ B2 )
= ( plus_plus_set_vec_a @ B2 @ A2 ) ) ) ) ).
% comm_add_vecset
thf(fact_446_set__times__intro,axiom,
! [A: mat_a,C2: set_mat_a,B: mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( member_mat_a @ B @ D )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_1230744552615602198_mat_a @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_447_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_448_ComplI,axiom,
! [C: vec_a,A2: set_vec_a] :
( ~ ( member_vec_a @ C @ A2 )
=> ( member_vec_a @ C @ ( uminus2769705506071317478_vec_a @ A2 ) ) ) ).
% ComplI
thf(fact_449_ComplI,axiom,
! [C: mat_a,A2: set_mat_a] :
( ~ ( member_mat_a @ C @ A2 )
=> ( member_mat_a @ C @ ( uminus1296375033039821146_mat_a @ A2 ) ) ) ).
% ComplI
thf(fact_450_ComplI,axiom,
! [C: nat,A2: set_nat] :
( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_451_Compl__iff,axiom,
! [C: vec_a,A2: set_vec_a] :
( ( member_vec_a @ C @ ( uminus2769705506071317478_vec_a @ A2 ) )
= ( ~ ( member_vec_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_452_Compl__iff,axiom,
! [C: mat_a,A2: set_mat_a] :
( ( member_mat_a @ C @ ( uminus1296375033039821146_mat_a @ A2 ) )
= ( ~ ( member_mat_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_453_Compl__iff,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_454_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_455_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_456_assoc__mult__mat,axiom,
! [A2: mat_a,N_1: nat,N_2: nat,B2: mat_a,N_3: nat,C2: mat_a,N_4: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N_3 @ N_4 ) )
=> ( ( times_times_mat_a @ ( times_times_mat_a @ A2 @ B2 ) @ C2 )
= ( times_times_mat_a @ A2 @ ( times_times_mat_a @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_457_assoc__mult__mat__vec,axiom,
! [A2: mat_a,N_1: nat,N_2: nat,B2: mat_a,N_3: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N_3 ) )
=> ( ( mult_mat_vec_a @ ( times_times_mat_a @ A2 @ B2 ) @ V )
= ( mult_mat_vec_a @ A2 @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% assoc_mult_mat_vec
thf(fact_458_ComplD,axiom,
! [C: vec_a,A2: set_vec_a] :
( ( member_vec_a @ C @ ( uminus2769705506071317478_vec_a @ A2 ) )
=> ~ ( member_vec_a @ C @ A2 ) ) ).
% ComplD
thf(fact_459_ComplD,axiom,
! [C: mat_a,A2: set_mat_a] :
( ( member_mat_a @ C @ ( uminus1296375033039821146_mat_a @ A2 ) )
=> ~ ( member_mat_a @ C @ A2 ) ) ).
% ComplD
thf(fact_460_ComplD,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_461_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_462_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_463_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_464_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_465_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_466_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_467_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_468_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_469_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_470_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_471_set__times__elim,axiom,
! [X: mat_a,A2: set_mat_a,B2: set_mat_a] :
( ( member_mat_a @ X @ ( times_1230744552615602198_mat_a @ A2 @ B2 ) )
=> ~ ! [A3: mat_a,B3: mat_a] :
( ( X
= ( times_times_mat_a @ A3 @ B3 ) )
=> ( ( member_mat_a @ A3 @ A2 )
=> ~ ( member_mat_a @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_472_set__times__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( times_times_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_473_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_474_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_475_mult__carrier__mat,axiom,
! [A2: mat_a,Nr: nat,N: nat,B2: mat_a,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( member_mat_a @ ( times_times_mat_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_476_right__mult__one__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ A2 @ ( one_mat_a @ Nc ) )
= A2 ) ) ).
% right_mult_one_mat
thf(fact_477_left__mult__one__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ ( one_mat_a @ Nr ) @ A2 )
= A2 ) ) ).
% left_mult_one_mat
thf(fact_478_left__mult__zero__mat,axiom,
! [A2: mat_a,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A2 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_479_right__mult__zero__mat,axiom,
! [A2: mat_a,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( times_times_mat_a @ A2 @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_480_transpose__mult,axiom,
! [A2: mat_a,Nr: nat,N: nat,B2: mat_a,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( transpose_mat_a @ ( times_times_mat_a @ A2 @ B2 ) )
= ( times_times_mat_a @ ( transpose_mat_a @ B2 ) @ ( transpose_mat_a @ A2 ) ) ) ) ) ).
% transpose_mult
thf(fact_481_mult__add__distrib__mat,axiom,
! [A2: mat_a,Nr: nat,N: nat,B2: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A2 @ ( plus_plus_mat_a @ B2 @ C2 ) )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A2 @ B2 ) @ ( times_times_mat_a @ A2 @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_482_add__mult__distrib__mat,axiom,
! [A2: mat_a,Nr: nat,N: nat,B2: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ C2 )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A2 @ C2 ) @ ( times_times_mat_a @ B2 @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_483_in__mono,axiom,
! [A2: set_vec_a,B2: set_vec_a,X: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
=> ( ( member_vec_a @ X @ A2 )
=> ( member_vec_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_484_in__mono,axiom,
! [A2: set_mat_a,B2: set_mat_a,X: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B2 )
=> ( ( member_mat_a @ X @ A2 )
=> ( member_mat_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_485_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_486_subsetD,axiom,
! [A2: set_vec_a,B2: set_vec_a,C: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
=> ( ( member_vec_a @ C @ A2 )
=> ( member_vec_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_487_subsetD,axiom,
! [A2: set_mat_a,B2: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B2 )
=> ( ( member_mat_a @ C @ A2 )
=> ( member_mat_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_488_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_489_subset__eq,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A5: set_vec_a,B5: set_vec_a] :
! [X2: vec_a] :
( ( member_vec_a @ X2 @ A5 )
=> ( member_vec_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_490_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A5: set_mat_a,B5: set_mat_a] :
! [X2: mat_a] :
( ( member_mat_a @ X2 @ A5 )
=> ( member_mat_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_491_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_492_subset__iff,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A5: set_vec_a,B5: set_vec_a] :
! [T: vec_a] :
( ( member_vec_a @ T @ A5 )
=> ( member_vec_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_493_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A5: set_mat_a,B5: set_mat_a] :
! [T: mat_a] :
( ( member_mat_a @ T @ A5 )
=> ( member_mat_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_494_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_495_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_496_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_497_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_498_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_499_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_500_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_501_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_502_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_503_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_504_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_505_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_506_mult__four__block__mat,axiom,
! [A1: mat_a,Nr1: nat,N1: nat,B1: mat_a,N2: nat,C1: mat_a,Nr2: nat,D1: mat_a,A22: mat_a,Nc1: nat,B22: mat_a,Nc2: nat,C22: mat_a,D22: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ N1 ) )
=> ( ( member_mat_a @ B1 @ ( carrier_mat_a @ Nr1 @ N2 ) )
=> ( ( member_mat_a @ C1 @ ( carrier_mat_a @ Nr2 @ N1 ) )
=> ( ( member_mat_a @ D1 @ ( carrier_mat_a @ Nr2 @ N2 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N1 @ Nc1 ) )
=> ( ( member_mat_a @ B22 @ ( carrier_mat_a @ N1 @ Nc2 ) )
=> ( ( member_mat_a @ C22 @ ( carrier_mat_a @ N2 @ Nc1 ) )
=> ( ( member_mat_a @ D22 @ ( carrier_mat_a @ N2 @ Nc2 ) )
=> ( ( times_times_mat_a @ ( four_block_mat_a @ A1 @ B1 @ C1 @ D1 ) @ ( four_block_mat_a @ A22 @ B22 @ C22 @ D22 ) )
= ( four_block_mat_a @ ( plus_plus_mat_a @ ( times_times_mat_a @ A1 @ A22 ) @ ( times_times_mat_a @ B1 @ C22 ) ) @ ( plus_plus_mat_a @ ( times_times_mat_a @ A1 @ B22 ) @ ( times_times_mat_a @ B1 @ D22 ) ) @ ( plus_plus_mat_a @ ( times_times_mat_a @ C1 @ A22 ) @ ( times_times_mat_a @ D1 @ C22 ) ) @ ( plus_plus_mat_a @ ( times_times_mat_a @ C1 @ B22 ) @ ( times_times_mat_a @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ) ).
% mult_four_block_mat
thf(fact_507_assoc__add__vecset,axiom,
! [A2: set_vec_a,N: nat,B2: set_vec_a,C2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ C2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A2 @ ( plus_plus_set_vec_a @ B2 @ C2 ) )
= ( plus_plus_set_vec_a @ ( plus_plus_set_vec_a @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% assoc_add_vecset
thf(fact_508_mat__mult__left__right__inverse,axiom,
! [A2: mat_a,N: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( ( times_times_mat_a @ A2 @ B2 )
= ( one_mat_a @ N ) )
=> ( ( times_times_mat_a @ B2 @ A2 )
= ( one_mat_a @ N ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_509_mult__mat__of__col,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( times_times_mat_a @ A2 @ ( missing_mat_of_col_a @ V ) )
= ( missing_mat_of_col_a @ ( mult_mat_vec_a @ A2 @ V ) ) ) ) ) ).
% mult_mat_of_col
thf(fact_510_gram__schmidt_OFarkas__Lemma_H,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( ? [X2: vec_a] :
( ( member_vec_a @ X2 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X2 ) @ B ) ) )
= ( ! [Y2: vec_a] :
( ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y2 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y2 )
= ( zero_vec_a @ Nc ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ Y2 @ B ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma'
thf(fact_511_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_512_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_513_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_514_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_515_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_516_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_517_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_518_add__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A )
= A ) ).
% add_0
thf(fact_519_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_520_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_521_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_522_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_523_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_524_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_525_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_526_add_Oright__neutral,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ zero_zero_set_nat )
= A ) ).
% add.right_neutral
thf(fact_527_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_528_set__zero__plus,axiom,
! [C2: set_nat] :
( ( set_elt_set_plus_nat @ zero_zero_nat @ C2 )
= C2 ) ).
% set_zero_plus
thf(fact_529_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_530_add__le__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel1
thf(fact_531_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_532_add__le__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel2
thf(fact_533_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_534_le__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_535_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_536_le__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_537_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_538_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_539_neg__less__eq__nonneg,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ A )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_540_less__eq__neg__nonpos,axiom,
! [A: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% less_eq_neg_nonpos
thf(fact_541_neg__le__0__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_le_0_iff_le
thf(fact_542_neg__0__le__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% neg_0_le_iff_le
thf(fact_543_scalar__prod__right__zero,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V @ ( zero_vec_a @ N ) )
= zero_zero_a ) ) ).
% scalar_prod_right_zero
thf(fact_544_scalar__prod__right__zero,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ V @ ( zero_vec_nat @ N ) )
= zero_zero_nat ) ) ).
% scalar_prod_right_zero
thf(fact_545_scalar__prod__left__zero,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( zero_vec_a @ N ) @ V )
= zero_zero_a ) ) ).
% scalar_prod_left_zero
thf(fact_546_scalar__prod__left__zero,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( zero_vec_nat @ N ) @ V )
= zero_zero_nat ) ) ).
% scalar_prod_left_zero
thf(fact_547_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_548_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_549_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_550_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_551_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_552_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_553_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_554_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_555_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_556_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_557_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_558_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_559_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_560_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_561_le__numeral__extra_I3_J,axiom,
ord_less_eq_a @ zero_zero_a @ zero_zero_a ).
% le_numeral_extra(3)
thf(fact_562_add_Ocomm__neutral,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ zero_zero_set_nat )
= A ) ).
% add.comm_neutral
thf(fact_563_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_564_comm__monoid__add__class_Oadd__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_565_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_566_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_567_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_568_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_569_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_570_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_571_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_572_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_573_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_574_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_a @ zero_zero_a @ one_one_a ).
% zero_less_one_class.zero_le_one
thf(fact_575_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_576_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_a @ zero_zero_a @ one_one_a ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_577_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_578_not__one__le__zero,axiom,
~ ( ord_less_eq_a @ one_one_a @ zero_zero_a ) ).
% not_one_le_zero
thf(fact_579_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_580_mult__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_581_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_582_mult__mono_H,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_583_zero__le__square,axiom,
! [A: a] : ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ A ) ) ).
% zero_le_square
thf(fact_584_split__mult__pos__le,axiom,
! [A: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_585_mult__left__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_586_mult__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_587_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_588_mult__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_589_mult__right__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_590_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_591_mult__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_592_mult__le__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_593_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_594_split__mult__neg__le,axiom,
! [A: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ).
% split_mult_neg_le
thf(fact_595_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_596_mult__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_597_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_598_mult__nonneg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos
thf(fact_599_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_600_mult__nonpos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% mult_nonpos_nonneg
thf(fact_601_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_602_mult__nonneg__nonpos2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_603_zero__le__mult__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) ) ) ).
% zero_le_mult_iff
thf(fact_604_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_605_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_606_mult__left__le__one__le,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ Y @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_607_mult__right__le__one__le,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ Y @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_608_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_609_mult__le__one,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ one_one_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ B @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ one_one_a ) ) ) ) ).
% mult_le_one
thf(fact_610_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_611_mult__left__le,axiom,
! [C: a,A: a] :
( ( ord_less_eq_a @ C @ one_one_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_612_sum__squares__ge__zero,axiom,
! [X: a,Y: a] : ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_613_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_614_add__decreasing,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_615_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_616_add__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_617_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_618_add__decreasing2,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_619_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_620_add__increasing2,axiom,
! [C: a,B: a,A: a] :
( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_621_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_622_add__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_623_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_624_add__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_625_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_626_add__nonneg__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_627_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_628_add__nonpos__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ zero_zero_a )
=> ( ( ord_less_eq_a @ Y @ zero_zero_a )
=> ( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_629_set__zero__plus2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( plus_plus_set_nat @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_630_convex__bound__le,axiom,
! [X: a,A: a,Y: a,U: a,V: a] :
( ( ord_less_eq_a @ X @ A )
=> ( ( ord_less_eq_a @ Y @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ U )
=> ( ( ord_less_eq_a @ zero_zero_a @ V )
=> ( ( ( plus_plus_a @ U @ V )
= one_one_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ U @ X ) @ ( times_times_a @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_631_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% le_minus_one_simps(3)
thf(fact_632_le__minus__one__simps_I1_J,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ zero_zero_a ).
% le_minus_one_simps(1)
thf(fact_633_mat__of__col__def,axiom,
( missing_mat_of_col_a
= ( ^ [V4: vec_a] : ( transpose_mat_a @ ( mat_of_row_a @ V4 ) ) ) ) ).
% mat_of_col_def
thf(fact_634_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_635_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_636_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_637_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_638_gram__schmidt_OFarkas__Lemma,axiom,
! [A2: mat_a,N: nat,Nr: nat,B: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ Nr ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ N ) )
=> ( ( ? [X2: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ X2 )
& ( ( mult_mat_vec_a @ A2 @ X2 )
= B ) ) )
= ( ! [Y2: vec_a] :
( ( member_vec_a @ Y2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y2 ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ Y2 @ B ) ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma
thf(fact_639_sum__squares__le__zero__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) @ zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_640_scalar__prod__ge__0,axiom,
! [X: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_641_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_642_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_643_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_644_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_645_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_646_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_647_square__lesseq__square,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ) ).
% square_lesseq_square
thf(fact_648_less__eq__fract__respect,axiom,
! [B: a,B6: a,D2: a,D3: a,A: a,A6: a,C: a,C5: a] :
( ( B != zero_zero_a )
=> ( ( B6 != zero_zero_a )
=> ( ( D2 != zero_zero_a )
=> ( ( D3 != zero_zero_a )
=> ( ( ( times_times_a @ A @ B6 )
= ( times_times_a @ A6 @ B ) )
=> ( ( ( times_times_a @ C @ D3 )
= ( times_times_a @ C5 @ D2 ) )
=> ( ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A @ D2 ) @ ( times_times_a @ B @ D2 ) ) @ ( times_times_a @ ( times_times_a @ C @ B ) @ ( times_times_a @ B @ D2 ) ) )
= ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A6 @ D3 ) @ ( times_times_a @ B6 @ D3 ) ) @ ( times_times_a @ ( times_times_a @ C5 @ B6 ) @ ( times_times_a @ B6 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_649_norm1__ge__0,axiom,
! [F2: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( norm1_a @ F2 ) ) ).
% norm1_ge_0
thf(fact_650_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_651_vec__inv,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( V
!= ( zero_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( schur_vec_inv_a @ V ) @ V )
= one_one_a ) ) ) ).
% vec_inv
thf(fact_652_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_653_linf__norm__vec__eq__0,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ( linf_norm_vec_a @ V )
= zero_zero_a )
= ( V
= ( zero_vec_a @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_654_vardim_Ounpadl__padl,axiom,
! [M: nat,V: vec_a] :
( ( matrix_unpadl_a @ M @ ( append_vec_a @ ( zero_vec_a @ M ) @ V ) )
= V ) ).
% vardim.unpadl_padl
thf(fact_655_linf__norm__vec__ge__0,axiom,
! [V: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( linf_norm_vec_a @ V ) ) ).
% linf_norm_vec_ge_0
thf(fact_656_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_a @ ( zero_vec_a @ N ) )
= zero_zero_a ) ).
% linf_norm_zero_vec
thf(fact_657_vec__inv__closed,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( schur_vec_inv_a @ V ) @ ( carrier_vec_a @ N ) ) ) ).
% vec_inv_closed
thf(fact_658_is__det__bound__ge__zero,axiom,
! [F2: nat > a > a,X: a,N: nat] :
( ( integr1439407231847824701ound_a @ F2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ord_less_eq_a @ zero_zero_a @ ( F2 @ N @ X ) ) ) ) ).
% is_det_bound_ge_zero
thf(fact_659_set__times__plus__distrib3,axiom,
! [A: nat,C2: set_nat,D: set_nat] : ( ord_less_eq_set_nat @ ( times_times_set_nat @ ( set_elt_set_plus_nat @ A @ C2 ) @ D ) @ ( plus_plus_set_nat @ ( set_el2933305810450955905es_nat @ A @ D ) @ ( times_times_set_nat @ C2 @ D ) ) ) ).
% set_times_plus_distrib3
thf(fact_660_swap__col__to__front__four__block,axiom,
! [A1: mat_a,N1: nat,M: nat,A22: mat_a,A32: mat_a,N2: nat,A42: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N1 @ M ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N1 @ one_one_nat ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ N2 @ M ) )
=> ( ( member_mat_a @ A42 @ ( carrier_mat_a @ N2 @ one_one_nat ) )
=> ( ( column2924081423933032910ront_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A42 ) @ M )
= ( four_block_mat_a @ A22 @ A1 @ A42 @ A32 ) ) ) ) ) ) ).
% swap_col_to_front_four_block
thf(fact_661_swap__row__to__front__four__block,axiom,
! [A1: mat_a,N: nat,M1: nat,A22: mat_a,M22: nat,A32: mat_a,A42: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ M1 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N @ M22 ) )
=> ( ( member_mat_a @ A32 @ ( carrier_mat_a @ one_one_nat @ M1 ) )
=> ( ( member_mat_a @ A42 @ ( carrier_mat_a @ one_one_nat @ M22 ) )
=> ( ( column973622294476583016ront_a @ ( four_block_mat_a @ A1 @ A22 @ A32 @ A42 ) @ N )
= ( four_block_mat_a @ A32 @ A42 @ A1 @ A22 ) ) ) ) ) ) ).
% swap_row_to_front_four_block
thf(fact_662_set__one__times,axiom,
! [C2: set_nat] :
( ( set_el2933305810450955905es_nat @ one_one_nat @ C2 )
= C2 ) ).
% set_one_times
thf(fact_663_set__times__intro2,axiom,
! [B: mat_a,C2: set_mat_a,A: mat_a] :
( ( member_mat_a @ B @ C2 )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B ) @ ( set_el1062546952344711308_mat_a @ A @ C2 ) ) ) ).
% set_times_intro2
thf(fact_664_set__times__intro2,axiom,
! [B: nat,C2: set_nat,A: nat] :
( ( member_nat @ B @ C2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( set_el2933305810450955905es_nat @ A @ C2 ) ) ) ).
% set_times_intro2
thf(fact_665_set__times__mono3,axiom,
! [A: mat_a,C2: set_mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ord_le3318621148231462513_mat_a @ ( set_el1062546952344711308_mat_a @ A @ D ) @ ( times_1230744552615602198_mat_a @ C2 @ D ) ) ) ).
% set_times_mono3
thf(fact_666_set__times__mono3,axiom,
! [A: nat,C2: set_nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ ( set_el2933305810450955905es_nat @ A @ D ) @ ( times_times_set_nat @ C2 @ D ) ) ) ).
% set_times_mono3
thf(fact_667_set__times__mono4,axiom,
! [A: nat,C2: set_nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ ( set_el2933305810450955905es_nat @ A @ D ) @ ( times_times_set_nat @ D @ C2 ) ) ) ).
% set_times_mono4
thf(fact_668_set__times__rearrange2,axiom,
! [A: nat,B: nat,C2: set_nat] :
( ( set_el2933305810450955905es_nat @ A @ ( set_el2933305810450955905es_nat @ B @ C2 ) )
= ( set_el2933305810450955905es_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ).
% set_times_rearrange2
thf(fact_669_set__times__plus__distrib2,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( set_el2933305810450955905es_nat @ A @ ( plus_plus_set_nat @ B2 @ C2 ) )
= ( plus_plus_set_nat @ ( set_el2933305810450955905es_nat @ A @ B2 ) @ ( set_el2933305810450955905es_nat @ A @ C2 ) ) ) ).
% set_times_plus_distrib2
thf(fact_670_set__times__rearrange,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( times_times_set_nat @ ( set_el2933305810450955905es_nat @ A @ C2 ) @ ( set_el2933305810450955905es_nat @ B @ D ) )
= ( set_el2933305810450955905es_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ).
% set_times_rearrange
thf(fact_671_set__times__plus__distrib,axiom,
! [A: nat,B: nat,C2: set_nat] :
( ( set_el2933305810450955905es_nat @ A @ ( set_elt_set_plus_nat @ B @ C2 ) )
= ( set_elt_set_plus_nat @ ( times_times_nat @ A @ B ) @ ( set_el2933305810450955905es_nat @ A @ C2 ) ) ) ).
% set_times_plus_distrib
thf(fact_672_set__times__mono5,axiom,
! [A: mat_a,C2: set_mat_a,B2: set_mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( ord_le3318621148231462513_mat_a @ B2 @ D )
=> ( ord_le3318621148231462513_mat_a @ ( set_el1062546952344711308_mat_a @ A @ B2 ) @ ( times_1230744552615602198_mat_a @ C2 @ D ) ) ) ) ).
% set_times_mono5
thf(fact_673_set__times__mono5,axiom,
! [A: nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( set_el2933305810450955905es_nat @ A @ B2 ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_mono5
thf(fact_674_linf__norm__vec__greater__0,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_a @ zero_zero_a @ ( linf_norm_vec_a @ V ) )
= ( V
!= ( zero_vec_a @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_675_mat__of__row__carrier_I2_J,axiom,
! [Y: vec_a,N: nat] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ N ) )
=> ( member_mat_a @ ( mat_of_row_a @ Y ) @ ( carrier_mat_a @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% mat_of_row_carrier(2)
thf(fact_676_vardim_Okernel__padr,axiom,
! [A: vec_a,A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,D: mat_a,Nr2: nat] :
( ( member_vec_a @ A @ ( matrix_mat_kernel_a @ A2 ) )
=> ( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( member_vec_a @ ( append_vec_a @ A @ ( zero_vec_a @ Nc2 ) ) @ ( matrix_mat_kernel_a @ ( four_block_mat_a @ A2 @ B2 @ ( zero_mat_a @ Nr2 @ Nc1 ) @ D ) ) ) ) ) ) ) ).
% vardim.kernel_padr
thf(fact_677_vardim_Okernel__padl,axiom,
! [D2: vec_a,D: mat_a,A2: mat_a,Nr1: nat,Nc1: nat,C2: mat_a,Nr2: nat,Nc2: nat] :
( ( member_vec_a @ D2 @ ( matrix_mat_kernel_a @ D ) )
=> ( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( member_vec_a @ ( append_vec_a @ ( zero_vec_a @ Nc1 ) @ D2 ) @ ( matrix_mat_kernel_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ Nr1 @ Nc2 ) @ C2 @ D ) ) ) ) ) ) ) ).
% vardim.kernel_padl
thf(fact_678_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_679_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_680_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_681_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_682_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_683_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_684_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_685_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_686_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_687_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_688_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_689_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_690_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_691_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_692_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_693_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_694_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_695_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_696_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_697_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_698_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_699_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_700_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_701_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_702_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_703_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_704_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_705_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_706_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_707_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_708_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_709_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_710_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_711_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_712_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_713_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_714_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_715_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_716_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_717_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_718_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_719_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_720_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_721_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_722_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_723_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_724_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_725_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_726_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_727_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_728_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_729_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_730_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_731_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_732_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_733_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_734_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_735_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_736_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_737_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_738_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_739_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_740_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_741_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_742_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_743_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_744_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_745_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_746_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_747_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_748_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_749_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_750_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_751_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_752_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_753_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_754_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_755_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_756_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_757_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_758_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_759_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_760_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_761_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_762_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_763_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_764_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_765_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_766_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_767_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_768_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_769_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_770_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_771_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_772_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_773_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_774_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_775_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_776_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_777_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_778_leD,axiom,
! [Y: vec_a,X: vec_a] :
( ( ord_less_eq_vec_a @ Y @ X )
=> ~ ( ord_less_vec_a @ X @ Y ) ) ).
% leD
thf(fact_779_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_780_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_781_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_782_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_783_nless__le,axiom,
! [A: vec_a,B: vec_a] :
( ( ~ ( ord_less_vec_a @ A @ B ) )
= ( ~ ( ord_less_eq_vec_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_784_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_785_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_786_antisym__conv1,axiom,
! [X: vec_a,Y: vec_a] :
( ~ ( ord_less_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_787_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_788_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_789_antisym__conv2,axiom,
! [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ~ ( ord_less_vec_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_790_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_791_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_792_dense__ge,axiom,
! [Z: a,Y: a] :
( ! [X3: a] :
( ( ord_less_a @ Z @ X3 )
=> ( ord_less_eq_a @ Y @ X3 ) )
=> ( ord_less_eq_a @ Y @ Z ) ) ).
% dense_ge
thf(fact_793_dense__le,axiom,
! [Y: a,Z: a] :
( ! [X3: a] :
( ( ord_less_a @ X3 @ Y )
=> ( ord_less_eq_a @ X3 @ Z ) )
=> ( ord_less_eq_a @ Y @ Z ) ) ).
% dense_le
thf(fact_794_less__le__not__le,axiom,
( ord_less_vec_a
= ( ^ [X2: vec_a,Y2: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
& ~ ( ord_less_eq_vec_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_795_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_796_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ~ ( ord_less_eq_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_797_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_798_not__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_eq_a @ Y @ X )
=> ( ord_less_a @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_799_order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [A4: vec_a,B4: vec_a] :
( ( ord_less_vec_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_800_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_801_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_802_order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [A4: vec_a,B4: vec_a] :
( ( ord_less_eq_vec_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_803_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_804_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_805_order_Ostrict__trans1,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_vec_a @ B @ C )
=> ( ord_less_vec_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_806_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_807_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_808_order_Ostrict__trans2,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ord_less_vec_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_809_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_810_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_811_order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [A4: vec_a,B4: vec_a] :
( ( ord_less_eq_vec_a @ A4 @ B4 )
& ~ ( ord_less_eq_vec_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_812_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_813_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ~ ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_814_dense__ge__bounded,axiom,
! [Z: a,X: a,Y: a] :
( ( ord_less_a @ Z @ X )
=> ( ! [W3: a] :
( ( ord_less_a @ Z @ W3 )
=> ( ( ord_less_a @ W3 @ X )
=> ( ord_less_eq_a @ Y @ W3 ) ) )
=> ( ord_less_eq_a @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_815_dense__le__bounded,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_a @ X @ Y )
=> ( ! [W3: a] :
( ( ord_less_a @ X @ W3 )
=> ( ( ord_less_a @ W3 @ Y )
=> ( ord_less_eq_a @ W3 @ Z ) ) )
=> ( ord_less_eq_a @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_816_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [B4: vec_a,A4: vec_a] :
( ( ord_less_vec_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_817_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_818_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_819_dual__order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [B4: vec_a,A4: vec_a] :
( ( ord_less_eq_vec_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_820_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_821_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_822_dual__order_Ostrict__trans1,axiom,
! [B: vec_a,A: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ B @ A )
=> ( ( ord_less_vec_a @ C @ B )
=> ( ord_less_vec_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_823_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_824_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_825_dual__order_Ostrict__trans2,axiom,
! [B: vec_a,A: vec_a,C: vec_a] :
( ( ord_less_vec_a @ B @ A )
=> ( ( ord_less_eq_vec_a @ C @ B )
=> ( ord_less_vec_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_826_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_827_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_828_dual__order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [B4: vec_a,A4: vec_a] :
( ( ord_less_eq_vec_a @ B4 @ A4 )
& ~ ( ord_less_eq_vec_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_829_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_830_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_831_order_Ostrict__implies__order,axiom,
! [A: vec_a,B: vec_a] :
( ( ord_less_vec_a @ A @ B )
=> ( ord_less_eq_vec_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_832_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_833_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_834_dual__order_Ostrict__implies__order,axiom,
! [B: vec_a,A: vec_a] :
( ( ord_less_vec_a @ B @ A )
=> ( ord_less_eq_vec_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_835_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_836_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_837_order__le__less,axiom,
( ord_less_eq_vec_a
= ( ^ [X2: vec_a,Y2: vec_a] :
( ( ord_less_vec_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_838_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_839_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_840_order__less__le,axiom,
( ord_less_vec_a
= ( ^ [X2: vec_a,Y2: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_841_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_842_order__less__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_843_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_844_linorder__not__le,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X @ Y ) )
= ( ord_less_a @ Y @ X ) ) ).
% linorder_not_le
thf(fact_845_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_846_linorder__not__less,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_not_less
thf(fact_847_order__less__imp__le,axiom,
! [X: vec_a,Y: vec_a] :
( ( ord_less_vec_a @ X @ Y )
=> ( ord_less_eq_vec_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_848_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_849_order__less__imp__le,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_850_order__le__neq__trans,axiom,
! [A: vec_a,B: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_vec_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_851_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_852_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_853_order__neq__le__trans,axiom,
! [A: vec_a,B: vec_a] :
( ( A != B )
=> ( ( ord_less_eq_vec_a @ A @ B )
=> ( ord_less_vec_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_854_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_855_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_856_order__le__less__trans,axiom,
! [X: vec_a,Y: vec_a,Z: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ord_less_vec_a @ Y @ Z )
=> ( ord_less_vec_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_857_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_858_order__le__less__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_859_order__less__le__trans,axiom,
! [X: vec_a,Y: vec_a,Z: vec_a] :
( ( ord_less_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ Y @ Z )
=> ( ord_less_vec_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_860_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_861_order__less__le__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_862_order__le__less__subst1,axiom,
! [A: vec_a,F2: nat > vec_a,B: nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_863_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_864_order__le__less__subst1,axiom,
! [A: a,F2: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_865_order__le__less__subst2,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_866_order__le__less__subst2,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_867_order__le__less__subst2,axiom,
! [A: vec_a,B: vec_a,F2: vec_a > a,C: a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_a @ ( F2 @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_868_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_869_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_870_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_871_order__le__less__subst2,axiom,
! [A: a,B: a,F2: a > vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_872_order__le__less__subst2,axiom,
! [A: a,B: a,F2: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_873_order__le__less__subst2,axiom,
! [A: a,B: a,F2: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F2 @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_874_order__less__le__subst1,axiom,
! [A: vec_a,F2: vec_a > vec_a,B: vec_a,C: vec_a] :
( ( ord_less_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_875_order__less__le__subst1,axiom,
! [A: nat,F2: vec_a > nat,B: vec_a,C: vec_a] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_876_order__less__le__subst1,axiom,
! [A: a,F2: vec_a > a,B: vec_a,C: vec_a] :
( ( ord_less_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_877_order__less__le__subst1,axiom,
! [A: vec_a,F2: nat > vec_a,B: nat,C: nat] :
( ( ord_less_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_878_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_879_order__less__le__subst1,axiom,
! [A: a,F2: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_880_order__less__le__subst1,axiom,
! [A: vec_a,F2: a > vec_a,B: a,C: a] :
( ( ord_less_vec_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_881_order__less__le__subst1,axiom,
! [A: nat,F2: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_882_order__less__le__subst1,axiom,
! [A: a,F2: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_883_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > vec_a,C: vec_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_vec_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_884_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_885_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_886_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_887_linorder__le__less__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_a @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_888_order__le__imp__less__or__eq,axiom,
! [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ord_less_vec_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_889_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_890_order__le__imp__less__or__eq,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_891_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_892_verit__comp__simplify1_I3_J,axiom,
! [B6: a,A6: a] :
( ( ~ ( ord_less_eq_a @ B6 @ A6 ) )
= ( ord_less_a @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_893_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_894_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_895_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_896_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_897_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_898_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_899_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_900_less__vec__def,axiom,
( ord_less_vec_a
= ( ^ [V4: vec_a,W4: vec_a] :
( ( ord_less_eq_vec_a @ V4 @ W4 )
& ~ ( ord_less_eq_vec_a @ W4 @ V4 ) ) ) ) ).
% less_vec_def
thf(fact_901_not__le__real,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ~ ( ord_less_eq_nat @ B @ A ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% not_le_real
thf(fact_902_not__le__real,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ~ ( ord_less_eq_a @ B @ A ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% not_le_real
thf(fact_903_not__less__real,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ~ ( ord_less_nat @ B @ A ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% not_less_real
thf(fact_904_not__less__real,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ~ ( ord_less_a @ B @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% not_less_real
thf(fact_905_nonneg__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_906_nonneg__linorder__cases,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_907_nonpos__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_908_nonpos__linorder__cases,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_909_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_910_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_911_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_912_real__add__less__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% real_add_less_cancel_right_pos
thf(fact_913_real__add__less__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% real_add_less_cancel_left_pos
thf(fact_914_add__pos__neg__is__real,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ) ).
% add_pos_neg_is_real
thf(fact_915_add__neg__pos__is__real,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ) ).
% add_neg_pos_is_real
thf(fact_916_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_917_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_918_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_919_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_920_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_921_add__less__le__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_922_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_923_add__le__less__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_924_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_925_add__mono__thms__linordered__field_I3_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_926_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_927_add__mono__thms__linordered__field_I4_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_928_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_929_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_930_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_931_lift__Suc__mono__le,axiom,
! [F2: nat > vec_a,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_vec_a @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_vec_a @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_932_lift__Suc__mono__le,axiom,
! [F2: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_933_lift__Suc__mono__le,axiom,
! [F2: nat > a,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_a @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_934_lift__Suc__antimono__le,axiom,
! [F2: nat > vec_a,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_vec_a @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_vec_a @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_935_lift__Suc__antimono__le,axiom,
! [F2: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_936_lift__Suc__antimono__le,axiom,
! [F2: nat > a,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_a @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_937_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_938_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_939_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_940_mat__kernelD_I1_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( matrix_mat_kernel_a @ A2 ) )
=> ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) ) ) ) ).
% mat_kernelD(1)
thf(fact_941_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_942_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_943_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_944_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_945_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_946_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_947_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_948_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F2 @ M4 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_949_real__mult__le__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_950_real__mult__le__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_951_real__mult__le__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_952_real__mult__le__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_953_mult__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_954_mult__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_955_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_956_mult__left__less__imp__less,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_957_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_958_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_959_mult__less__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_960_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_961_mult__right__less__imp__less,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_962_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_963_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_964_mult__less__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_965_mult__le__cancel__left__neg,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ C @ zero_zero_a )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_966_mult__le__cancel__left__pos,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_967_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_968_mult__left__le__imp__le,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_969_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_970_mult__right__le__imp__le,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_971_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_972_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_973_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_974_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_975_square__less__square,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= ( ord_less_a @ X @ Y ) ) ) ) ).
% square_less_square
thf(fact_976_real__add__le__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% real_add_le_cancel_left_pos
thf(fact_977_real__add__le__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% real_add_le_cancel_left_pos
thf(fact_978_real__add__le__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% real_add_le_cancel_right_pos
thf(fact_979_real__add__le__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% real_add_le_cancel_right_pos
thf(fact_980_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_981_add__strict__increasing2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_982_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_983_add__strict__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_984_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_985_add__pos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_986_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_987_add__nonpos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_neg
thf(fact_988_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_989_add__nonneg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_990_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_991_add__neg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_nonpos
thf(fact_992_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_993_mat__kernel__carrier,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ord_le4791951621262958845_vec_a @ ( matrix_mat_kernel_a @ A2 ) @ ( carrier_vec_a @ Nc ) ) ) ).
% mat_kernel_carrier
thf(fact_994_mat__kernel__mult__subset,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,N: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N @ Nr ) )
=> ( ord_le4791951621262958845_vec_a @ ( matrix_mat_kernel_a @ A2 ) @ ( matrix_mat_kernel_a @ ( times_times_mat_a @ B2 @ A2 ) ) ) ) ) ).
% mat_kernel_mult_subset
thf(fact_995_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_996_mat__kernel__mult__eq,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nr ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nr ) )
=> ( ( ( times_times_mat_a @ C2 @ B2 )
= ( one_mat_a @ Nr ) )
=> ( ( matrix_mat_kernel_a @ ( times_times_mat_a @ B2 @ A2 ) )
= ( matrix_mat_kernel_a @ A2 ) ) ) ) ) ) ).
% mat_kernel_mult_eq
thf(fact_997_mat__kernelD_I2_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( matrix_mat_kernel_a @ A2 ) )
=> ( ( mult_mat_vec_a @ A2 @ V )
= ( zero_vec_a @ Nr ) ) ) ) ).
% mat_kernelD(2)
thf(fact_998_mult__less__cancel__right2,axiom,
! [A: a,C: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ C )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ one_one_a ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ one_one_a @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_999_mult__less__cancel__right1,axiom,
! [C: a,B: a] :
( ( ord_less_a @ C @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ one_one_a @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ one_one_a ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1000_mult__less__cancel__left2,axiom,
! [C: a,A: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ C )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ one_one_a ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ one_one_a @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1001_mult__less__cancel__left1,axiom,
! [C: a,B: a] :
( ( ord_less_a @ C @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ one_one_a @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ one_one_a ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1002_mult__le__cancel__right2,axiom,
! [A: a,C: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ C )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ one_one_a ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ one_one_a @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1003_mult__le__cancel__right1,axiom,
! [C: a,B: a] :
( ( ord_less_eq_a @ C @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ one_one_a @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ one_one_a ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1004_mult__le__cancel__left2,axiom,
! [C: a,A: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ C )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ one_one_a ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ one_one_a @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1005_mult__le__cancel__left1,axiom,
! [C: a,B: a] :
( ( ord_less_eq_a @ C @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ one_one_a @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ one_one_a ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1006_mat__kernelI,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( ( mult_mat_vec_a @ A2 @ V )
= ( zero_vec_a @ Nr ) )
=> ( member_vec_a @ V @ ( matrix_mat_kernel_a @ A2 ) ) ) ) ) ).
% mat_kernelI
thf(fact_1007_vardim_Omat__kernel__split_I1_J,axiom,
! [A2: mat_a,N: nat,D: mat_a,M: nat,K: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_vec_a @ K @ ( matrix_mat_kernel_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ D ) ) )
=> ( member_vec_a @ ( vec_first_a @ K @ N ) @ ( matrix_mat_kernel_a @ A2 ) ) ) ) ) ).
% vardim.mat_kernel_split(1)
thf(fact_1008_vardim_Omat__kernel__split_I2_J,axiom,
! [A2: mat_a,N: nat,D: mat_a,M: nat,K: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_vec_a @ K @ ( matrix_mat_kernel_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ D ) ) )
=> ( member_vec_a @ ( vec_last_a @ K @ M ) @ ( matrix_mat_kernel_a @ D ) ) ) ) ) ).
% vardim.mat_kernel_split(2)
thf(fact_1009_convex__bound__lt,axiom,
! [X: a,A: a,Y: a,U: a,V: a] :
( ( ord_less_a @ X @ A )
=> ( ( ord_less_a @ Y @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ U )
=> ( ( ord_less_eq_a @ zero_zero_a @ V )
=> ( ( ( plus_plus_a @ U @ V )
= one_one_a )
=> ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ U @ X ) @ ( times_times_a @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1010_field__le__mult__one__interval,axiom,
! [X: a,Y: a] :
( ! [Z3: a] :
( ( ord_less_a @ zero_zero_a @ Z3 )
=> ( ( ord_less_a @ Z3 @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_1011_less__1__mult_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ one_one_nat @ B )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% less_1_mult'
thf(fact_1012_less__1__mult_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ one_one_a @ A )
=> ( ( ord_less_eq_a @ one_one_a @ B )
=> ( ord_less_a @ one_one_a @ ( times_times_a @ A @ B ) ) ) ) ).
% less_1_mult'
thf(fact_1013_field__le__epsilon,axiom,
! [X: a,Y: a] :
( ! [E3: a] :
( ( ord_less_a @ zero_zero_a @ E3 )
=> ( ord_less_eq_a @ X @ ( plus_plus_a @ Y @ E3 ) ) )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1014_ordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono
thf(fact_1015_ordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono
thf(fact_1016_ordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1017_ordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1018_ordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1019_ordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1020_ordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono'
thf(fact_1021_ordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono'
thf(fact_1022_mult__le__cancel__iff2,axiom,
! [Z: a,X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ Z )
=> ( ( ord_less_eq_a @ ( times_times_a @ Z @ X ) @ ( times_times_a @ Z @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1023_mult__le__cancel__iff1,axiom,
! [Z: a,X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ Z )
=> ( ( ord_less_eq_a @ ( times_times_a @ X @ Z ) @ ( times_times_a @ Y @ Z ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1024_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M6 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1025_for__all__Suc,axiom,
! [P: nat > $o,I: nat] :
( ( P @ I )
=> ( ( ! [J3: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J3 )
=> ( P @ J3 ) ) )
= ( ! [J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( P @ J3 ) ) ) ) ) ).
% for_all_Suc
thf(fact_1026_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1027_inf__pigeonhole__principle,axiom,
! [N: nat,F2: nat > nat > $o] :
( ! [K3: nat] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( F2 @ K3 @ I3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F2 @ K5 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1028__092_060open_062dim__vec_A_I0_092_060_094sub_062v_A_Inr_A_L_A1_A_L_A_Inc_A_L_Anc_J_A_L_Anr_J_J_A_061_Adim__vec_Aulv_A_092_060and_062_A_I_092_060forall_062i_060dim__vec_Aulv_O_A0_092_060_094sub_062v_A_Inr_A_L_A1_A_L_A_Inc_A_L_Anc_J_A_L_Anr_J_A_E_Ai_A_092_060le_062_Aulv_A_E_Ai_J_092_060close_062,axiom,
( ( ( dim_vec_a @ ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) )
= ( dim_vec_a @ ulv ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_a @ ulv ) )
=> ( ord_less_eq_a @ ( vec_index_a @ ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) @ I3 ) @ ( vec_index_a @ ulv @ I3 ) ) ) ) ).
% \<open>dim_vec (0\<^sub>v (nr + 1 + (nc + nc) + nr)) = dim_vec ulv \<and> (\<forall>i<dim_vec ulv. 0\<^sub>v (nr + 1 + (nc + nc) + nr) $ i \<le> ulv $ i)\<close>
thf(fact_1029_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_1030_minf_I8_J,axiom,
! [T2: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ~ ( ord_less_eq_a @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_1031_carrier__vecD,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( dim_vec_a @ V )
= N ) ) ).
% carrier_vecD
thf(fact_1032_index__zero__vec_I2_J,axiom,
! [N: nat] :
( ( dim_vec_a @ ( zero_vec_a @ N ) )
= N ) ).
% index_zero_vec(2)
thf(fact_1033_index__add__vec_I2_J,axiom,
! [V_1: vec_a,V_2: vec_a] :
( ( dim_vec_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) )
= ( dim_vec_a @ V_2 ) ) ).
% index_add_vec(2)
thf(fact_1034_index__uminus__vec_I2_J,axiom,
! [V: vec_a] :
( ( dim_vec_a @ ( uminus_uminus_vec_a @ V ) )
= ( dim_vec_a @ V ) ) ).
% index_uminus_vec(2)
thf(fact_1035_dim__vec__first,axiom,
! [V: vec_a,N: nat] :
( ( dim_vec_a @ ( vec_first_a @ V @ N ) )
= N ) ).
% dim_vec_first
thf(fact_1036_dim__vec__last,axiom,
! [V: vec_a,N: nat] :
( ( dim_vec_a @ ( vec_last_a @ V @ N ) )
= N ) ).
% dim_vec_last
thf(fact_1037_vec__of__dim__0,axiom,
! [V: vec_a] :
( ( ( dim_vec_a @ V )
= zero_zero_nat )
= ( V
= ( zero_vec_a @ zero_zero_nat ) ) ) ).
% vec_of_dim_0
thf(fact_1038_eq__vecI,axiom,
! [W: vec_a,V: vec_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_vec_a @ W ) )
=> ( ( vec_index_a @ V @ I2 )
= ( vec_index_a @ W @ I2 ) ) )
=> ( ( ( dim_vec_a @ V )
= ( dim_vec_a @ W ) )
=> ( V = W ) ) ) ).
% eq_vecI
thf(fact_1039_index__append__vec_I2_J,axiom,
! [V: vec_a,W: vec_a] :
( ( dim_vec_a @ ( append_vec_a @ V @ W ) )
= ( plus_plus_nat @ ( dim_vec_a @ V ) @ ( dim_vec_a @ W ) ) ) ).
% index_append_vec(2)
thf(fact_1040_index__zero__vec_I1_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_a @ ( zero_vec_a @ N ) @ I )
= zero_zero_a ) ) ).
% index_zero_vec(1)
thf(fact_1041_index__zero__vec_I1_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_nat @ ( zero_vec_nat @ N ) @ I )
= zero_zero_nat ) ) ).
% index_zero_vec(1)
thf(fact_1042_scalar__prod__uminus__left,axiom,
! [V: vec_a,W: vec_a] :
( ( ( dim_vec_a @ V )
= ( dim_vec_a @ W ) )
=> ( ( scalar_prod_a @ ( uminus_uminus_vec_a @ V ) @ W )
= ( uminus_uminus_a @ ( scalar_prod_a @ V @ W ) ) ) ) ).
% scalar_prod_uminus_left
thf(fact_1043_scalar__prod__uminus__right,axiom,
! [V: vec_a,W: vec_a] :
( ( ( dim_vec_a @ V )
= ( dim_vec_a @ W ) )
=> ( ( scalar_prod_a @ V @ ( uminus_uminus_vec_a @ W ) )
= ( uminus_uminus_a @ ( scalar_prod_a @ V @ W ) ) ) ) ).
% scalar_prod_uminus_right
thf(fact_1044_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_nat,V_1: vec_nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ V_2 ) )
=> ( ( vec_index_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ I )
= ( plus_plus_nat @ ( vec_index_nat @ V_1 @ I ) @ ( vec_index_nat @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1045_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_set_nat,V_1: vec_set_nat] :
( ( ord_less_nat @ I @ ( dim_vec_set_nat @ V_2 ) )
=> ( ( vec_index_set_nat @ ( plus_p6405046879299855402et_nat @ V_1 @ V_2 ) @ I )
= ( plus_plus_set_nat @ ( vec_index_set_nat @ V_1 @ I ) @ ( vec_index_set_nat @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1046_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_mat_a,V_1: vec_mat_a] :
( ( ord_less_nat @ I @ ( dim_vec_mat_a @ V_2 ) )
=> ( ( vec_index_mat_a @ ( plus_plus_vec_mat_a @ V_1 @ V_2 ) @ I )
= ( plus_plus_mat_a @ ( vec_index_mat_a @ V_1 @ I ) @ ( vec_index_mat_a @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1047_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_set_mat_a,V_1: vec_set_mat_a] :
( ( ord_less_nat @ I @ ( dim_vec_set_mat_a @ V_2 ) )
=> ( ( vec_index_set_mat_a @ ( plus_p8204160854631895605_mat_a @ V_1 @ V_2 ) @ I )
= ( plus_plus_set_mat_a @ ( vec_index_set_mat_a @ V_1 @ I ) @ ( vec_index_set_mat_a @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1048_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_vec_a,V_1: vec_vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_vec_a @ V_2 ) )
=> ( ( vec_index_vec_a @ ( plus_plus_vec_vec_a @ V_1 @ V_2 ) @ I )
= ( plus_plus_vec_a @ ( vec_index_vec_a @ V_1 @ I ) @ ( vec_index_vec_a @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1049_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_set_vec_a,V_1: vec_set_vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_set_vec_a @ V_2 ) )
=> ( ( vec_index_set_vec_a @ ( plus_p5241491716512693953_vec_a @ V_1 @ V_2 ) @ I )
= ( plus_plus_set_vec_a @ ( vec_index_set_vec_a @ V_1 @ I ) @ ( vec_index_set_vec_a @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1050_index__add__vec_I1_J,axiom,
! [I: nat,V_2: vec_a,V_1: vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_a @ V_2 ) )
=> ( ( vec_index_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ I )
= ( plus_plus_a @ ( vec_index_a @ V_1 @ I ) @ ( vec_index_a @ V_2 @ I ) ) ) ) ).
% index_add_vec(1)
thf(fact_1051_index__uminus__vec_I1_J,axiom,
! [I: nat,V: vec_vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_vec_a @ V ) )
=> ( ( vec_index_vec_a @ ( uminus8262787361227035083_vec_a @ V ) @ I )
= ( uminus_uminus_vec_a @ ( vec_index_vec_a @ V @ I ) ) ) ) ).
% index_uminus_vec(1)
thf(fact_1052_index__uminus__vec_I1_J,axiom,
! [I: nat,V: vec_mat_a] :
( ( ord_less_nat @ I @ ( dim_vec_mat_a @ V ) )
=> ( ( vec_index_mat_a @ ( uminus6789456888195538751_mat_a @ V ) @ I )
= ( uminus_uminus_mat_a @ ( vec_index_mat_a @ V @ I ) ) ) ) ).
% index_uminus_vec(1)
thf(fact_1053_index__uminus__vec_I1_J,axiom,
! [I: nat,V: vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_a @ V ) )
=> ( ( vec_index_a @ ( uminus_uminus_vec_a @ V ) @ I )
= ( uminus_uminus_a @ ( vec_index_a @ V @ I ) ) ) ) ).
% index_uminus_vec(1)
thf(fact_1054_vec__eq__iff,axiom,
( ( ^ [Y5: vec_a,Z2: vec_a] : ( Y5 = Z2 ) )
= ( ^ [X2: vec_a,Y2: vec_a] :
( ( ( dim_vec_a @ X2 )
= ( dim_vec_a @ Y2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_a @ Y2 ) )
=> ( ( vec_index_a @ X2 @ I4 )
= ( vec_index_a @ Y2 @ I4 ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_1055_less__eq__vec__def,axiom,
( ord_le4012615358376148468_vec_a
= ( ^ [V4: vec_vec_a,W4: vec_vec_a] :
( ( ( dim_vec_vec_a @ V4 )
= ( dim_vec_vec_a @ W4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_vec_a @ W4 ) )
=> ( ord_less_eq_vec_a @ ( vec_index_vec_a @ V4 @ I4 ) @ ( vec_index_vec_a @ W4 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_1056_less__eq__vec__def,axiom,
( ord_less_eq_vec_nat
= ( ^ [V4: vec_nat,W4: vec_nat] :
( ( ( dim_vec_nat @ V4 )
= ( dim_vec_nat @ W4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_nat @ W4 ) )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V4 @ I4 ) @ ( vec_index_nat @ W4 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_1057_less__eq__vec__def,axiom,
( ord_less_eq_vec_a
= ( ^ [V4: vec_a,W4: vec_a] :
( ( ( dim_vec_a @ V4 )
= ( dim_vec_a @ W4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_a @ W4 ) )
=> ( ord_less_eq_a @ ( vec_index_a @ V4 @ I4 ) @ ( vec_index_a @ W4 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_1058_vec__first__index,axiom,
! [N: nat,V: vec_a,I: nat] :
( ( ord_less_eq_nat @ N @ ( dim_vec_a @ V ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( vec_index_a @ ( vec_first_a @ V @ N ) @ I )
= ( vec_index_a @ V @ I ) ) ) ) ).
% vec_first_index
thf(fact_1059_carrier__vec__dim__vec,axiom,
! [V: vec_a] : ( member_vec_a @ V @ ( carrier_vec_a @ ( dim_vec_a @ V ) ) ) ).
% carrier_vec_dim_vec
thf(fact_1060_carrier__dim__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
= ( ( dim_vec_a @ V )
= N ) ) ).
% carrier_dim_vec
thf(fact_1061_carrier__vecI,axiom,
! [V: vec_a,N: nat] :
( ( ( dim_vec_a @ V )
= N )
=> ( member_vec_a @ V @ ( carrier_vec_a @ N ) ) ) ).
% carrier_vecI
thf(fact_1062_vec__add__mono,axiom,
! [B: vec_a,D2: vec_a,A: vec_a,C: vec_a] :
( ( ( dim_vec_a @ B )
= ( dim_vec_a @ D2 ) )
=> ( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ C @ D2 )
=> ( ord_less_eq_vec_a @ ( plus_plus_vec_a @ A @ C ) @ ( plus_plus_vec_a @ B @ D2 ) ) ) ) ) ).
% vec_add_mono
thf(fact_1063_vec__first__add,axiom,
! [N: nat,X: vec_a,Y: vec_a] :
( ( ord_less_eq_nat @ N @ ( dim_vec_a @ X ) )
=> ( ( ord_less_eq_nat @ N @ ( dim_vec_a @ Y ) )
=> ( ( vec_first_a @ ( plus_plus_vec_a @ X @ Y ) @ N )
= ( plus_plus_vec_a @ ( vec_first_a @ X @ N ) @ ( vec_first_a @ Y @ N ) ) ) ) ) ).
% vec_first_add
thf(fact_1064_mat__of__col__dim_I3_J,axiom,
! [V: vec_a] : ( member_mat_a @ ( missing_mat_of_col_a @ V ) @ ( carrier_mat_a @ ( dim_vec_a @ V ) @ one_one_nat ) ) ).
% mat_of_col_dim(3)
thf(fact_1065_lesseq__vecI,axiom,
! [V: vec_vec_a,N: nat,W: vec_vec_a] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N ) )
=> ( ( member_vec_vec_a @ W @ ( carrier_vec_vec_a @ N ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_vec_a @ ( vec_index_vec_a @ V @ I2 ) @ ( vec_index_vec_a @ W @ I2 ) ) )
=> ( ord_le4012615358376148468_vec_a @ V @ W ) ) ) ) ).
% lesseq_vecI
thf(fact_1066_lesseq__vecI,axiom,
! [V: vec_nat,N: nat,W: vec_nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ W @ ( carrier_vec_nat @ N ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V @ I2 ) @ ( vec_index_nat @ W @ I2 ) ) )
=> ( ord_less_eq_vec_nat @ V @ W ) ) ) ) ).
% lesseq_vecI
thf(fact_1067_lesseq__vecI,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_a @ ( vec_index_a @ V @ I2 ) @ ( vec_index_a @ W @ I2 ) ) )
=> ( ord_less_eq_vec_a @ V @ W ) ) ) ) ).
% lesseq_vecI
thf(fact_1068_lesseq__vecD,axiom,
! [W: vec_vec_a,N: nat,V: vec_vec_a,I: nat] :
( ( member_vec_vec_a @ W @ ( carrier_vec_vec_a @ N ) )
=> ( ( ord_le4012615358376148468_vec_a @ V @ W )
=> ( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_vec_a @ ( vec_index_vec_a @ V @ I ) @ ( vec_index_vec_a @ W @ I ) ) ) ) ) ).
% lesseq_vecD
thf(fact_1069_lesseq__vecD,axiom,
! [W: vec_nat,N: nat,V: vec_nat,I: nat] :
( ( member_vec_nat @ W @ ( carrier_vec_nat @ N ) )
=> ( ( ord_less_eq_vec_nat @ V @ W )
=> ( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V @ I ) @ ( vec_index_nat @ W @ I ) ) ) ) ) ).
% lesseq_vecD
thf(fact_1070_lesseq__vecD,axiom,
! [W: vec_a,N: nat,V: vec_a,I: nat] :
( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_eq_vec_a @ V @ W )
=> ( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_a @ ( vec_index_a @ V @ I ) @ ( vec_index_a @ W @ I ) ) ) ) ) ).
% lesseq_vecD
thf(fact_1071_vec__last__index,axiom,
! [V: vec_a,N: nat,M: nat,I: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ I @ M )
=> ( ( vec_index_a @ ( vec_last_a @ V @ M ) @ I )
= ( vec_index_a @ V @ ( plus_plus_nat @ N @ I ) ) ) ) ) ).
% vec_last_index
thf(fact_1072_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_1073_pinf_I6_J,axiom,
! [T2: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_1074_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_1075_pinf_I8_J,axiom,
! [T2: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ord_less_eq_a @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_1076_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_1077_minf_I6_J,axiom,
! [T2: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ord_less_eq_a @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_1078_index__update__vec1,axiom,
! [I: nat,V: vec_a,A: a] :
( ( ord_less_nat @ I @ ( dim_vec_a @ V ) )
=> ( ( vec_index_a @ ( update_vec_a @ V @ I @ A ) @ I )
= A ) ) ).
% index_update_vec1
thf(fact_1079_index__component__mult,axiom,
! [I: nat,V: vec_a,W: vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_a @ V ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_a @ W ) )
=> ( ( vec_index_a @ ( component_mult_a @ V @ W ) @ I )
= ( times_times_a @ ( vec_index_a @ V @ I ) @ ( vec_index_a @ W @ I ) ) ) ) ) ).
% index_component_mult
thf(fact_1080_index__component__mult,axiom,
! [I: nat,V: vec_nat,W: vec_nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ V ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_nat @ W ) )
=> ( ( vec_index_nat @ ( component_mult_nat @ V @ W ) @ I )
= ( times_times_nat @ ( vec_index_nat @ V @ I ) @ ( vec_index_nat @ W @ I ) ) ) ) ) ).
% index_component_mult
thf(fact_1081_vec__of__scal__dim__1,axiom,
! [V: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ one_one_nat ) )
= ( V
= ( missin5951511974119752530scal_a @ ( vec_index_a @ V @ zero_zero_nat ) ) ) ) ).
% vec_of_scal_dim_1
thf(fact_1082_dim__update__vec,axiom,
! [V: vec_a,I: nat,A: a] :
( ( dim_vec_a @ ( update_vec_a @ V @ I @ A ) )
= ( dim_vec_a @ V ) ) ).
% dim_update_vec
thf(fact_1083_index__update__vec2,axiom,
! [I5: nat,I: nat,V: vec_a,A: a] :
( ( I5 != I )
=> ( ( vec_index_a @ ( update_vec_a @ V @ I @ A ) @ I5 )
= ( vec_index_a @ V @ I5 ) ) ) ).
% index_update_vec2
thf(fact_1084_index__vec__of__scal,axiom,
! [X: a] :
( ( vec_index_a @ ( missin5951511974119752530scal_a @ X ) @ zero_zero_nat )
= X ) ).
% index_vec_of_scal
thf(fact_1085_vec__of__scal__dim_I1_J,axiom,
! [X: a] :
( ( dim_vec_a @ ( missin5951511974119752530scal_a @ X ) )
= one_one_nat ) ).
% vec_of_scal_dim(1)
thf(fact_1086_vec__of__scal__dim_I2_J,axiom,
! [X: a] : ( member_vec_a @ ( missin5951511974119752530scal_a @ X ) @ ( carrier_vec_a @ one_one_nat ) ) ).
% vec_of_scal_dim(2)
thf(fact_1087_row__mat__of__col,axiom,
! [I: nat,V: vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_a @ V ) )
=> ( ( row_a @ ( missing_mat_of_col_a @ V ) @ I )
= ( missin5951511974119752530scal_a @ ( vec_index_a @ V @ I ) ) ) ) ).
% row_mat_of_col
thf(fact_1088_index__append__vec_I1_J,axiom,
! [I: nat,V: vec_a,W: vec_a] :
( ( ord_less_nat @ I @ ( plus_plus_nat @ ( dim_vec_a @ V ) @ ( dim_vec_a @ W ) ) )
=> ( ( ( ord_less_nat @ I @ ( dim_vec_a @ V ) )
=> ( ( vec_index_a @ ( append_vec_a @ V @ W ) @ I )
= ( vec_index_a @ V @ I ) ) )
& ( ~ ( ord_less_nat @ I @ ( dim_vec_a @ V ) )
=> ( ( vec_index_a @ ( append_vec_a @ V @ W ) @ I )
= ( vec_index_a @ W @ ( minus_minus_nat @ I @ ( dim_vec_a @ V ) ) ) ) ) ) ) ).
% index_append_vec(1)
thf(fact_1089_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_1090_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1091_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_diff__ge__0__iff__ge,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1099_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_1100_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_1101_le__add__diff__inverse2,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1102_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_1103_le__add__diff__inverse,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( plus_plus_a @ B @ ( minus_minus_a @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1104_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_1105_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_1106_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1107_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_1108_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_1109_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_1110_row__mat__of__row,axiom,
! [Y: vec_a] :
( ( row_a @ ( mat_of_row_a @ Y ) @ zero_zero_nat )
= Y ) ).
% row_mat_of_row
thf(fact_1111_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1112_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1113_index__minus__vec_I1_J,axiom,
! [I: nat,V_2: vec_a,V_1: vec_a] :
( ( ord_less_nat @ I @ ( dim_vec_a @ V_2 ) )
=> ( ( vec_index_a @ ( minus_minus_vec_a @ V_1 @ V_2 ) @ I )
= ( minus_minus_a @ ( vec_index_a @ V_1 @ I ) @ ( vec_index_a @ V_2 @ I ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1114_index__minus__vec_I1_J,axiom,
! [I: nat,V_2: vec_nat,V_1: vec_nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ V_2 ) )
=> ( ( vec_index_nat @ ( minus_minus_vec_nat @ V_1 @ V_2 ) @ I )
= ( minus_minus_nat @ ( vec_index_nat @ V_1 @ I ) @ ( vec_index_nat @ V_2 @ I ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1115_row__zero,axiom,
! [I: nat,Nr: nat,Nc: nat] :
( ( ord_less_nat @ I @ Nr )
=> ( ( row_a @ ( zero_mat_a @ Nr @ Nc ) @ I )
= ( zero_vec_a @ Nc ) ) ) ).
% row_zero
thf(fact_1116_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1117_row__add_I1_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,I: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I @ Nr )
=> ( ( row_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ I )
= ( plus_plus_vec_a @ ( row_a @ A2 @ I ) @ ( row_a @ B2 @ I ) ) ) ) ) ) ).
% row_add(1)
thf(fact_1118_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_1119_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_1120_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_1121_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_1122_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1123_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_1124_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_1125_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_1126_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1127_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1128_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_1129_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_1130_scalar__prod__minus__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ ( minus_minus_vec_a @ V_2 @ V_3 ) )
= ( minus_minus_a @ ( scalar_prod_a @ V_1 @ V_2 ) @ ( scalar_prod_a @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_minus_distrib
thf(fact_1131_minus__scalar__prod__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( minus_minus_vec_a @ V_1 @ V_2 ) @ V_3 )
= ( minus_minus_a @ ( scalar_prod_a @ V_1 @ V_3 ) @ ( scalar_prod_a @ V_2 @ V_3 ) ) ) ) ) ) ).
% minus_scalar_prod_distrib
thf(fact_1132_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_1133_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_1134_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_1135_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_1136_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_1137_diff__mono,axiom,
! [A: a,B: a,D2: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ D2 @ C )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1138_diff__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1139_diff__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1140_diff__eq__diff__less__eq,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D2 ) )
=> ( ( ord_less_eq_a @ A @ B )
= ( ord_less_eq_a @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1141_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_1142_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_1143_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1144_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_1145_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_1146_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_1147_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_1148_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_le__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ ( minus_minus_a @ C @ B ) )
= ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1160_diff__le__eq,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ ( minus_minus_a @ A @ B ) @ C )
= ( ord_less_eq_a @ A @ ( plus_plus_a @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1161_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_1162_add__le__add__imp__diff__le,axiom,
! [I: a,K: a,N: a,J: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ( ord_less_eq_a @ N @ ( plus_plus_a @ J @ K ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ( ord_less_eq_a @ N @ ( plus_plus_a @ J @ K ) )
=> ( ord_less_eq_a @ ( minus_minus_a @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1163_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_1164_add__le__imp__le__diff,axiom,
! [I: a,K: a,N: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ord_less_eq_a @ I @ ( minus_minus_a @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1165_le__iff__diff__le__0,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B4: a] : ( ord_less_eq_a @ ( minus_minus_a @ A4 @ B4 ) @ zero_zero_a ) ) ) ).
% le_iff_diff_le_0
thf(fact_1166_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_1167_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_1168_row__four__block__mat_I2_J,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D: mat_a,I: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ~ ( ord_less_nat @ I @ Nr1 )
=> ( ( ord_less_nat @ I @ ( plus_plus_nat @ Nr1 @ Nr2 ) )
=> ( ( row_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) @ I )
= ( append_vec_a @ ( row_a @ C2 @ ( minus_minus_nat @ I @ Nr1 ) ) @ ( row_a @ D @ ( minus_minus_nat @ I @ Nr1 ) ) ) ) ) ) ) ) ) ) ).
% row_four_block_mat(2)
thf(fact_1169_append__rows__nth_I2_J,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat,I: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( ord_less_eq_nat @ Nr1 @ I )
=> ( ( ord_less_nat @ I @ ( plus_plus_nat @ Nr1 @ Nr2 ) )
=> ( ( row_a @ ( append_rows_a @ A2 @ B2 ) @ I )
= ( row_a @ B2 @ ( minus_minus_nat @ I @ Nr1 ) ) ) ) ) ) ) ).
% append_rows_nth(2)
thf(fact_1170_ordered__ring__class_Ole__add__iff1,axiom,
! [A: a,E2: a,C: a,B: a,D2: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1171_ordered__ring__class_Ole__add__iff2,axiom,
! [A: a,E2: a,C: a,B: a,D2: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_a @ C @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1172_row__carrier__vec,axiom,
! [I: nat,Nr: nat,A2: mat_a,Nc: nat] :
( ( ord_less_nat @ I @ Nr )
=> ( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_vec_a @ ( row_a @ A2 @ I ) @ ( carrier_vec_a @ Nc ) ) ) ) ).
% row_carrier_vec
thf(fact_1173_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1174_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1175_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_1176_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_1177_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_1178_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_1179_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_1180_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_1181_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_1182_append__rows__nth_I1_J,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat,I: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( ord_less_nat @ I @ Nr1 )
=> ( ( row_a @ ( append_rows_a @ A2 @ B2 ) @ I )
= ( row_a @ A2 @ I ) ) ) ) ) ).
% append_rows_nth(1)
thf(fact_1183_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1184_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1185_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1186_row__four__block__mat_I1_J,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D: mat_a,I: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( ord_less_nat @ I @ Nr1 )
=> ( ( row_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) @ I )
= ( append_vec_a @ ( row_a @ A2 @ I ) @ ( row_a @ B2 @ I ) ) ) ) ) ) ) ) ).
% row_four_block_mat(1)
thf(fact_1187_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_1188_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_1189_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1190_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I4: nat,J3: nat,P4: nat > nat,I6: nat] : ( if_nat @ ( ord_less_nat @ I6 @ I4 ) @ ( if_nat @ ( ord_less_nat @ ( P4 @ I6 ) @ J3 ) @ ( P4 @ I6 ) @ ( suc @ ( P4 @ I6 ) ) ) @ ( if_nat @ ( I6 = I4 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P4 @ ( minus_minus_nat @ I6 @ one_one_nat ) ) @ J3 ) @ ( P4 @ ( minus_minus_nat @ I6 @ one_one_nat ) ) @ ( suc @ ( P4 @ ( minus_minus_nat @ I6 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1191_minus__r__inv__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A2 @ A2 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% minus_r_inv_mat
thf(fact_1192_index__minus__vec_I2_J,axiom,
! [V_1: vec_a,V_2: vec_a] :
( ( dim_vec_a @ ( minus_minus_vec_a @ V_1 @ V_2 ) )
= ( dim_vec_a @ V_2 ) ) ).
% index_minus_vec(2)
thf(fact_1193_minus__zero__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ V @ ( zero_vec_a @ N ) )
= V ) ) ).
% minus_zero_vec
thf(fact_1194_minus__cancel__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ V @ V )
= ( zero_vec_a @ N ) ) ) ).
% minus_cancel_vec
thf(fact_1195_zero__minus__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ ( zero_vec_a @ N ) @ V )
= ( uminus_uminus_vec_a @ V ) ) ) ).
% zero_minus_vec
thf(fact_1196_add__diff__eq__vec,axiom,
! [Y: vec_a,N: nat,X: vec_a,Z: vec_a] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ Z @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ Y @ ( minus_minus_vec_a @ X @ Z ) )
= ( minus_minus_vec_a @ ( plus_plus_vec_a @ Y @ X ) @ Z ) ) ) ) ) ).
% add_diff_eq_vec
thf(fact_1197_add__diff__cancel__right__vec,axiom,
! [A: vec_a,N: nat,B: vec_a] :
( ( member_vec_a @ A @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ ( plus_plus_vec_a @ A @ B ) @ B )
= A ) ) ) ).
% add_diff_cancel_right_vec
thf(fact_1198_minus__add__minus__vec,axiom,
! [U: vec_a,N: nat,V: vec_a,W: vec_a] :
( ( member_vec_a @ U @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ U @ ( plus_plus_vec_a @ V @ W ) )
= ( minus_minus_vec_a @ ( minus_minus_vec_a @ U @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_vec
thf(fact_1199_transpose__minus,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( minus_minus_mat_a @ A2 @ B2 ) )
= ( minus_minus_mat_a @ ( transpose_mat_a @ A2 ) @ ( transpose_mat_a @ B2 ) ) ) ) ) ).
% transpose_minus
thf(fact_1200_minus__add__minus__mat,axiom,
! [U: mat_a,Nr: nat,Nc: nat,V: mat_a,W: mat_a] :
( ( member_mat_a @ U @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ V @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ W @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ U @ ( plus_plus_mat_a @ V @ W ) )
= ( minus_minus_mat_a @ ( minus_minus_mat_a @ U @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_1201_minus__mult__distrib__mat,axiom,
! [A2: mat_a,Nr: nat,N: nat,B2: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( minus_minus_mat_a @ A2 @ B2 ) @ C2 )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A2 @ C2 ) @ ( times_times_mat_a @ B2 @ C2 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_1202_mult__minus__distrib__mat,axiom,
! [A2: mat_a,Nr: nat,N: nat,B2: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A2 @ ( minus_minus_mat_a @ B2 @ C2 ) )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A2 @ B2 ) @ ( times_times_mat_a @ A2 @ C2 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_1203_minus__mult__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( minus_minus_mat_a @ A2 @ B2 ) @ V )
= ( minus_minus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% minus_mult_distrib_mat_vec
thf(fact_1204_minus__carrier__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( minus_minus_vec_a @ V_1 @ V_2 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% minus_carrier_vec
thf(fact_1205_minus__carrier__mat,axiom,
! [B2: mat_a,Nr: nat,Nc: nat,A2: mat_a] :
( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_1206_mult__minus__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a,W: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ A2 @ ( minus_minus_vec_a @ V @ W ) )
= ( minus_minus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ A2 @ W ) ) ) ) ) ) ).
% mult_minus_distrib_mat_vec
thf(fact_1207_uminus__add__minus__vec,axiom,
! [L: vec_a,N: nat,R: vec_a] :
( ( member_vec_a @ L @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ R @ ( carrier_vec_a @ N ) )
=> ( ( uminus_uminus_vec_a @ ( plus_plus_vec_a @ L @ R ) )
= ( minus_minus_vec_a @ ( uminus_uminus_vec_a @ L ) @ R ) ) ) ) ).
% uminus_add_minus_vec
thf(fact_1208_minus__add__uminus__vec,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ V @ W )
= ( plus_plus_vec_a @ V @ ( uminus_uminus_vec_a @ W ) ) ) ) ) ).
% minus_add_uminus_vec
thf(fact_1209_vec__le__iff__diff__le__0,axiom,
( ord_less_eq_vec_a
= ( ^ [A4: vec_a,B4: vec_a] : ( ord_less_eq_vec_a @ ( minus_minus_vec_a @ A4 @ B4 ) @ ( zero_vec_a @ ( dim_vec_a @ A4 ) ) ) ) ) ).
% vec_le_iff_diff_le_0
thf(fact_1210_add__uminus__minus__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A2 @ ( uminus_uminus_mat_a @ B2 ) )
= ( minus_minus_mat_a @ A2 @ B2 ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_1211_minus__add__uminus__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A2 @ B2 )
= ( plus_plus_mat_a @ A2 @ ( uminus_uminus_mat_a @ B2 ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_1212_uminus__add__minus__mat,axiom,
! [L: mat_a,Nr: nat,Nc: nat,R: mat_a] :
( ( member_mat_a @ L @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ R @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ L @ R ) )
= ( minus_minus_mat_a @ ( uminus_uminus_mat_a @ L ) @ R ) ) ) ) ).
% uminus_add_minus_mat
thf(fact_1213_row__echelon__form__imp__1__or__0__row,axiom,
! [A2: mat_a,N: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( gauss_5855338539171749649form_a @ A2 )
=> ( ( A2
= ( one_mat_a @ N ) )
| ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( row_a @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= ( zero_vec_a @ N ) ) ) ) ) ) ).
% row_echelon_form_imp_1_or_0_row
thf(fact_1214_row__echelon__form__one__dim0,axiom,
gauss_5855338539171749649form_a @ ( one_mat_a @ zero_zero_nat ) ).
% row_echelon_form_one_dim0
thf(fact_1215_row__echelon__form__dim0__col,axiom,
! [A2: mat_a,N: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ zero_zero_nat ) )
=> ( gauss_5855338539171749649form_a @ A2 ) ) ).
% row_echelon_form_dim0_col
thf(fact_1216_row__echelon__form__dim0__row,axiom,
! [A2: mat_a,N: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ zero_zero_nat @ N ) )
=> ( gauss_5855338539171749649form_a @ A2 ) ) ).
% row_echelon_form_dim0_row
thf(fact_1217_find__base__vector__not__1_I3_J,axiom,
! [A2: mat_a,N: nat] :
( ( gauss_5855338539171749649form_a @ A2 )
=> ( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A2
!= ( one_mat_a @ N ) )
=> ( ( mult_mat_vec_a @ A2 @ ( gauss_6280258074615264798ctor_a @ A2 ) )
= ( zero_vec_a @ N ) ) ) ) ) ).
% find_base_vector_not_1(3)
thf(fact_1218_gauss__jordan__single_I1_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,C2: mat_a,X: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A2 )
= C2 )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( ( mult_mat_vec_a @ A2 @ X )
= ( zero_vec_a @ Nr ) )
= ( ( mult_mat_vec_a @ C2 @ X )
= ( zero_vec_a @ Nr ) ) ) ) ) ) ).
% gauss_jordan_single(1)
thf(fact_1219_gauss__jordan__single_I2_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A2 )
= C2 )
=> ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% gauss_jordan_single(2)
thf(fact_1220_gauss__jordan__single_I3_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A2 )
= C2 )
=> ( gauss_5855338539171749649form_a @ C2 ) ) ) ).
% gauss_jordan_single(3)
thf(fact_1221_gauss__jordan__single_I4_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A2 )
= C2 )
=> ? [P5: mat_a,Q3: mat_a] :
( ( C2
= ( times_times_mat_a @ P5 @ A2 ) )
& ( member_mat_a @ P5 @ ( carrier_mat_a @ Nr @ Nr ) )
& ( member_mat_a @ Q3 @ ( carrier_mat_a @ Nr @ Nr ) )
& ( ( times_times_mat_a @ P5 @ Q3 )
= ( one_mat_a @ Nr ) )
& ( ( times_times_mat_a @ Q3 @ P5 )
= ( one_mat_a @ Nr ) ) ) ) ) ).
% gauss_jordan_single(4)
thf(fact_1222_find__base__vector__not__1_I1_J,axiom,
! [A2: mat_a,N: nat] :
( ( gauss_5855338539171749649form_a @ A2 )
=> ( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A2
!= ( one_mat_a @ N ) )
=> ( member_vec_a @ ( gauss_6280258074615264798ctor_a @ A2 ) @ ( carrier_vec_a @ N ) ) ) ) ) ).
% find_base_vector_not_1(1)
thf(fact_1223_find__base__vector__not__1_I2_J,axiom,
! [A2: mat_a,N: nat] :
( ( gauss_5855338539171749649form_a @ A2 )
=> ( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A2
!= ( one_mat_a @ N ) )
=> ( ( gauss_6280258074615264798ctor_a @ A2 )
!= ( zero_vec_a @ N ) ) ) ) ) ).
% find_base_vector_not_1(2)
thf(fact_1224_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D5: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D5 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1225_addrow__mat__inv,axiom,
! [K: nat,N: nat,L: nat,A: a] :
( ( ord_less_nat @ K @ N )
=> ( ( ord_less_nat @ L @ N )
=> ( ( K != L )
=> ( ( times_times_mat_a @ ( gauss_8159914756388622152_mat_a @ N @ A @ K @ L ) @ ( gauss_8159914756388622152_mat_a @ N @ ( uminus_uminus_a @ A ) @ K @ L ) )
= ( one_mat_a @ N ) ) ) ) ) ).
% addrow_mat_inv
thf(fact_1226_index__mult__mat__vec,axiom,
! [I: nat,A2: mat_a,V: vec_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( vec_index_a @ ( mult_mat_vec_a @ A2 @ V ) @ I )
= ( scalar_prod_a @ ( row_a @ A2 @ I ) @ V ) ) ) ).
% index_mult_mat_vec
thf(fact_1227_carrier__matD_I1_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_a @ A2 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_1228_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_a @ ( one_mat_a @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_1229_index__zero__mat_I2_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_row_a @ ( zero_mat_a @ Nr @ Nc ) )
= Nr ) ).
% index_zero_mat(2)
thf(fact_1230_index__add__mat_I2_J,axiom,
! [A2: mat_a,B2: mat_a] :
( ( dim_row_a @ ( plus_plus_mat_a @ A2 @ B2 ) )
= ( dim_row_a @ B2 ) ) ).
% index_add_mat(2)
thf(fact_1231_index__uminus__mat_I2_J,axiom,
! [A2: mat_a] :
( ( dim_row_a @ ( uminus_uminus_mat_a @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_uminus_mat(2)
thf(fact_1232_index__mat__four__block_I2_J,axiom,
! [A2: mat_a,B2: mat_a,C2: mat_a,D: mat_a] :
( ( dim_row_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) )
= ( plus_plus_nat @ ( dim_row_a @ A2 ) @ ( dim_row_a @ D ) ) ) ).
% index_mat_four_block(2)
thf(fact_1233_left__mult__one__mat_H,axiom,
! [A2: mat_a,N: nat] :
( ( ( dim_row_a @ A2 )
= N )
=> ( ( times_times_mat_a @ ( one_mat_a @ N ) @ A2 )
= A2 ) ) ).
% left_mult_one_mat'
thf(fact_1234_dim__mult__mat__vec,axiom,
! [A2: mat_a,V: vec_a] :
( ( dim_vec_a @ ( mult_mat_vec_a @ A2 @ V ) )
= ( dim_row_a @ A2 ) ) ).
% dim_mult_mat_vec
thf(fact_1235_mat__of__row__dim_I1_J,axiom,
! [Y: vec_a] :
( ( dim_row_a @ ( mat_of_row_a @ Y ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_1236_mat__of__col__dim_I1_J,axiom,
! [V: vec_a] :
( ( dim_row_a @ ( missing_mat_of_col_a @ V ) )
= ( dim_vec_a @ V ) ) ).
% mat_of_col_dim(1)
thf(fact_1237_row__uminus,axiom,
! [I: nat,A2: mat_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( row_a @ ( uminus_uminus_mat_a @ A2 ) @ I )
= ( uminus_uminus_vec_a @ ( row_a @ A2 @ I ) ) ) ) ).
% row_uminus
thf(fact_1238_addrow__mat__carrier,axiom,
! [N: nat,A: a,K: nat,L: nat] : ( member_mat_a @ ( gauss_8159914756388622152_mat_a @ N @ A @ K @ L ) @ ( carrier_mat_a @ N @ N ) ) ).
% addrow_mat_carrier
thf(fact_1239_mat__of__row__dim__row__1,axiom,
! [A2: mat_a] :
( ( ( dim_row_a @ A2 )
= one_one_nat )
= ( A2
= ( mat_of_row_a @ ( row_a @ A2 @ zero_zero_nat ) ) ) ) ).
% mat_of_row_dim_row_1
thf(fact_1240_append__rows__def,axiom,
( append_rows_a
= ( ^ [A5: mat_a,B5: mat_a] : ( four_block_mat_a @ A5 @ ( zero_mat_a @ ( dim_row_a @ A5 ) @ zero_zero_nat ) @ B5 @ ( zero_mat_a @ ( dim_row_a @ B5 ) @ zero_zero_nat ) ) ) ) ).
% append_rows_def
thf(fact_1241_mat__row__first__last__append,axiom,
! [A2: mat_a,M: nat,N: nat] :
( ( ( dim_row_a @ A2 )
= ( plus_plus_nat @ M @ N ) )
=> ( ( append_rows_a @ ( missin3040492613037353666irst_a @ A2 @ M ) @ ( missin5577565584678110354last_a @ A2 @ N ) )
= A2 ) ) ).
% mat_row_first_last_append
thf(fact_1242_addrow__mat,axiom,
! [A2: mat_a,N: nat,Nc: nat,L: nat,A: a,K: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A2 )
= ( times_times_mat_a @ ( gauss_8159914756388622152_mat_a @ N @ A @ K @ L ) @ A2 ) ) ) ) ).
% addrow_mat
thf(fact_1243_addrow__mat,axiom,
! [A2: mat_nat,N: nat,Nc: nat,L: nat,A: nat,K: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A2 )
= ( times_times_mat_nat @ ( gauss_6496870380031412486at_nat @ N @ A @ K @ L ) @ A2 ) ) ) ) ).
% addrow_mat
thf(fact_1244_addrow__carrier,axiom,
! [Ad: a > a > a,Mul: a > a > a,A: a,K: nat,L: nat,A2: mat_a,N: nat,Nc: nat] :
( ( member_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A2 ) @ ( carrier_mat_a @ N @ Nc ) )
= ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ Nc ) ) ) ).
% addrow_carrier
thf(fact_1245_mat__delete__carrier,axiom,
! [A2: mat_a,M: nat,N: nat,I: nat,J: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ M @ N ) )
=> ( member_mat_a @ ( mat_delete_a @ A2 @ I @ J ) @ ( carrier_mat_a @ ( minus_minus_nat @ M @ one_one_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% mat_delete_carrier
thf(fact_1246_inverts__mat__def,axiom,
( inverts_mat_a
= ( ^ [A5: mat_a,B5: mat_a] :
( ( times_times_mat_a @ A5 @ B5 )
= ( one_mat_a @ ( dim_row_a @ A5 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_1247_system__if__mat__kernel,axiom,
! [Q: mat_a,V: vec_a] :
( ( ( dim_row_a @ Q )
= ( dim_col_a @ Q ) )
=> ( ( ( dim_row_a @ Q )
= ( dim_vec_a @ V ) )
=> ( ( ( mult_mat_vec_a @ ( transpose_mat_a @ Q ) @ V )
= V )
= ( member_vec_a @ V @ ( matrix_mat_kernel_a @ ( transpose_mat_a @ ( minus_minus_mat_a @ Q @ ( one_mat_a @ ( dim_row_a @ Q ) ) ) ) ) ) ) ) ) ).
% system_if_mat_kernel
thf(fact_1248_carrier__matD_I2_J,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_col_a @ A2 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_1249_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_a @ ( one_mat_a @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_1250_index__zero__mat_I3_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_col_a @ ( zero_mat_a @ Nr @ Nc ) )
= Nc ) ).
% index_zero_mat(3)
thf(fact_1251_index__add__mat_I3_J,axiom,
! [A2: mat_a,B2: mat_a] :
( ( dim_col_a @ ( plus_plus_mat_a @ A2 @ B2 ) )
= ( dim_col_a @ B2 ) ) ).
% index_add_mat(3)
thf(fact_1252_index__uminus__mat_I3_J,axiom,
! [A2: mat_a] :
( ( dim_col_a @ ( uminus_uminus_mat_a @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_uminus_mat(3)
thf(fact_1253_carrier__matI,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_a @ A2 )
= Nr )
=> ( ( ( dim_col_a @ A2 )
= Nc )
=> ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_1254_index__mat__four__block_I3_J,axiom,
! [A2: mat_a,B2: mat_a,C2: mat_a,D: mat_a] :
( ( dim_col_a @ ( four_block_mat_a @ A2 @ B2 @ C2 @ D ) )
= ( plus_plus_nat @ ( dim_col_a @ A2 ) @ ( dim_col_a @ D ) ) ) ).
% index_mat_four_block(3)
thf(fact_1255_right__mult__one__mat_H,axiom,
! [A2: mat_a,N: nat] :
( ( ( dim_col_a @ A2 )
= N )
=> ( ( times_times_mat_a @ A2 @ ( one_mat_a @ N ) )
= A2 ) ) ).
% right_mult_one_mat'
thf(fact_1256_index__transpose__mat_I3_J,axiom,
! [A2: mat_a] :
( ( dim_col_a @ ( transpose_mat_a @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_transpose_mat(3)
thf(fact_1257_index__transpose__mat_I2_J,axiom,
! [A2: mat_a] :
( ( dim_row_a @ ( transpose_mat_a @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_transpose_mat(2)
thf(fact_1258_index__row_I2_J,axiom,
! [A2: mat_a,I: nat] :
( ( dim_vec_a @ ( row_a @ A2 @ I ) )
= ( dim_col_a @ A2 ) ) ).
% index_row(2)
thf(fact_1259_mat__of__row__dim_I2_J,axiom,
! [Y: vec_a] :
( ( dim_col_a @ ( mat_of_row_a @ Y ) )
= ( dim_vec_a @ Y ) ) ).
% mat_of_row_dim(2)
thf(fact_1260_right__mult__zero__mat_H,axiom,
! [A2: mat_a,N: nat,Nc: nat] :
( ( ( dim_col_a @ A2 )
= N )
=> ( ( times_times_mat_a @ A2 @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ ( dim_row_a @ A2 ) @ Nc ) ) ) ).
% right_mult_zero_mat'
thf(fact_1261_left__mult__zero__mat_H,axiom,
! [A2: mat_a,N: nat,Nr: nat] :
( ( ( dim_row_a @ A2 )
= N )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A2 )
= ( zero_mat_a @ Nr @ ( dim_col_a @ A2 ) ) ) ) ).
% left_mult_zero_mat'
thf(fact_1262_uminus__mult__left__mat,axiom,
! [A2: mat_a,B2: mat_a] :
( ( ( dim_col_a @ A2 )
= ( dim_row_a @ B2 ) )
=> ( ( times_times_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ B2 )
= ( uminus_uminus_mat_a @ ( times_times_mat_a @ A2 @ B2 ) ) ) ) ).
% uminus_mult_left_mat
thf(fact_1263_uminus__mult__right__mat,axiom,
! [A2: mat_a,B2: mat_a] :
( ( ( dim_col_a @ A2 )
= ( dim_row_a @ B2 ) )
=> ( ( times_times_mat_a @ A2 @ ( uminus_uminus_mat_a @ B2 ) )
= ( uminus_uminus_mat_a @ ( times_times_mat_a @ A2 @ B2 ) ) ) ) ).
% uminus_mult_right_mat
thf(fact_1264_uminus__mult__mat__vec,axiom,
! [V: vec_a,A2: mat_a] :
( ( ( dim_vec_a @ V )
= ( dim_col_a @ A2 ) )
=> ( ( mult_mat_vec_a @ ( uminus_uminus_mat_a @ A2 ) @ V )
= ( uminus_uminus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) ) ) ) ).
% uminus_mult_mat_vec
thf(fact_1265_row__add_I2_J,axiom,
! [I: nat,A2: mat_a,B2: mat_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ( dim_row_a @ B2 )
= ( dim_row_a @ A2 ) )
=> ( ( ( dim_col_a @ B2 )
= ( dim_col_a @ A2 ) )
=> ( ( row_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ I )
= ( plus_plus_vec_a @ ( row_a @ A2 @ I ) @ ( row_a @ B2 @ I ) ) ) ) ) ) ).
% row_add(2)
thf(fact_1266_assoc__four__block__mat,axiom,
! [A2: mat_a,B2: mat_a,C2: mat_a] :
( ( four_block_mat_a @ A2 @ ( zero_mat_a @ ( dim_row_a @ A2 ) @ ( dim_col_a @ ( four_block_mat_a @ B2 @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ B2 ) ) @ C2 ) ) ) @ ( zero_mat_a @ ( dim_row_a @ ( four_block_mat_a @ B2 @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ B2 ) ) @ C2 ) ) @ ( dim_col_a @ A2 ) ) @ ( four_block_mat_a @ B2 @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ B2 ) ) @ C2 ) )
= ( four_block_mat_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ ( dim_row_a @ A2 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ A2 ) ) @ B2 ) @ ( zero_mat_a @ ( dim_row_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ ( dim_row_a @ A2 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ A2 ) ) @ B2 ) ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ ( dim_row_a @ A2 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ A2 ) ) @ B2 ) ) ) @ C2 ) ) ).
% assoc_four_block_mat
thf(fact_1267_carrier__mat__triv,axiom,
! [M: mat_a] : ( member_mat_a @ M @ ( carrier_mat_a @ ( dim_row_a @ M ) @ ( dim_col_a @ M ) ) ) ).
% carrier_mat_triv
thf(fact_1268_row__carrier,axiom,
! [A2: mat_a,I: nat] : ( member_vec_a @ ( row_a @ A2 @ I ) @ ( carrier_vec_a @ ( dim_col_a @ A2 ) ) ) ).
% row_carrier
thf(fact_1269_mat__row__first__carrier,axiom,
! [A2: mat_a,N: nat] : ( member_mat_a @ ( missin3040492613037353666irst_a @ A2 @ N ) @ ( carrier_mat_a @ N @ ( dim_col_a @ A2 ) ) ) ).
% mat_row_first_carrier
thf(fact_1270_mat__row__last__carrier,axiom,
! [A2: mat_a,N: nat] : ( member_mat_a @ ( missin5577565584678110354last_a @ A2 @ N ) @ ( carrier_mat_a @ N @ ( dim_col_a @ A2 ) ) ) ).
% mat_row_last_carrier
thf(fact_1271_transpose__minus__1,axiom,
! [Q: mat_a] :
( ( ( dim_row_a @ Q )
= ( dim_col_a @ Q ) )
=> ( ( transpose_mat_a @ ( minus_minus_mat_a @ Q @ ( one_mat_a @ ( dim_row_a @ Q ) ) ) )
= ( minus_minus_mat_a @ ( transpose_mat_a @ Q ) @ ( one_mat_a @ ( dim_row_a @ Q ) ) ) ) ) ).
% transpose_minus_1
thf(fact_1272_system__iff,axiom,
! [Q: mat_a,V: vec_a] :
( ( ( dim_row_a @ Q )
= ( dim_col_a @ Q ) )
=> ( ( ( dim_row_a @ Q )
= ( dim_vec_a @ V ) )
=> ( ( ( mult_mat_vec_a @ ( transpose_mat_a @ Q ) @ V )
= V )
= ( ( mult_mat_vec_a @ ( minus_minus_mat_a @ ( transpose_mat_a @ Q ) @ ( one_mat_a @ ( dim_row_a @ Q ) ) ) @ V )
= ( zero_vec_a @ ( dim_vec_a @ V ) ) ) ) ) ) ).
% system_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_vec_a @ u3 @ ( carrier_vec_a @ ( plus_plus_nat @ nc @ nc ) ) ).
%------------------------------------------------------------------------------