TPTP Problem File: SLH0004^1.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_00201_009103__28817388_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1460 ( 510 unt; 187 typ; 0 def)
% Number of atoms : 3704 (1152 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12582 ( 208 ~; 105 |; 154 &;10452 @)
% ( 0 <=>;1663 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 371 ( 371 >; 0 *; 0 +; 0 <<)
% Number of symbols : 170 ( 167 usr; 28 con; 0-6 aty)
% Number of variables : 3577 ( 195 ^;3340 !; 42 ?;3577 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:58:20.810
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
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__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__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__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 (167)
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__Matrix__Ovec_Itf__a_J_J,type,
minus_3631651556841400635_vec_a: vec_vec_a > vec_vec_a > vec_vec_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_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
minus_4757590266979429866_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
minus_6230920740010926198_vec_a: set_vec_a > set_vec_a > set_vec_a ).
thf(sy_c_Groups_Ominus__class_Ominus_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__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_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__Matrix__Ovec_Itf__a_J_J_J,type,
plus_p8188967515152927083_vec_a: set_vec_vec_a > set_vec_vec_a > set_vec_vec_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_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_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_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
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__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_Ocol_001t__Nat__Onat,type,
col_nat: mat_nat > nat > vec_nat ).
thf(sy_c_Matrix_Ocol_001tf__a,type,
col_a: mat_a > nat > vec_a ).
thf(sy_c_Matrix_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_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_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_001tf__a,type,
dim_vec_a: vec_a > 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_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_Oorthogonal__mat_001t__Nat__Onat,type,
orthogonal_mat_nat: mat_nat > $o ).
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_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_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_001t__Nat__Onat,type,
matrix_unpadl_nat: nat > vec_nat > vec_nat ).
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_001t__Nat__Onat,type,
matrix_unpadr_nat: nat > vec_nat > vec_nat ).
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_001tf__a,type,
missin386308114684349109cols_a: mat_a > mat_a > mat_a ).
thf(sy_c_Missing__Matrix_Omat__col__first_001tf__a,type,
missin6964418172097587752irst_a: mat_a > nat > mat_a ).
thf(sy_c_Missing__Matrix_Omat__col__last_001tf__a,type,
missin1527294172917009324last_a: mat_a > nat > 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_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_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_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_less_set_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_less_set_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_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_It__Matrix__Ovec_Itf__a_J_J_J,type,
ord_le8068522077793703124_vec_a: set_vec_vec_a > set_vec_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
ord_le7536782659060323133ec_nat: set_vec_nat > set_vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4791951621262958845_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_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__Matrix__Ovec_Itf__a_J_J,type,
collect_vec_vec_a: ( vec_vec_a > $o ) > set_vec_vec_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_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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_L____,type,
l: vec_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_u____,type,
u: vec_a ).
thf(sy_v_ulv____,type,
ulv: vec_a ).
thf(sy_v_v____,type,
v: vec_a ).
thf(sy_v_vec1____,type,
vec1: vec_a ).
thf(sy_v_vec2____,type,
vec2: vec_a ).
thf(sy_v_vec3____,type,
vec3: vec_a ).
thf(sy_v_w____,type,
w: vec_a ).
% Relevant facts (1269)
thf(fact_0__C02_C,axiom,
( ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t )
= ( zero_vec_a @ nr ) ) ).
% "02"
thf(fact_1_add__diff__cancel,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_2_diff__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_3_add__diff__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4_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_5_add__diff__cancel__left_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_6_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_7_add__diff__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_8_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_9_add__diff__cancel__right_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_10_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_11_vec2,axiom,
member_vec_a @ vec2 @ ( carrier_vec_a @ nr ) ).
% vec2
thf(fact_12_vec3,axiom,
member_vec_a @ vec3 @ ( carrier_vec_a @ nr ) ).
% vec3
thf(fact_13_t,axiom,
member_vec_a @ t @ ( carrier_vec_a @ nr ) ).
% t
thf(fact_14_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_15_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_16_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_17_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_18_b,axiom,
member_vec_a @ b @ ( carrier_vec_a @ nr ) ).
% b
thf(fact_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_add_Ocommute,axiom,
( plus_plus_set_nat
= ( ^ [A2: set_nat,B2: set_nat] : ( plus_plus_set_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_27_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A2: a,B2: a] : ( plus_plus_a @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_28_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_29_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_30_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_31_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_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_group__cancel_Oadd2,axiom,
! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B3
= ( plus_plus_set_nat @ K @ B ) )
=> ( ( plus_plus_set_nat @ A @ B3 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_35_group__cancel_Oadd2,axiom,
! [B3: a,K: a,B: a,A: a] :
( ( B3
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B3 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_36_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_37_group__cancel_Oadd1,axiom,
! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A3
= ( plus_plus_set_nat @ K @ A ) )
=> ( ( plus_plus_set_nat @ A3 @ B )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_38_group__cancel_Oadd1,axiom,
! [A3: a,K: a,A: a,B: a] :
( ( A3
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A3 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_39_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_40_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_41_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_42_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_43_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_44_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_45_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
= ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_46_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_47_diff__eq__diff__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_48_diff__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_49_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_50_add__implies__diff,axiom,
! [C: a,B: a,A: a] :
( ( ( plus_plus_a @ C @ B )
= A )
=> ( C
= ( minus_minus_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_51_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_52_diff__add__eq__diff__diff__swap,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_53_diff__add__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_54_diff__diff__eq2,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_55_add__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_56_eq__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( A
= ( minus_minus_a @ C @ B ) )
= ( ( plus_plus_a @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_57_diff__eq__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( minus_minus_a @ A @ B )
= C )
= ( A
= ( plus_plus_a @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_58_group__cancel_Osub1,axiom,
! [A3: a,K: a,A: a,B: a] :
( ( A3
= ( plus_plus_a @ K @ A ) )
=> ( ( minus_minus_a @ A3 @ B )
= ( plus_plus_a @ K @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_59_u,axiom,
member_vec_a @ u @ ( carrier_vec_a @ nr ) ).
% u
thf(fact_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_add__diff__cancel__right__vec,axiom,
! [A: vec_nat,N: nat,B: vec_nat] :
( ( member_vec_nat @ A @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ B @ ( carrier_vec_nat @ N ) )
=> ( ( minus_minus_vec_nat @ ( plus_plus_vec_nat @ A @ B ) @ B )
= A ) ) ) ).
% add_diff_cancel_right_vec
thf(fact_69_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_70_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_71_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_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_73_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_74_mem__Collect__eq,axiom,
! [A: vec_vec_a,P: vec_vec_a > $o] :
( ( member_vec_vec_a @ A @ ( collect_vec_vec_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_75_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_76_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_77_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A3: set_mat_nat] :
( ( collect_mat_nat
@ ^ [X2: mat_nat] : ( member_mat_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A3: set_vec_nat] :
( ( collect_vec_nat
@ ^ [X2: vec_nat] : ( member_vec_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A3: set_vec_vec_a] :
( ( collect_vec_vec_a
@ ^ [X2: vec_vec_a] : ( member_vec_vec_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A3: set_vec_a] :
( ( collect_vec_a
@ ^ [X2: vec_a] : ( member_vec_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A3: set_mat_a] :
( ( collect_mat_a
@ ^ [X2: mat_a] : ( member_mat_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_83_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_84_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_85_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_86_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_87_set__plus__intro,axiom,
! [A: mat_nat,C2: set_mat_nat,B: mat_nat,D2: set_mat_nat] :
( ( member_mat_nat @ A @ C2 )
=> ( ( member_mat_nat @ B @ D2 )
=> ( member_mat_nat @ ( plus_plus_mat_nat @ A @ B ) @ ( plus_p2215855510709889632at_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_88_set__plus__intro,axiom,
! [A: vec_nat,C2: set_vec_nat,B: vec_nat,D2: set_vec_nat] :
( ( member_vec_nat @ A @ C2 )
=> ( ( member_vec_nat @ B @ D2 )
=> ( member_vec_nat @ ( plus_plus_vec_nat @ A @ B ) @ ( plus_p1963516127331757268ec_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_89_set__plus__intro,axiom,
! [A: vec_vec_a,C2: set_vec_vec_a,B: vec_vec_a,D2: set_vec_vec_a] :
( ( member_vec_vec_a @ A @ C2 )
=> ( ( member_vec_vec_a @ B @ D2 )
=> ( member_vec_vec_a @ ( plus_plus_vec_vec_a @ A @ B ) @ ( plus_p8188967515152927083_vec_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_90_set__plus__intro,axiom,
! [A: set_vec_a,C2: set_set_vec_a,B: set_vec_a,D2: set_set_vec_a] :
( ( member_set_vec_a @ A @ C2 )
=> ( ( member_set_vec_a @ B @ D2 )
=> ( member_set_vec_a @ ( plus_plus_set_vec_a @ A @ B ) @ ( plus_p5225466182533350236_vec_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_91_set__plus__intro,axiom,
! [A: set_mat_a,C2: set_set_mat_a,B: set_mat_a,D2: set_set_mat_a] :
( ( member_set_mat_a @ A @ C2 )
=> ( ( member_set_mat_a @ B @ D2 )
=> ( member_set_mat_a @ ( plus_plus_set_mat_a @ A @ B ) @ ( plus_p8188135320652551888_mat_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_92_set__plus__intro,axiom,
! [A: set_nat,C2: set_set_nat,B: set_nat,D2: set_set_nat] :
( ( member_set_nat @ A @ C2 )
=> ( ( member_set_nat @ B @ D2 )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_93_set__plus__intro,axiom,
! [A: a,C2: set_a,B: a,D2: set_a] :
( ( member_a @ A @ C2 )
=> ( ( member_a @ B @ D2 )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_94_set__plus__intro,axiom,
! [A: mat_a,C2: set_mat_a,B: mat_a,D2: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( member_mat_a @ B @ D2 )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( plus_plus_set_mat_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_95_set__plus__intro,axiom,
! [A: vec_a,C2: set_vec_a,B: vec_a,D2: set_vec_a] :
( ( member_vec_a @ A @ C2 )
=> ( ( member_vec_a @ B @ D2 )
=> ( member_vec_a @ ( plus_plus_vec_a @ A @ B ) @ ( plus_plus_set_vec_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_96_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_97_minus__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 @ ( minus_3631651556841400635_vec_a @ V_1 @ V_2 ) @ ( carrier_vec_vec_a @ N ) ) ) ) ).
% minus_carrier_vec
thf(fact_98_minus__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 @ ( minus_minus_vec_nat @ V_1 @ V_2 ) @ ( carrier_vec_nat @ N ) ) ) ) ).
% minus_carrier_vec
thf(fact_99_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_100_set__plus__elim,axiom,
! [X: mat_nat,A3: set_mat_nat,B3: set_mat_nat] :
( ( member_mat_nat @ X @ ( plus_p2215855510709889632at_nat @ A3 @ B3 ) )
=> ~ ! [A4: mat_nat,B4: mat_nat] :
( ( X
= ( plus_plus_mat_nat @ A4 @ B4 ) )
=> ( ( member_mat_nat @ A4 @ A3 )
=> ~ ( member_mat_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_101_set__plus__elim,axiom,
! [X: vec_nat,A3: set_vec_nat,B3: set_vec_nat] :
( ( member_vec_nat @ X @ ( plus_p1963516127331757268ec_nat @ A3 @ B3 ) )
=> ~ ! [A4: vec_nat,B4: vec_nat] :
( ( X
= ( plus_plus_vec_nat @ A4 @ B4 ) )
=> ( ( member_vec_nat @ A4 @ A3 )
=> ~ ( member_vec_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_102_set__plus__elim,axiom,
! [X: vec_vec_a,A3: set_vec_vec_a,B3: set_vec_vec_a] :
( ( member_vec_vec_a @ X @ ( plus_p8188967515152927083_vec_a @ A3 @ B3 ) )
=> ~ ! [A4: vec_vec_a,B4: vec_vec_a] :
( ( X
= ( plus_plus_vec_vec_a @ A4 @ B4 ) )
=> ( ( member_vec_vec_a @ A4 @ A3 )
=> ~ ( member_vec_vec_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_103_set__plus__elim,axiom,
! [X: set_vec_a,A3: set_set_vec_a,B3: set_set_vec_a] :
( ( member_set_vec_a @ X @ ( plus_p5225466182533350236_vec_a @ A3 @ B3 ) )
=> ~ ! [A4: set_vec_a,B4: set_vec_a] :
( ( X
= ( plus_plus_set_vec_a @ A4 @ B4 ) )
=> ( ( member_set_vec_a @ A4 @ A3 )
=> ~ ( member_set_vec_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_104_set__plus__elim,axiom,
! [X: set_mat_a,A3: set_set_mat_a,B3: set_set_mat_a] :
( ( member_set_mat_a @ X @ ( plus_p8188135320652551888_mat_a @ A3 @ B3 ) )
=> ~ ! [A4: set_mat_a,B4: set_mat_a] :
( ( X
= ( plus_plus_set_mat_a @ A4 @ B4 ) )
=> ( ( member_set_mat_a @ A4 @ A3 )
=> ~ ( member_set_mat_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_105_set__plus__elim,axiom,
! [X: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ X @ ( plus_p4817606893110106565et_nat @ A3 @ B3 ) )
=> ~ ! [A4: set_nat,B4: set_nat] :
( ( X
= ( plus_plus_set_nat @ A4 @ B4 ) )
=> ( ( member_set_nat @ A4 @ A3 )
=> ~ ( member_set_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_106_set__plus__elim,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ( member_a @ X @ ( plus_plus_set_a @ A3 @ B3 ) )
=> ~ ! [A4: a,B4: a] :
( ( X
= ( plus_plus_a @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ A3 )
=> ~ ( member_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_107_set__plus__elim,axiom,
! [X: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ X @ ( plus_plus_set_mat_a @ A3 @ B3 ) )
=> ~ ! [A4: mat_a,B4: mat_a] :
( ( X
= ( plus_plus_mat_a @ A4 @ B4 ) )
=> ( ( member_mat_a @ A4 @ A3 )
=> ~ ( member_mat_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_108_set__plus__elim,axiom,
! [X: vec_a,A3: set_vec_a,B3: set_vec_a] :
( ( member_vec_a @ X @ ( plus_plus_set_vec_a @ A3 @ B3 ) )
=> ~ ! [A4: vec_a,B4: vec_a] :
( ( X
= ( plus_plus_vec_a @ A4 @ B4 ) )
=> ( ( member_vec_a @ A4 @ A3 )
=> ~ ( member_vec_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_109_set__plus__elim,axiom,
! [X: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A3 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A3 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_110_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_nat @ ( zero_vec_nat @ N ) @ ( carrier_vec_nat @ N ) ) ).
% zero_carrier_vec
thf(fact_111_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_a @ ( zero_vec_a @ N ) @ ( carrier_vec_a @ N ) ) ).
% zero_carrier_vec
thf(fact_112_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_113_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_114_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_115_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_116_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_117__092_060open_062vec1_A_061_A0_092_060_094sub_062v_Anc_A_092_060and_062_Avec3_A_L_Avec2_A_N_At_A_061_A0_092_060_094sub_062v_Anr_092_060close_062,axiom,
( ( vec1
= ( zero_vec_a @ nc ) )
& ( ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t )
= ( zero_vec_a @ nr ) ) ) ).
% \<open>vec1 = 0\<^sub>v nc \<and> vec3 + vec2 - t = 0\<^sub>v nr\<close>
thf(fact_118_add__diff__add,axiom,
! [A: a,C: a,B: a,D: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) )
= ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ ( minus_minus_a @ C @ D ) ) ) ).
% add_diff_add
thf(fact_119_u__def,axiom,
( u
= ( vec_first_a @ u2 @ nr ) ) ).
% u_def
thf(fact_120_t__def,axiom,
( t
= ( vec_last_a @ ulv @ nr ) ) ).
% t_def
thf(fact_121__092_060open_062vec1_A_064_092_060_094sub_062v_Avec3_A_L_Avec2_A_N_At_A_061_A0_092_060_094sub_062v_A_Inc_A_L_Anr_J_092_060close_062,axiom,
( ( append_vec_a @ vec1 @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) )
= ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) ) ) ).
% \<open>vec1 @\<^sub>v vec3 + vec2 - t = 0\<^sub>v (nc + nr)\<close>
thf(fact_122_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_123_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_124_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_125_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_126_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_127_diff__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_128_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_129_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_130__C01_C,axiom,
( vec1
= ( zero_vec_a @ nc ) ) ).
% "01"
thf(fact_131_vec1,axiom,
member_vec_a @ vec1 @ ( carrier_vec_a @ nc ) ).
% vec1
thf(fact_132_uminus__uminus__vec,axiom,
! [V: vec_a] :
( ( uminus_uminus_vec_a @ ( uminus_uminus_vec_a @ V ) )
= V ) ).
% uminus_uminus_vec
thf(fact_133_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_134_c,axiom,
member_vec_a @ c @ ( carrier_vec_a @ nc ) ).
% c
thf(fact_135__092_060open_0620_092_060_094sub_062v_A_Inc_A_L_Anr_J_A_061_A0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A0_092_060_094sub_062v_Anr_092_060close_062,axiom,
( ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) )
= ( append_vec_a @ ( zero_vec_a @ nc ) @ ( zero_vec_a @ nr ) ) ) ).
% \<open>0\<^sub>v (nc + nr) = 0\<^sub>v nc @\<^sub>v 0\<^sub>v nr\<close>
thf(fact_136_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_137_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_138_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_139_minus__diff__eq,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
= ( minus_minus_a @ B @ A ) ) ).
% minus_diff_eq
thf(fact_140_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_141_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_142_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_143_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_144_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_145_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_146_uminus__zero__vec,axiom,
! [N: nat] :
( ( uminus_uminus_vec_a @ ( zero_vec_a @ N ) )
= ( zero_vec_a @ N ) ) ).
% uminus_zero_vec
thf(fact_147_uminus__add__conv__diff,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B )
= ( minus_minus_a @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_148_diff__minus__eq__add,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ A @ ( uminus_uminus_a @ B ) )
= ( plus_plus_a @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_149_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_150_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_151_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_152__092_060open_062vec1_A_064_092_060_094sub_062v_Avec3_A_L_A_I0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_Avec2_J_A_L_A_I0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A_N_At_J_A_061_Avec1_A_064_092_060_094sub_062v_Avec3_A_L_Avec2_A_N_At_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( plus_plus_vec_a @ ( append_vec_a @ vec1 @ vec3 ) @ ( append_vec_a @ ( zero_vec_a @ nc ) @ vec2 ) ) @ ( append_vec_a @ ( zero_vec_a @ nc ) @ ( uminus_uminus_vec_a @ t ) ) )
= ( append_vec_a @ vec1 @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) ) ) ).
% \<open>vec1 @\<^sub>v vec3 + (0\<^sub>v nc @\<^sub>v vec2) + (0\<^sub>v nc @\<^sub>v - t) = vec1 @\<^sub>v vec3 + vec2 - t\<close>
thf(fact_153_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_154_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_155_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_156_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_157_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_158_minus__diff__minus,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) )
= ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_159_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_160_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_161_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_162_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_163_group__cancel_Oneg1,axiom,
! [A3: a,K: a,A: a] :
( ( A3
= ( plus_plus_a @ K @ A ) )
=> ( ( uminus_uminus_a @ A3 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( uminus_uminus_a @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_164_minus__diff__commute,axiom,
! [B: a,A: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
= ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_a
= ( ^ [A2: a,B2: a] : ( plus_plus_a @ A2 @ ( uminus_uminus_a @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_175_diff__conv__add__uminus,axiom,
( minus_minus_a
= ( ^ [A2: a,B2: a] : ( plus_plus_a @ A2 @ ( uminus_uminus_a @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_176_group__cancel_Osub2,axiom,
! [B3: a,K: a,B: a,A: a] :
( ( B3
= ( plus_plus_a @ K @ B ) )
=> ( ( minus_minus_a @ A @ B3 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_ulvid,axiom,
( ulv
= ( append_vec_a @ u1 @ t ) ) ).
% ulvid
thf(fact_185_u2id,axiom,
( u2
= ( append_vec_a @ u @ l ) ) ).
% u2id
thf(fact_186_u3,axiom,
member_vec_a @ u3 @ ( carrier_vec_a @ ( plus_plus_nat @ nc @ nc ) ) ).
% u3
thf(fact_187_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_188_v,axiom,
member_vec_a @ v @ ( carrier_vec_a @ nc ) ).
% v
thf(fact_189_w,axiom,
member_vec_a @ w @ ( carrier_vec_a @ nc ) ).
% w
thf(fact_190_u2,axiom,
member_vec_a @ u2 @ ( carrier_vec_a @ ( plus_plus_nat @ nr @ one_one_nat ) ) ).
% u2
thf(fact_191_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_192_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_193_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_194_class__ring_Ominus__eq,axiom,
( minus_minus_a
= ( ^ [X2: a,Y2: a] : ( plus_plus_a @ X2 @ ( uminus_uminus_a @ Y2 ) ) ) ) ).
% class_ring.minus_eq
thf(fact_195_L,axiom,
member_vec_a @ l @ ( carrier_vec_a @ one_one_nat ) ).
% L
thf(fact_196_L__def,axiom,
( l
= ( vec_last_a @ u2 @ one_one_nat ) ) ).
% L_def
thf(fact_197_w__def,axiom,
( w
= ( vec_last_a @ u3 @ nc ) ) ).
% w_def
thf(fact_198_v__def,axiom,
( v
= ( vec_first_a @ u3 @ nc ) ) ).
% v_def
thf(fact_199_u3id,axiom,
( u3
= ( append_vec_a @ v @ w ) ) ).
% u3id
thf(fact_200_u1id,axiom,
( u1
= ( append_vec_a @ u2 @ u3 ) ) ).
% u1id
thf(fact_201_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_202_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_203_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_204_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_205_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_206_u3__def,axiom,
( u3
= ( vec_last_a @ u1 @ ( plus_plus_nat @ nc @ nc ) ) ) ).
% u3_def
thf(fact_207_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_208_u2__def,axiom,
( u2
= ( vec_first_a @ u1 @ ( plus_plus_nat @ nr @ one_one_nat ) ) ) ).
% u2_def
thf(fact_209_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_210_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_211_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_212_one__reorient,axiom,
! [X: a] :
( ( one_one_a = X )
= ( X = one_one_a ) ) ).
% one_reorient
thf(fact_213_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_214_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_215_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_216_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_217_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_218_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_219_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_220_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_221_add__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_222_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_223_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_224_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_225_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_226_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_227_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_228_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_229_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_230_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_231_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_232_diff__mono,axiom,
! [A: a,B: a,D: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ D @ C )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_233_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_234_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_235_diff__eq__diff__less__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( ord_less_eq_a @ A @ B )
= ( ord_less_eq_a @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_append__vec__le,axiom,
! [V: vec_nat,N: nat,W: vec_nat,V2: vec_nat,W2: vec_nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ W @ ( carrier_vec_nat @ N ) )
=> ( ( ord_less_eq_vec_nat @ ( append_vec_nat @ V @ V2 ) @ ( append_vec_nat @ W @ W2 ) )
= ( ( ord_less_eq_vec_nat @ V @ W )
& ( ord_less_eq_vec_nat @ V2 @ W2 ) ) ) ) ) ).
% append_vec_le
thf(fact_252_append__vec__le,axiom,
! [V: vec_vec_a,N: nat,W: vec_vec_a,V2: vec_vec_a,W2: vec_vec_a] :
( ( member_vec_vec_a @ V @ ( carrier_vec_vec_a @ N ) )
=> ( ( member_vec_vec_a @ W @ ( carrier_vec_vec_a @ N ) )
=> ( ( ord_le4012615358376148468_vec_a @ ( append_vec_vec_a @ V @ V2 ) @ ( append_vec_vec_a @ W @ W2 ) )
= ( ( ord_le4012615358376148468_vec_a @ V @ W )
& ( ord_le4012615358376148468_vec_a @ V2 @ W2 ) ) ) ) ) ).
% append_vec_le
thf(fact_253_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_254_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: a,B: a,C: a,D: a] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_a @ A @ C )
= ( plus_plus_a @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_255_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_set__plus__mono2,axiom,
! [C2: set_vec_a,D2: set_vec_a,E: set_vec_a,F: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ C2 @ D2 )
=> ( ( ord_le4791951621262958845_vec_a @ E @ F )
=> ( ord_le4791951621262958845_vec_a @ ( plus_plus_set_vec_a @ C2 @ E ) @ ( plus_plus_set_vec_a @ D2 @ F ) ) ) ) ).
% set_plus_mono2
thf(fact_273_set__plus__mono2,axiom,
! [C2: set_mat_a,D2: set_mat_a,E: set_mat_a,F: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ D2 )
=> ( ( ord_le3318621148231462513_mat_a @ E @ F )
=> ( ord_le3318621148231462513_mat_a @ ( plus_plus_set_mat_a @ C2 @ E ) @ ( plus_plus_set_mat_a @ D2 @ F ) ) ) ) ).
% set_plus_mono2
thf(fact_274_set__plus__mono2,axiom,
! [C2: set_nat,D2: set_nat,E: set_nat,F: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ( ord_less_eq_set_nat @ E @ F )
=> ( ord_less_eq_set_nat @ ( plus_plus_set_nat @ C2 @ E ) @ ( plus_plus_set_nat @ D2 @ F ) ) ) ) ).
% set_plus_mono2
thf(fact_275_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_276_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_277_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_278_vec__first__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( vec_first_nat @ ( zero_vec_nat @ N ) @ M )
= ( zero_vec_nat @ M ) ) ) ).
% vec_first_zero
thf(fact_279_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_280_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_281_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_282_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_283_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_284_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_285_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_286_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_287_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_288_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_289_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_290_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_308_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_309_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_310_le__numeral__extra_I4_J,axiom,
ord_less_eq_a @ one_one_a @ one_one_a ).
% le_numeral_extra(4)
thf(fact_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_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_319_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_320_one__neq__neg__one,axiom,
( one_one_a
!= ( uminus_uminus_a @ one_one_a ) ) ).
% one_neq_neg_one
thf(fact_321_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_322_sum__carrier__vec,axiom,
! [A3: set_vec_nat,N: nat,B3: set_vec_nat] :
( ( ord_le7536782659060323133ec_nat @ A3 @ ( carrier_vec_nat @ N ) )
=> ( ( ord_le7536782659060323133ec_nat @ B3 @ ( carrier_vec_nat @ N ) )
=> ( ord_le7536782659060323133ec_nat @ ( plus_p1963516127331757268ec_nat @ A3 @ B3 ) @ ( carrier_vec_nat @ N ) ) ) ) ).
% sum_carrier_vec
thf(fact_323_sum__carrier__vec,axiom,
! [A3: set_vec_vec_a,N: nat,B3: set_vec_vec_a] :
( ( ord_le8068522077793703124_vec_a @ A3 @ ( carrier_vec_vec_a @ N ) )
=> ( ( ord_le8068522077793703124_vec_a @ B3 @ ( carrier_vec_vec_a @ N ) )
=> ( ord_le8068522077793703124_vec_a @ ( plus_p8188967515152927083_vec_a @ A3 @ B3 ) @ ( carrier_vec_vec_a @ N ) ) ) ) ).
% sum_carrier_vec
thf(fact_324_sum__carrier__vec,axiom,
! [A3: set_vec_a,N: nat,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A3 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B3 @ ( carrier_vec_a @ N ) )
=> ( ord_le4791951621262958845_vec_a @ ( plus_plus_set_vec_a @ A3 @ B3 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% sum_carrier_vec
thf(fact_325_verit__minus__simplify_I4_J,axiom,
! [B: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_326_order__refl,axiom,
! [X: set_vec_a] : ( ord_le4791951621262958845_vec_a @ X @ X ) ).
% order_refl
thf(fact_327_order__refl,axiom,
! [X: set_mat_a] : ( ord_le3318621148231462513_mat_a @ X @ X ) ).
% order_refl
thf(fact_328_order__refl,axiom,
! [X: vec_nat] : ( ord_less_eq_vec_nat @ X @ X ) ).
% order_refl
thf(fact_329_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_330_order__refl,axiom,
! [X: vec_vec_a] : ( ord_le4012615358376148468_vec_a @ X @ X ) ).
% order_refl
thf(fact_331_order__refl,axiom,
! [X: vec_a] : ( ord_less_eq_vec_a @ X @ X ) ).
% order_refl
thf(fact_332_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_333_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_334_dual__order_Orefl,axiom,
! [A: set_vec_a] : ( ord_le4791951621262958845_vec_a @ A @ A ) ).
% dual_order.refl
thf(fact_335_dual__order_Orefl,axiom,
! [A: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A @ A ) ).
% dual_order.refl
thf(fact_336_dual__order_Orefl,axiom,
! [A: vec_nat] : ( ord_less_eq_vec_nat @ A @ A ) ).
% dual_order.refl
thf(fact_337_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_338_dual__order_Orefl,axiom,
! [A: vec_vec_a] : ( ord_le4012615358376148468_vec_a @ A @ A ) ).
% dual_order.refl
thf(fact_339_dual__order_Orefl,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% dual_order.refl
thf(fact_340_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_341_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_342_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_343_vardim_Ounpadl__padl,axiom,
! [M: nat,V: vec_nat] :
( ( matrix_unpadl_nat @ M @ ( append_vec_nat @ ( zero_vec_nat @ M ) @ V ) )
= V ) ).
% vardim.unpadl_padl
thf(fact_344_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_345_vardim_Ounpadr__padr,axiom,
! [M: nat,V: vec_nat] :
( ( matrix_unpadr_nat @ M @ ( append_vec_nat @ V @ ( zero_vec_nat @ M ) ) )
= V ) ).
% vardim.unpadr_padr
thf(fact_346_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_347__092_060open_062vec1_A_L_Amat__of__col_Ac_A_K_092_060_094sub_062v_AL_A_061_Amat__of__col_Ac_A_K_092_060_094sub_062v_AL_092_060close_062,axiom,
( ( plus_plus_vec_a @ vec1 @ ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) )
= ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) ) ).
% \<open>vec1 + mat_of_col c *\<^sub>v L = mat_of_col c *\<^sub>v L\<close>
thf(fact_348_vec3__def,axiom,
( vec3
= ( mult_mat_vec_a @ ( missing_mat_of_col_a @ b ) @ l ) ) ).
% vec3_def
thf(fact_349__092_060open_062A_A_K_092_060_094sub_062v_Av_A_L_A_N_AA_A_K_092_060_094sub_062v_Aw_A_061_Avec2_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( mult_mat_vec_a @ a2 @ v ) @ ( mult_mat_vec_a @ ( uminus_uminus_mat_a @ a2 ) @ w ) )
= vec2 ) ).
% \<open>A *\<^sub>v v + - A *\<^sub>v w = vec2\<close>
thf(fact_350_vec2__def,axiom,
( vec2
= ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) ) ).
% vec2_def
thf(fact_351_A,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ nr @ nc ) ).
% A
thf(fact_352_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_353_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_354_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_355_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_356_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_357_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_358_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_359_order__antisym__conv,axiom,
! [Y: set_vec_a,X: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ Y @ X )
=> ( ( ord_le4791951621262958845_vec_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_360_order__antisym__conv,axiom,
! [Y: set_mat_a,X: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ Y @ X )
=> ( ( ord_le3318621148231462513_mat_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_361_order__antisym__conv,axiom,
! [Y: vec_nat,X: vec_nat] :
( ( ord_less_eq_vec_nat @ Y @ X )
=> ( ( ord_less_eq_vec_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_362_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_363_order__antisym__conv,axiom,
! [Y: vec_vec_a,X: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ Y @ X )
=> ( ( ord_le4012615358376148468_vec_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_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_374_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_375_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_376_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_377_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_378_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_ord__eq__le__subst,axiom,
! [A: vec_nat,F2: nat > vec_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_vec_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_389_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_390_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_391_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_392_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_393_order__eq__refl,axiom,
! [X: set_vec_a,Y: set_vec_a] :
( ( X = Y )
=> ( ord_le4791951621262958845_vec_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_394_order__eq__refl,axiom,
! [X: set_mat_a,Y: set_mat_a] :
( ( X = Y )
=> ( ord_le3318621148231462513_mat_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_395_order__eq__refl,axiom,
! [X: vec_nat,Y: vec_nat] :
( ( X = Y )
=> ( ord_less_eq_vec_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_396_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_397_order__eq__refl,axiom,
! [X: vec_vec_a,Y: vec_vec_a] :
( ( X = Y )
=> ( ord_le4012615358376148468_vec_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_398_order__eq__refl,axiom,
! [X: vec_a,Y: vec_a] :
( ( X = Y )
=> ( ord_less_eq_vec_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_399_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_400_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_vec_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_vec_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_order__subst1,axiom,
! [A: nat,F2: vec_nat > nat,B: vec_nat,C: vec_nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ! [X3: vec_nat,Y3: vec_nat] :
( ( ord_less_eq_vec_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_421_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_vec_a,Z2: set_vec_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
& ( ord_le4791951621262958845_vec_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_422_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_mat_a,Z2: set_mat_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B2 )
& ( ord_le3318621148231462513_mat_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_423_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_nat,Z2: vec_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: vec_nat,B2: vec_nat] :
( ( ord_less_eq_vec_nat @ A2 @ B2 )
& ( ord_less_eq_vec_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_424_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_425_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_vec_a,Z2: vec_vec_a] : ( Y5 = Z2 ) )
= ( ^ [A2: vec_vec_a,B2: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ A2 @ B2 )
& ( ord_le4012615358376148468_vec_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_426_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: vec_a,Z2: vec_a] : ( Y5 = Z2 ) )
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ A2 @ B2 )
& ( ord_less_eq_vec_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_427_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_428_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_429_antisym,axiom,
! [A: set_vec_a,B: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_430_antisym,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_431_antisym,axiom,
! [A: vec_nat,B: vec_nat] :
( ( ord_less_eq_vec_nat @ A @ B )
=> ( ( ord_less_eq_vec_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_432_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_433_antisym,axiom,
! [A: vec_vec_a,B: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ A @ B )
=> ( ( ord_le4012615358376148468_vec_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_434_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_435_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_436_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_437_dual__order_Otrans,axiom,
! [B: set_vec_a,A: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B @ A )
=> ( ( ord_le4791951621262958845_vec_a @ C @ B )
=> ( ord_le4791951621262958845_vec_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_438_dual__order_Otrans,axiom,
! [B: set_mat_a,A: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( ( ord_le3318621148231462513_mat_a @ C @ B )
=> ( ord_le3318621148231462513_mat_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_439_dual__order_Otrans,axiom,
! [B: vec_nat,A: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ B @ A )
=> ( ( ord_less_eq_vec_nat @ C @ B )
=> ( ord_less_eq_vec_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_440_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_441_dual__order_Otrans,axiom,
! [B: vec_vec_a,A: vec_vec_a,C: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ B @ A )
=> ( ( ord_le4012615358376148468_vec_a @ C @ B )
=> ( ord_le4012615358376148468_vec_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_442_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_443_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_444_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_445_dual__order_Oantisym,axiom,
! [B: set_vec_a,A: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B @ A )
=> ( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_446_dual__order_Oantisym,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_447_dual__order_Oantisym,axiom,
! [B: vec_nat,A: vec_nat] :
( ( ord_less_eq_vec_nat @ B @ A )
=> ( ( ord_less_eq_vec_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_448_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_449_dual__order_Oantisym,axiom,
! [B: vec_vec_a,A: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ B @ A )
=> ( ( ord_le4012615358376148468_vec_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_450_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_451_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_452_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_453_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_vec_a,Z2: set_vec_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B2 @ A2 )
& ( ord_le4791951621262958845_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_454_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_mat_a,Z2: set_mat_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B2 @ A2 )
& ( ord_le3318621148231462513_mat_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_455_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: vec_nat,Z2: vec_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: vec_nat,B2: vec_nat] :
( ( ord_less_eq_vec_nat @ B2 @ A2 )
& ( ord_less_eq_vec_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_456_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_457_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: vec_vec_a,Z2: vec_vec_a] : ( Y5 = Z2 ) )
= ( ^ [A2: vec_vec_a,B2: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ B2 @ A2 )
& ( ord_le4012615358376148468_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_458_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: vec_a,Z2: vec_a] : ( Y5 = Z2 ) )
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ B2 @ A2 )
& ( ord_less_eq_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_459_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_460_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_461_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_462_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_463_order__trans,axiom,
! [X: set_vec_a,Y: set_vec_a,Z: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X @ Y )
=> ( ( ord_le4791951621262958845_vec_a @ Y @ Z )
=> ( ord_le4791951621262958845_vec_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_464_order__trans,axiom,
! [X: set_mat_a,Y: set_mat_a,Z: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X @ Y )
=> ( ( ord_le3318621148231462513_mat_a @ Y @ Z )
=> ( ord_le3318621148231462513_mat_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_465_order__trans,axiom,
! [X: vec_nat,Y: vec_nat,Z: vec_nat] :
( ( ord_less_eq_vec_nat @ X @ Y )
=> ( ( ord_less_eq_vec_nat @ Y @ Z )
=> ( ord_less_eq_vec_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_466_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_467_order__trans,axiom,
! [X: vec_vec_a,Y: vec_vec_a,Z: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ X @ Y )
=> ( ( ord_le4012615358376148468_vec_a @ Y @ Z )
=> ( ord_le4012615358376148468_vec_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_468_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_469_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_470_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_471_order_Otrans,axiom,
! [A: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ord_le4791951621262958845_vec_a @ A @ C ) ) ) ).
% order.trans
thf(fact_472_order_Otrans,axiom,
! [A: set_mat_a,B: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ord_le3318621148231462513_mat_a @ B @ C )
=> ( ord_le3318621148231462513_mat_a @ A @ C ) ) ) ).
% order.trans
thf(fact_473_order_Otrans,axiom,
! [A: vec_nat,B: vec_nat,C: vec_nat] :
( ( ord_less_eq_vec_nat @ A @ B )
=> ( ( ord_less_eq_vec_nat @ B @ C )
=> ( ord_less_eq_vec_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_474_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_475_order_Otrans,axiom,
! [A: vec_vec_a,B: vec_vec_a,C: vec_vec_a] :
( ( ord_le4012615358376148468_vec_a @ A @ B )
=> ( ( ord_le4012615358376148468_vec_a @ B @ C )
=> ( ord_le4012615358376148468_vec_a @ A @ C ) ) ) ).
% order.trans
thf(fact_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_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_491_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_492_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_493_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_494_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_495_verit__comp__simplify1_I2_J,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_496_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_497_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_498_assoc__add__vecset,axiom,
! [A3: set_vec_a,N: nat,B3: set_vec_a,C2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A3 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B3 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ C2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A3 @ ( plus_plus_set_vec_a @ B3 @ C2 ) )
= ( plus_plus_set_vec_a @ ( plus_plus_set_vec_a @ A3 @ B3 ) @ C2 ) ) ) ) ) ).
% assoc_add_vecset
thf(fact_499_comm__add__vecset,axiom,
! [A3: set_vec_a,N: nat,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A3 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B3 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A3 @ B3 )
= ( plus_plus_set_vec_a @ B3 @ A3 ) ) ) ) ).
% comm_add_vecset
thf(fact_500_vec1__def,axiom,
( vec1
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ u ) @ ( mult_mat_vec_a @ ( missing_mat_of_col_a @ ( uminus_uminus_vec_a @ c ) ) @ l ) ) ) ).
% vec1_def
thf(fact_501__092_060open_062vec1_A_L_Amat__of__col_Ac_A_K_092_060_094sub_062v_AL_A_061_AA_092_060_094sup_062T_A_K_092_060_094sub_062v_Au_092_060close_062,axiom,
( ( plus_plus_vec_a @ vec1 @ ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) )
= ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ u ) ) ).
% \<open>vec1 + mat_of_col c *\<^sub>v L = A\<^sup>T *\<^sub>v u\<close>
thf(fact_502_As,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ u )
= ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) ) ).
% As
thf(fact_503__092_060open_0620_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Av_A_L_A0_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Aw_A_061_A0_092_060_094sub_062v_Anc_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ v ) @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ w ) )
= ( zero_vec_a @ nc ) ) ).
% \<open>0\<^sub>m nc nc *\<^sub>v v + 0\<^sub>m nc nc *\<^sub>v w = 0\<^sub>v nc\<close>
thf(fact_504_Mulv,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ m ) @ ulv )
= ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) ) ) ).
% Mulv
thf(fact_505__092_060open_062four__block__mat_A_I0_092_060_094sub_062m_Anc_Anc_J_A_I0_092_060_094sub_062m_Anc_Anc_J_AA_A_I_N_AA_J_A_K_092_060_094sub_062v_Au3_A_061_A_I0_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Av_A_L_A0_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Aw_J_A_064_092_060_094sub_062v_AA_A_K_092_060_094sub_062v_Av_A_L_A_N_AA_A_K_092_060_094sub_062v_Aw_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( four_block_mat_a @ ( zero_mat_a @ nc @ nc ) @ ( zero_mat_a @ nc @ nc ) @ a2 @ ( uminus_uminus_mat_a @ a2 ) ) @ u3 )
= ( append_vec_a @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ v ) @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ w ) ) @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ a2 @ v ) @ ( mult_mat_vec_a @ ( uminus_uminus_mat_a @ a2 ) @ w ) ) ) ) ).
% \<open>four_block_mat (0\<^sub>m nc nc) (0\<^sub>m nc nc) A (- A) *\<^sub>v u3 = (0\<^sub>m nc nc *\<^sub>v v + 0\<^sub>m nc nc *\<^sub>v w) @\<^sub>v A *\<^sub>v v + - A *\<^sub>v w\<close>
thf(fact_506_subsetI,axiom,
! [A3: set_vec_a,B3: set_vec_a] :
( ! [X3: vec_a] :
( ( member_vec_a @ X3 @ A3 )
=> ( member_vec_a @ X3 @ B3 ) )
=> ( ord_le4791951621262958845_vec_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_507_subsetI,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ! [X3: mat_a] :
( ( member_mat_a @ X3 @ A3 )
=> ( member_mat_a @ X3 @ B3 ) )
=> ( ord_le3318621148231462513_mat_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_508_assoc__add__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a,C2: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ C2 )
= ( plus_plus_mat_a @ A3 @ ( plus_plus_mat_a @ B3 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_509_Matrix_Otranspose__transpose,axiom,
! [A3: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A3 ) )
= A3 ) ).
% Matrix.transpose_transpose
thf(fact_510_transpose__mat__eq,axiom,
! [A3: mat_a,B3: mat_a] :
( ( ( transpose_mat_a @ A3 )
= ( transpose_mat_a @ B3 ) )
= ( A3 = B3 ) ) ).
% transpose_mat_eq
thf(fact_511_left__add__zero__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ A3 )
= A3 ) ) ).
% left_add_zero_mat
thf(fact_512_right__add__zero__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A3 @ ( zero_mat_a @ Nr @ Nc ) )
= A3 ) ) ).
% right_add_zero_mat
thf(fact_513_minus__r__inv__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A3 @ A3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% minus_r_inv_mat
thf(fact_514_transpose__carrier__mat,axiom,
! [A3: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A3 ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_515_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_516_uminus__carrier__iff__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ ( uminus_uminus_mat_a @ A3 ) @ ( carrier_mat_a @ Nr @ Nc ) )
= ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_517_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_518_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_519_uminus__l__inv__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ A3 ) @ A3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% uminus_l_inv_mat
thf(fact_520_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_521__092_060open_062four__block__mat_AA_092_060_094sup_062T_A_Imat__of__col_A_I_N_Ac_J_J_A_I0_092_060_094sub_062m_Anr_Anr_J_A_Imat__of__col_Ab_J_A_K_092_060_094sub_062v_Au2_A_061_Avec1_A_064_092_060_094sub_062v_Avec3_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( four_block_mat_a @ ( transpose_mat_a @ a2 ) @ ( missing_mat_of_col_a @ ( uminus_uminus_vec_a @ c ) ) @ ( zero_mat_a @ nr @ nr ) @ ( missing_mat_of_col_a @ b ) ) @ u2 )
= ( append_vec_a @ vec1 @ vec3 ) ) ).
% \<open>four_block_mat A\<^sup>T (mat_of_col (- c)) (0\<^sub>m nr nr) (mat_of_col b) *\<^sub>v u2 = vec1 @\<^sub>v vec3\<close>
thf(fact_522_add__uminus__minus__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A3 @ ( uminus_uminus_mat_a @ B3 ) )
= ( minus_minus_mat_a @ A3 @ B3 ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_523_minus__add__uminus__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A3 @ B3 )
= ( plus_plus_mat_a @ A3 @ ( uminus_uminus_mat_a @ B3 ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_524_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_525_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_526_comm__add__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A3 @ B3 )
= ( plus_plus_mat_a @ B3 @ A3 ) ) ) ) ).
% comm_add_mat
thf(fact_527_transpose__add,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( plus_plus_mat_a @ A3 @ B3 ) )
= ( plus_plus_mat_a @ ( transpose_mat_a @ A3 ) @ ( transpose_mat_a @ B3 ) ) ) ) ) ).
% transpose_add
thf(fact_528_add__carrier__mat,axiom,
! [B3: mat_a,Nr: nat,Nc: nat,A3: mat_a] :
( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_529_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_530_add__inv__exists__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ? [X3: mat_a] :
( ( member_mat_a @ X3 @ ( carrier_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ X3 @ A3 )
= ( zero_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ A3 @ X3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ) ).
% add_inv_exists_mat
thf(fact_531_four__block__carrier__mat,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,D2: mat_a,Nr2: nat,Nc2: nat,B3: mat_a,C2: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( member_mat_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% four_block_carrier_mat
thf(fact_532_uminus__add__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ A3 @ B3 ) )
= ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ B3 ) @ ( uminus_uminus_mat_a @ A3 ) ) ) ) ) ).
% uminus_add_mat
thf(fact_533_uminus__carrier__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( uminus_uminus_mat_a @ A3 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_534_transpose__uminus,axiom,
! [A3: mat_a] :
( ( transpose_mat_a @ ( uminus_uminus_mat_a @ A3 ) )
= ( uminus_uminus_mat_a @ ( transpose_mat_a @ A3 ) ) ) ).
% transpose_uminus
thf(fact_535_transpose__four__block__mat,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,B3: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D2: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( transpose_mat_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) )
= ( four_block_mat_a @ ( transpose_mat_a @ A3 ) @ ( transpose_mat_a @ C2 ) @ ( transpose_mat_a @ B3 ) @ ( transpose_mat_a @ D2 ) ) ) ) ) ) ) ).
% transpose_four_block_mat
thf(fact_536_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_537_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_538_minus__carrier__mat,axiom,
! [B3: mat_a,Nr: nat,Nc: nat,A3: mat_a] :
( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A3 @ B3 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_539_transpose__minus,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( minus_minus_mat_a @ A3 @ B3 ) )
= ( minus_minus_mat_a @ ( transpose_mat_a @ A3 ) @ ( transpose_mat_a @ B3 ) ) ) ) ) ).
% transpose_minus
thf(fact_540_mult__mat__vec__split,axiom,
! [A3: mat_a,N: nat,D2: mat_a,M: nat,A: vec_a,D: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ D @ ( carrier_vec_a @ M ) )
=> ( ( mult_mat_vec_a @ ( four_block_mat_a @ A3 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ D2 ) @ ( append_vec_a @ A @ D ) )
= ( append_vec_a @ ( mult_mat_vec_a @ A3 @ A ) @ ( mult_mat_vec_a @ D2 @ D ) ) ) ) ) ) ) ).
% mult_mat_vec_split
thf(fact_541_four__block__mat__mult__vec,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,B3: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D2: mat_a,A: vec_a,D: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ Nc1 ) )
=> ( ( member_vec_a @ D @ ( carrier_vec_a @ Nc2 ) )
=> ( ( mult_mat_vec_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) @ ( append_vec_a @ A @ D ) )
= ( append_vec_a @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ A3 @ A ) @ ( mult_mat_vec_a @ B3 @ D ) ) @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ C2 @ A ) @ ( mult_mat_vec_a @ D2 @ D ) ) ) ) ) ) ) ) ) ) ).
% four_block_mat_mult_vec
thf(fact_542_mult__mat__vec__carrier,axiom,
! [A3: mat_a,Nr: nat,N: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_543_in__mono,axiom,
! [A3: set_vec_a,B3: set_vec_a,X: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A3 @ B3 )
=> ( ( member_vec_a @ X @ A3 )
=> ( member_vec_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_544_in__mono,axiom,
! [A3: set_mat_a,B3: set_mat_a,X: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( member_mat_a @ X @ A3 )
=> ( member_mat_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_545_subsetD,axiom,
! [A3: set_vec_a,B3: set_vec_a,C: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A3 @ B3 )
=> ( ( member_vec_a @ C @ A3 )
=> ( member_vec_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_546_subsetD,axiom,
! [A3: set_mat_a,B3: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( member_mat_a @ C @ A3 )
=> ( member_mat_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_547_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_548_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_549_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_550_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_551_add__mult__distrib__mat__vec,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ V )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ ( mult_mat_vec_a @ B3 @ V ) ) ) ) ) ) ).
% add_mult_distrib_mat_vec
thf(fact_552_mult__add__distrib__mat__vec,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,V_1: vec_a,V_2: vec_a] :
( ( member_mat_a @ A3 @ ( 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 @ A3 @ ( plus_plus_vec_a @ V_1 @ V_2 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A3 @ V_1 ) @ ( mult_mat_vec_a @ A3 @ V_2 ) ) ) ) ) ) ).
% mult_add_distrib_mat_vec
thf(fact_553_minus__mult__distrib__mat__vec,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( minus_minus_mat_a @ A3 @ B3 ) @ V )
= ( minus_minus_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ ( mult_mat_vec_a @ B3 @ V ) ) ) ) ) ) ).
% minus_mult_distrib_mat_vec
thf(fact_554_mult__minus__distrib__mat__vec,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,V: vec_a,W: vec_a] :
( ( member_mat_a @ A3 @ ( 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 @ A3 @ ( minus_minus_vec_a @ V @ W ) )
= ( minus_minus_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ ( mult_mat_vec_a @ A3 @ W ) ) ) ) ) ) ).
% mult_minus_distrib_mat_vec
thf(fact_555_M__last,axiom,
member_mat_a @ m_last @ ( carrier_mat_a @ nr @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_last
thf(fact_556__092_060open_062M__up_092_060_094sup_062T_A_061_Afour__block__mat_AA_092_060_094sup_062T_A_Imat__of__col_A_I_N_Ac_J_J_A_I0_092_060_094sub_062m_Anr_Anr_J_A_Imat__of__col_Ab_J_092_060close_062,axiom,
( ( transpose_mat_a @ m_up )
= ( four_block_mat_a @ ( transpose_mat_a @ a2 ) @ ( missing_mat_of_col_a @ ( uminus_uminus_vec_a @ c ) ) @ ( zero_mat_a @ nr @ nr ) @ ( missing_mat_of_col_a @ b ) ) ) ).
% \<open>M_up\<^sup>T = four_block_mat A\<^sup>T (mat_of_col (- c)) (0\<^sub>m nr nr) (mat_of_col b)\<close>
thf(fact_557__092_060open_062M__low_092_060_094sup_062T_A_061_Afour__block__mat_A_I0_092_060_094sub_062m_Anc_Anc_J_A_I0_092_060_094sub_062m_Anc_Anc_J_AA_A_I_N_AA_J_092_060close_062,axiom,
( ( transpose_mat_a @ m_low )
= ( four_block_mat_a @ ( zero_mat_a @ nc @ nc ) @ ( zero_mat_a @ nc @ nc ) @ a2 @ ( uminus_uminus_mat_a @ a2 ) ) ) ).
% \<open>M_low\<^sup>T = four_block_mat (0\<^sub>m nc nc) (0\<^sub>m nc nc) A (- A)\<close>
thf(fact_558_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_559_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_560_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_561_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_562_Diff__iff,axiom,
! [C: vec_a,A3: set_vec_a,B3: set_vec_a] :
( ( member_vec_a @ C @ ( minus_6230920740010926198_vec_a @ A3 @ B3 ) )
= ( ( member_vec_a @ C @ A3 )
& ~ ( member_vec_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_563_Diff__iff,axiom,
! [C: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ C @ ( minus_4757590266979429866_mat_a @ A3 @ B3 ) )
= ( ( member_mat_a @ C @ A3 )
& ~ ( member_mat_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_564_DiffI,axiom,
! [C: vec_a,A3: set_vec_a,B3: set_vec_a] :
( ( member_vec_a @ C @ A3 )
=> ( ~ ( member_vec_a @ C @ B3 )
=> ( member_vec_a @ C @ ( minus_6230920740010926198_vec_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_565_DiffI,axiom,
! [C: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ C @ A3 )
=> ( ~ ( member_mat_a @ C @ B3 )
=> ( member_mat_a @ C @ ( minus_4757590266979429866_mat_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_566_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_567_DiffD2,axiom,
! [C: vec_a,A3: set_vec_a,B3: set_vec_a] :
( ( member_vec_a @ C @ ( minus_6230920740010926198_vec_a @ A3 @ B3 ) )
=> ~ ( member_vec_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_568_DiffD2,axiom,
! [C: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ C @ ( minus_4757590266979429866_mat_a @ A3 @ B3 ) )
=> ~ ( member_mat_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_569_DiffD1,axiom,
! [C: vec_a,A3: set_vec_a,B3: set_vec_a] :
( ( member_vec_a @ C @ ( minus_6230920740010926198_vec_a @ A3 @ B3 ) )
=> ( member_vec_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_570_DiffD1,axiom,
! [C: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ C @ ( minus_4757590266979429866_mat_a @ A3 @ B3 ) )
=> ( member_mat_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_571_DiffE,axiom,
! [C: vec_a,A3: set_vec_a,B3: set_vec_a] :
( ( member_vec_a @ C @ ( minus_6230920740010926198_vec_a @ A3 @ B3 ) )
=> ~ ( ( member_vec_a @ C @ A3 )
=> ( member_vec_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_572_DiffE,axiom,
! [C: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ C @ ( minus_4757590266979429866_mat_a @ A3 @ B3 ) )
=> ~ ( ( member_mat_a @ C @ A3 )
=> ( member_mat_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_573_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_574_mat__of__col__def,axiom,
( missing_mat_of_col_a
= ( ^ [V3: vec_a] : ( transpose_mat_a @ ( mat_of_row_a @ V3 ) ) ) ) ).
% mat_of_col_def
thf(fact_575__092_060open_062M_092_060_094sup_062T_A_K_092_060_094sub_062v_Aulv_A_061_A_IM__up_092_060_094sup_062T_A_064_092_060_094sub_062c_AM__low_092_060_094sup_062T_J_A_K_092_060_094sub_062v_Au1_A_L_AM__last_092_060_094sup_062T_A_K_092_060_094sub_062v_At_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ m ) @ ulv )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( missin386308114684349109cols_a @ ( transpose_mat_a @ m_up ) @ ( transpose_mat_a @ m_low ) ) @ u1 ) @ ( mult_mat_vec_a @ ( transpose_mat_a @ m_last ) @ t ) ) ) ).
% \<open>M\<^sup>T *\<^sub>v ulv = (M_up\<^sup>T @\<^sub>c M_low\<^sup>T) *\<^sub>v u1 + M_last\<^sup>T *\<^sub>v t\<close>
thf(fact_576__092_060open_062_IM__up_092_060_094sup_062T_A_064_092_060_094sub_062c_AM__low_092_060_094sup_062T_J_A_K_092_060_094sub_062v_Au1_A_061_AM__up_092_060_094sup_062T_A_K_092_060_094sub_062v_Au2_A_L_AM__low_092_060_094sup_062T_A_K_092_060_094sub_062v_Au3_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( missin386308114684349109cols_a @ ( transpose_mat_a @ m_up ) @ ( transpose_mat_a @ m_low ) ) @ u1 )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ m_up ) @ u2 ) @ ( mult_mat_vec_a @ ( transpose_mat_a @ m_low ) @ u3 ) ) ) ).
% \<open>(M_up\<^sup>T @\<^sub>c M_low\<^sup>T) *\<^sub>v u1 = M_up\<^sup>T *\<^sub>v u2 + M_low\<^sup>T *\<^sub>v u3\<close>
thf(fact_577_Mt,axiom,
( ( transpose_mat_a @ m )
= ( missin386308114684349109cols_a @ ( missin386308114684349109cols_a @ ( transpose_mat_a @ m_up ) @ ( transpose_mat_a @ m_low ) ) @ ( transpose_mat_a @ m_last ) ) ) ).
% Mt
thf(fact_578_M__def,axiom,
( m
= ( append_rows_a @ ( append_rows_a @ m_up @ m_low ) @ m_last ) ) ).
% M_def
thf(fact_579__092_060open_062_I0_092_060_094sub_062m_Anc_Anr_A_064_092_060_094sub_062r_A_N_A1_092_060_094sub_062m_Anr_J_A_K_092_060_094sub_062v_At_A_061_A0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A_N_At_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( append_rows_a @ ( zero_mat_a @ nc @ nr ) @ ( uminus_uminus_mat_a @ ( one_mat_a @ nr ) ) ) @ t )
= ( append_vec_a @ ( zero_vec_a @ nc ) @ ( uminus_uminus_vec_a @ t ) ) ) ).
% \<open>(0\<^sub>m nc nr @\<^sub>r - 1\<^sub>m nr) *\<^sub>v t = 0\<^sub>v nc @\<^sub>v - t\<close>
thf(fact_580_mat__mult__append__cols,axiom,
! [A3: mat_a,Nr: nat,Nc1: nat,B3: mat_a,Nc2: nat,V1: vec_a,V22: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( 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 @ A3 @ B3 ) @ ( append_vec_a @ V1 @ V22 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A3 @ V1 ) @ ( mult_mat_vec_a @ B3 @ V22 ) ) ) ) ) ) ) ).
% mat_mult_append_cols
thf(fact_581_transpose__one,axiom,
! [N: nat] :
( ( transpose_mat_a @ ( one_mat_a @ N ) )
= ( one_mat_a @ N ) ) ).
% transpose_one
thf(fact_582_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_583_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_584_carrier__append__rows,axiom,
! [A3: mat_a,Nr1: nat,Nc: nat,B3: mat_a,Nr2: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( member_mat_a @ ( append_rows_a @ A3 @ B3 ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ Nc ) ) ) ) ).
% carrier_append_rows
thf(fact_585_carrier__append__cols,axiom,
! [A3: mat_a,Nr: nat,Nc1: nat,B3: mat_a,Nc2: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( member_mat_a @ ( missin386308114684349109cols_a @ A3 @ B3 ) @ ( carrier_mat_a @ Nr @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% carrier_append_cols
thf(fact_586__092_060open_062M__last_092_060_094sup_062T_A_061_A0_092_060_094sub_062m_Anc_Anr_A_064_092_060_094sub_062r_A_N_A1_092_060_094sub_062m_Anr_092_060close_062,axiom,
( ( transpose_mat_a @ m_last )
= ( append_rows_a @ ( zero_mat_a @ nc @ nr ) @ ( uminus_uminus_mat_a @ ( one_mat_a @ nr ) ) ) ) ).
% \<open>M_last\<^sup>T = 0\<^sub>m nc nr @\<^sub>r - 1\<^sub>m nr\<close>
thf(fact_587_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_588_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_a @ ( one_mat_a @ N ) @ ( carrier_mat_a @ N @ N ) ) ).
% one_carrier_mat
thf(fact_589_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_590_mat__mult__append,axiom,
! [A3: mat_a,Nr1: nat,Nc: nat,B3: mat_a,Nr2: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( append_rows_a @ A3 @ B3 ) @ V )
= ( append_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ ( mult_mat_vec_a @ B3 @ V ) ) ) ) ) ) ).
% mat_mult_append
thf(fact_591_append__rows__le,axiom,
! [A3: mat_a,Nr1: nat,Nc: nat,B3: mat_a,Nr2: nat,A: vec_a,V: vec_a,B: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( 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 @ A3 @ B3 ) @ V ) @ ( append_vec_a @ A @ B ) )
= ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ A )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ B3 @ V ) @ B ) ) ) ) ) ) ) ).
% append_rows_le
thf(fact_592_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_593_weak__duality__theorem,axiom,
! [A3: mat_nat,Nr: nat,Nc: nat,B: vec_nat,C: vec_nat,X: vec_nat,Y: vec_nat] :
( ( member_mat_nat @ A3 @ ( 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 @ A3 @ X ) @ B )
=> ( ( ord_less_eq_vec_nat @ ( zero_vec_nat @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_nat @ ( transpose_mat_nat @ A3 ) @ Y )
= C )
=> ( ord_less_eq_nat @ ( scalar_prod_nat @ C @ X ) @ ( scalar_prod_nat @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_594_weak__duality__theorem,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a,X: vec_a,Y: vec_a] :
( ( member_mat_a @ A3 @ ( 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 @ A3 @ X ) @ B )
=> ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_a @ ( transpose_mat_a @ A3 ) @ Y )
= C )
=> ( ord_less_eq_a @ ( scalar_prod_a @ C @ X ) @ ( scalar_prod_a @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_595_unbounded__dual__solutions,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ! [V4: a] :
? [Y4: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y4 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A3 ) @ Y4 )
= C )
& ( ord_less_eq_a @ ( scalar_prod_a @ B @ Y4 ) @ V4 ) )
=> ~ ? [X4: vec_a] :
( ( member_vec_a @ X4 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A3 @ X4 ) @ B ) ) ) ) ) ) ).
% unbounded_dual_solutions
thf(fact_596_unbounded__primal__solutions,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ! [V4: a] :
? [X4: vec_a] :
( ( member_vec_a @ X4 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A3 @ X4 ) @ B )
& ( ord_less_eq_a @ V4 @ ( 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 @ A3 ) @ Y4 )
= C ) ) ) ) ) ) ).
% unbounded_primal_solutions
thf(fact_597_set__times__intro,axiom,
! [A: mat_a,C2: set_mat_a,B: mat_a,D2: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( member_mat_a @ B @ D2 )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_1230744552615602198_mat_a @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_598_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_599_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_600_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_601_assoc__mult__mat,axiom,
! [A3: mat_a,N_1: nat,N_2: nat,B3: mat_a,N_3: nat,C2: mat_a,N_4: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B3 @ ( 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 @ A3 @ B3 ) @ C2 )
= ( times_times_mat_a @ A3 @ ( times_times_mat_a @ B3 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_602_assoc__mult__mat__vec,axiom,
! [A3: mat_a,N_1: nat,N_2: nat,B3: mat_a,N_3: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N_3 ) )
=> ( ( mult_mat_vec_a @ ( times_times_mat_a @ A3 @ B3 ) @ V )
= ( mult_mat_vec_a @ A3 @ ( mult_mat_vec_a @ B3 @ V ) ) ) ) ) ) ).
% assoc_mult_mat_vec
thf(fact_603_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_604_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_605_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_606_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_607_set__times__elim,axiom,
! [X: mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( member_mat_a @ X @ ( times_1230744552615602198_mat_a @ A3 @ B3 ) )
=> ~ ! [A4: mat_a,B4: mat_a] :
( ( X
= ( times_times_mat_a @ A4 @ B4 ) )
=> ( ( member_mat_a @ A4 @ A3 )
=> ~ ( member_mat_a @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_608_set__times__elim,axiom,
! [X: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A3 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A3 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_609_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_610_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_611_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_612_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_613_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_614_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_615_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_616_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_617_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_618_mult__carrier__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( member_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_619_left__mult__zero__mat,axiom,
! [A3: mat_a,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_620_right__mult__zero__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( times_times_mat_a @ A3 @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_621_left__mult__one__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ ( one_mat_a @ Nr ) @ A3 )
= A3 ) ) ).
% left_mult_one_mat
thf(fact_622_right__mult__one__mat,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ A3 @ ( one_mat_a @ Nc ) )
= A3 ) ) ).
% right_mult_one_mat
thf(fact_623_transpose__mult,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( transpose_mat_a @ ( times_times_mat_a @ A3 @ B3 ) )
= ( times_times_mat_a @ ( transpose_mat_a @ B3 ) @ ( transpose_mat_a @ A3 ) ) ) ) ) ).
% transpose_mult
thf(fact_624_mult__add__distrib__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A3 @ ( plus_plus_mat_a @ B3 @ C2 ) )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ ( times_times_mat_a @ A3 @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_625_add__mult__distrib__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ C2 )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A3 @ C2 ) @ ( times_times_mat_a @ B3 @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_626_minus__mult__distrib__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( minus_minus_mat_a @ A3 @ B3 ) @ C2 )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A3 @ C2 ) @ ( times_times_mat_a @ B3 @ C2 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_627_mult__minus__distrib__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A3 @ ( minus_minus_mat_a @ B3 @ C2 ) )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ ( times_times_mat_a @ A3 @ C2 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_628_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_629_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_630_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_631_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_632_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_633_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_634_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_635_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_636_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_637_ordered__ring__class_Ole__add__iff1,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_638_ordered__ring__class_Ole__add__iff2,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ord_less_eq_a @ C @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_639_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_640_transpose__vec__mult__scalar,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,X: vec_a,Y: vec_a] :
( ( member_mat_a @ A3 @ ( 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 @ A3 ) @ Y ) @ X )
= ( scalar_prod_a @ Y @ ( mult_mat_vec_a @ A3 @ X ) ) ) ) ) ) ).
% transpose_vec_mult_scalar
thf(fact_641_mult__mat__of__col,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( times_times_mat_a @ A3 @ ( missing_mat_of_col_a @ V ) )
= ( missing_mat_of_col_a @ ( mult_mat_vec_a @ A3 @ V ) ) ) ) ) ).
% mult_mat_of_col
thf(fact_642_gram__schmidt_OFarkas__Lemma,axiom,
! [A3: mat_a,N: nat,Nr: nat,B: vec_a] :
( ( member_mat_a @ A3 @ ( 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 @ A3 @ 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 @ A3 ) @ Y2 ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ Y2 @ B ) ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma
thf(fact_643_gram__schmidt_OFarkas__Lemma_H,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B: vec_a] :
( ( member_mat_a @ A3 @ ( 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 @ A3 @ X2 ) @ B ) ) )
= ( ! [Y2: vec_a] :
( ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y2 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A3 ) @ Y2 )
= ( zero_vec_a @ Nc ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ Y2 @ B ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma'
thf(fact_644_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_645_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_646_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_647_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_648_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_649_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_650_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_651_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_652_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_653_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_654_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_655_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_656_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_657_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_658_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_659_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_660_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_661_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_662_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_663_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_664_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_665_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_666_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_667_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_668_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_669_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_694_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_695_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_696_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_697_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_698_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_699_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_700_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_701_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_702_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_703_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_704_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_705_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_706_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_707_le__numeral__extra_I3_J,axiom,
ord_less_eq_a @ zero_zero_a @ zero_zero_a ).
% le_numeral_extra(3)
thf(fact_708_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_709_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_710_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_711_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_712_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_713_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_714_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_715_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_716_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_717_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_718_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_719_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_720_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_721_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_722_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_723_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_724_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_725_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_726_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_727_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_728_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_729_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_730_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_731_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_732_not__one__le__zero,axiom,
~ ( ord_less_eq_a @ one_one_a @ zero_zero_a ) ).
% not_one_le_zero
thf(fact_733_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_734_mult__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_735_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_736_mult__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_737_zero__le__square,axiom,
! [A: a] : ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ A ) ) ).
% zero_le_square
thf(fact_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_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_772_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_773_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_774_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_775_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_776_le__iff__diff__le__0,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] : ( ord_less_eq_a @ ( minus_minus_a @ A2 @ B2 ) @ zero_zero_a ) ) ) ).
% le_iff_diff_le_0
thf(fact_777_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_778_set__zero__plus2,axiom,
! [A3: set_nat,B3: set_nat] :
( ( member_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_set_nat @ B3 @ ( plus_plus_set_nat @ A3 @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_793_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_794_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_795_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_796_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_806_less__eq__fract__respect,axiom,
! [B: a,B6: a,D: a,D3: a,A: a,A6: a,C: a,C5: a] :
( ( B != zero_zero_a )
=> ( ( B6 != zero_zero_a )
=> ( ( D != 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 @ D ) )
=> ( ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A @ D ) @ ( times_times_a @ B @ D ) ) @ ( times_times_a @ ( times_times_a @ C @ B ) @ ( times_times_a @ B @ D ) ) )
= ( 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_807_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_808_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_809_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_810_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_811_norm1__ge__0,axiom,
! [F2: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( norm1_a @ F2 ) ) ).
% norm1_ge_0
thf(fact_812_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_813_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_814_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_815_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_a @ ( zero_vec_a @ N ) )
= zero_zero_a ) ).
% linf_norm_zero_vec
thf(fact_816_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_817_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_818_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_819_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_820_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_821_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_822_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_823_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_824_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_825_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_826_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_827_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_828_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_leD,axiom,
! [Y: vec_a,X: vec_a] :
( ( ord_less_eq_vec_a @ Y @ X )
=> ~ ( ord_less_vec_a @ X @ Y ) ) ).
% leD
thf(fact_846_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_847_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_848_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_849_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_850_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_851_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_852_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_853_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_854_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_855_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_856_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_857_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_858_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_859_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_860_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_861_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_862_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_863_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_864_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_865_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_866_order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_vec_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_867_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_868_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_869_order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_870_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_871_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_872_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_873_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_874_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_875_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_876_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_877_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_878_order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ A2 @ B2 )
& ~ ( ord_less_eq_vec_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_879_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_880_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_881_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_882_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_883_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [B2: vec_a,A2: vec_a] :
( ( ord_less_vec_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_884_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_885_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_886_dual__order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [B2: vec_a,A2: vec_a] :
( ( ord_less_eq_vec_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_887_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_888_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_889_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_890_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_891_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_892_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_893_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_894_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_895_dual__order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [B2: vec_a,A2: vec_a] :
( ( ord_less_eq_vec_a @ B2 @ A2 )
& ~ ( ord_less_eq_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_896_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_897_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_906_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_907_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_908_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_909_order__less__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_910_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_911_linorder__not__le,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X @ Y ) )
= ( ord_less_a @ Y @ X ) ) ).
% linorder_not_le
thf(fact_912_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_913_linorder__not__less,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_not_less
thf(fact_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_959_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_960_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_961_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_962_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_963_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_964_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_965_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_966_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_967_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_968_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_969_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_978_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_979_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_980_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_981_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_982_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_983_psubset__imp__ex__mem,axiom,
! [A3: set_vec_a,B3: set_vec_a] :
( ( ord_less_set_vec_a @ A3 @ B3 )
=> ? [B4: vec_a] : ( member_vec_a @ B4 @ ( minus_6230920740010926198_vec_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_984_psubset__imp__ex__mem,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( ord_less_set_mat_a @ A3 @ B3 )
=> ? [B4: mat_a] : ( member_mat_a @ B4 @ ( minus_4757590266979429866_mat_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_985_less__vec__def,axiom,
( ord_less_vec_a
= ( ^ [V3: vec_a,W4: vec_a] :
( ( ord_less_eq_vec_a @ V3 @ W4 )
& ~ ( ord_less_eq_vec_a @ W4 @ V3 ) ) ) ) ).
% less_vec_def
thf(fact_986_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_987_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_988_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_989_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_990_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_991_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_992_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_993_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_994_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_995_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_996_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1007_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_1008_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1009_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1010_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1011_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1012_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1013_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1014_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1015_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1016_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1017_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1027_add__less__le__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1028_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1029_add__le__less__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1030_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_1031_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_1032_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_1033_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_1034_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_1035_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1036_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_1037_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_1038_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1039_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_1040_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_1041_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_1042_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1043_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_1044_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_1045_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_1046_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1047_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1048_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_1049_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_1050_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_1051_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_1052_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_1053_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1054_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1055_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_1056_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_1057_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_1058_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1059_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1068_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1069_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1070_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1112_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1113_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1114_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_1115_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1116_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1117_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1118_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1119_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_1120_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_1121_ordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono
thf(fact_1122_ordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono
thf(fact_1123_ordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono'
thf(fact_1124_ordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono'
thf(fact_1125_ordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1126_ordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1127_ordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1128_ordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1129_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1130_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_1131_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_1132_diff__diff__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N ) ) )
= ( ( ord_less_nat @ I @ M )
& ( ord_less_nat @ I @ N ) ) ) ).
% diff_diff_less
thf(fact_1133_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_1134__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_1135_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_1136_index__zero__vec_I2_J,axiom,
! [N: nat] :
( ( dim_vec_a @ ( zero_vec_a @ N ) )
= N ) ).
% index_zero_vec(2)
thf(fact_1137_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_1138_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_1139_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_1140_dim__vec__first,axiom,
! [V: vec_a,N: nat] :
( ( dim_vec_a @ ( vec_first_a @ V @ N ) )
= N ) ).
% dim_vec_first
thf(fact_1141_dim__vec__last,axiom,
! [V: vec_a,N: nat] :
( ( dim_vec_a @ ( vec_last_a @ V @ N ) )
= N ) ).
% dim_vec_last
thf(fact_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_index__minus__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 @ ( minus_3631651556841400635_vec_a @ V_1 @ V_2 ) @ I )
= ( minus_minus_vec_a @ ( vec_index_vec_a @ V_1 @ I ) @ ( vec_index_vec_a @ V_2 @ I ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_vec__add__mono,axiom,
! [B: vec_a,D: vec_a,A: vec_a,C: vec_a] :
( ( ( dim_vec_a @ B )
= ( dim_vec_a @ D ) )
=> ( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ C @ D )
=> ( ord_less_eq_vec_a @ ( plus_plus_vec_a @ A @ C ) @ ( plus_plus_vec_a @ B @ D ) ) ) ) ) ).
% vec_add_mono
thf(fact_1163_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_1164_less__eq__vec__def,axiom,
( ord_le4012615358376148468_vec_a
= ( ^ [V3: vec_vec_a,W4: vec_vec_a] :
( ( ( dim_vec_vec_a @ V3 )
= ( 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 @ V3 @ I4 ) @ ( vec_index_vec_a @ W4 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_1165_less__eq__vec__def,axiom,
( ord_less_eq_vec_nat
= ( ^ [V3: vec_nat,W4: vec_nat] :
( ( ( dim_vec_nat @ V3 )
= ( dim_vec_nat @ W4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_nat @ W4 ) )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V3 @ I4 ) @ ( vec_index_nat @ W4 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_1166_less__eq__vec__def,axiom,
( ord_less_eq_vec_a
= ( ^ [V3: vec_a,W4: vec_a] :
( ( ( dim_vec_a @ V3 )
= ( dim_vec_a @ W4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_a @ W4 ) )
=> ( ord_less_eq_a @ ( vec_index_a @ V3 @ I4 ) @ ( vec_index_a @ W4 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_1167_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_1168_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_1169_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_1170_vec__le__iff__diff__le__0,axiom,
( ord_less_eq_vec_a
= ( ^ [A2: vec_a,B2: vec_a] : ( ord_less_eq_vec_a @ ( minus_minus_vec_a @ A2 @ B2 ) @ ( zero_vec_a @ ( dim_vec_a @ A2 ) ) ) ) ) ).
% vec_le_iff_diff_le_0
thf(fact_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_index__vec__of__scal,axiom,
! [X: a] :
( ( vec_index_a @ ( missin5951511974119752530scal_a @ X ) @ zero_zero_nat )
= X ) ).
% index_vec_of_scal
thf(fact_1186_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_1187_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_1188_append__cols__nth_I2_J,axiom,
! [A3: mat_a,Nr: nat,Nc1: nat,B3: mat_a,Nc2: nat,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( ( ord_less_eq_nat @ Nc1 @ I )
=> ( ( ord_less_nat @ I @ ( plus_plus_nat @ Nc1 @ Nc2 ) )
=> ( ( col_a @ ( missin386308114684349109cols_a @ A3 @ B3 ) @ I )
= ( col_a @ B3 @ ( minus_minus_nat @ I @ Nc1 ) ) ) ) ) ) ) ).
% append_cols_nth(2)
thf(fact_1189_col__mat__of__col,axiom,
! [V: vec_a] :
( ( col_a @ ( missing_mat_of_col_a @ V ) @ zero_zero_nat )
= V ) ).
% col_mat_of_col
thf(fact_1190_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_1191_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_1192_col__zero,axiom,
! [J: nat,Nc: nat,Nr: nat] :
( ( ord_less_nat @ J @ Nc )
=> ( ( col_a @ ( zero_mat_a @ Nr @ Nc ) @ J )
= ( zero_vec_a @ Nr ) ) ) ).
% col_zero
thf(fact_1193_col__mult2,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat,J: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( col_a @ ( times_times_mat_a @ A3 @ B3 ) @ J )
= ( mult_mat_vec_a @ A3 @ ( col_a @ B3 @ J ) ) ) ) ) ) ).
% col_mult2
thf(fact_1194_row__add_I1_J,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I @ Nr )
=> ( ( row_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ I )
= ( plus_plus_vec_a @ ( row_a @ A3 @ I ) @ ( row_a @ B3 @ I ) ) ) ) ) ) ).
% row_add(1)
thf(fact_1195_col__add,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a,J: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( col_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ J )
= ( plus_plus_vec_a @ ( col_a @ A3 @ J ) @ ( col_a @ B3 @ J ) ) ) ) ) ) ).
% col_add
thf(fact_1196_row__carrier__vec,axiom,
! [I: nat,Nr: nat,A3: mat_a,Nc: nat] :
( ( ord_less_nat @ I @ Nr )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_vec_a @ ( row_a @ A3 @ I ) @ ( carrier_vec_a @ Nc ) ) ) ) ).
% row_carrier_vec
thf(fact_1197_col__carrier__vec,axiom,
! [J: nat,Nc: nat,A3: mat_a,Nr: nat] :
( ( ord_less_nat @ J @ Nc )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_vec_a @ ( col_a @ A3 @ J ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% col_carrier_vec
thf(fact_1198_append__rows__nth_I1_J,axiom,
! [A3: mat_a,Nr1: nat,Nc: nat,B3: mat_a,Nr2: nat,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( ord_less_nat @ I @ Nr1 )
=> ( ( row_a @ ( append_rows_a @ A3 @ B3 ) @ I )
= ( row_a @ A3 @ I ) ) ) ) ) ).
% append_rows_nth(1)
thf(fact_1199_append__cols__nth_I1_J,axiom,
! [A3: mat_a,Nr: nat,Nc1: nat,B3: mat_a,Nc2: nat,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( ( ord_less_nat @ I @ Nc1 )
=> ( ( col_a @ ( missin386308114684349109cols_a @ A3 @ B3 ) @ I )
= ( col_a @ A3 @ I ) ) ) ) ) ).
% append_cols_nth(1)
thf(fact_1200_col__four__block__mat_I1_J,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,B3: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D2: mat_a,J: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( ord_less_nat @ J @ Nc1 )
=> ( ( col_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) @ J )
= ( append_vec_a @ ( col_a @ A3 @ J ) @ ( col_a @ C2 @ J ) ) ) ) ) ) ) ) ).
% col_four_block_mat(1)
thf(fact_1201_row__four__block__mat_I1_J,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,B3: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D2: mat_a,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( ord_less_nat @ I @ Nr1 )
=> ( ( row_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) @ I )
= ( append_vec_a @ ( row_a @ A3 @ I ) @ ( row_a @ B3 @ I ) ) ) ) ) ) ) ) ).
% row_four_block_mat(1)
thf(fact_1202_col__four__block__mat_I2_J,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,B3: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D2: mat_a,J: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ~ ( ord_less_nat @ J @ Nc1 )
=> ( ( ord_less_nat @ J @ ( plus_plus_nat @ Nc1 @ Nc2 ) )
=> ( ( col_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) @ J )
= ( append_vec_a @ ( col_a @ B3 @ ( minus_minus_nat @ J @ Nc1 ) ) @ ( col_a @ D2 @ ( minus_minus_nat @ J @ Nc1 ) ) ) ) ) ) ) ) ) ) ).
% col_four_block_mat(2)
thf(fact_1203_row__four__block__mat_I2_J,axiom,
! [A3: mat_a,Nr1: nat,Nc1: nat,B3: mat_a,Nc2: nat,C2: mat_a,Nr2: nat,D2: mat_a,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( 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 @ A3 @ B3 @ C2 @ D2 ) @ I )
= ( append_vec_a @ ( row_a @ C2 @ ( minus_minus_nat @ I @ Nr1 ) ) @ ( row_a @ D2 @ ( minus_minus_nat @ I @ Nr1 ) ) ) ) ) ) ) ) ) ) ).
% row_four_block_mat(2)
thf(fact_1204_append__rows__nth_I2_J,axiom,
! [A3: mat_a,Nr1: nat,Nc: nat,B3: mat_a,Nr2: nat,I: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( 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 @ A3 @ B3 ) @ I )
= ( row_a @ B3 @ ( minus_minus_nat @ I @ Nr1 ) ) ) ) ) ) ) ).
% append_rows_nth(2)
thf(fact_1205_row__echelon__form__imp__1__or__0__row,axiom,
! [A3: mat_a,N: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( gauss_5855338539171749649form_a @ A3 )
=> ( ( A3
= ( one_mat_a @ N ) )
| ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( row_a @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) )
= ( zero_vec_a @ N ) ) ) ) ) ) ).
% row_echelon_form_imp_1_or_0_row
thf(fact_1206_find__base__vector__not__1_I3_J,axiom,
! [A3: mat_a,N: nat] :
( ( gauss_5855338539171749649form_a @ A3 )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A3
!= ( one_mat_a @ N ) )
=> ( ( mult_mat_vec_a @ A3 @ ( gauss_6280258074615264798ctor_a @ A3 ) )
= ( zero_vec_a @ N ) ) ) ) ) ).
% find_base_vector_not_1(3)
thf(fact_1207_gauss__jordan__single_I1_J,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,C2: mat_a,X: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A3 )
= C2 )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( ( mult_mat_vec_a @ A3 @ X )
= ( zero_vec_a @ Nr ) )
= ( ( mult_mat_vec_a @ C2 @ X )
= ( zero_vec_a @ Nr ) ) ) ) ) ) ).
% gauss_jordan_single(1)
thf(fact_1208_find__base__vector__not__1_I1_J,axiom,
! [A3: mat_a,N: nat] :
( ( gauss_5855338539171749649form_a @ A3 )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A3
!= ( one_mat_a @ N ) )
=> ( member_vec_a @ ( gauss_6280258074615264798ctor_a @ A3 ) @ ( carrier_vec_a @ N ) ) ) ) ) ).
% find_base_vector_not_1(1)
thf(fact_1209_find__base__vector__not__1_I2_J,axiom,
! [A3: mat_a,N: nat] :
( ( gauss_5855338539171749649form_a @ A3 )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( A3
!= ( one_mat_a @ N ) )
=> ( ( gauss_6280258074615264798ctor_a @ A3 )
!= ( zero_vec_a @ N ) ) ) ) ) ).
% find_base_vector_not_1(2)
thf(fact_1210_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_1211_index__mult__mat__vec,axiom,
! [I: nat,A3: mat_a,V: vec_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A3 ) )
=> ( ( vec_index_a @ ( mult_mat_vec_a @ A3 @ V ) @ I )
= ( scalar_prod_a @ ( row_a @ A3 @ I ) @ V ) ) ) ).
% index_mult_mat_vec
thf(fact_1212_carrier__matD_I1_J,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_a @ A3 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_1213_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_1214_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_a @ ( one_mat_a @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_1215_index__uminus__mat_I2_J,axiom,
! [A3: mat_a] :
( ( dim_row_a @ ( uminus_uminus_mat_a @ A3 ) )
= ( dim_row_a @ A3 ) ) ).
% index_uminus_mat(2)
thf(fact_1216_index__mat__four__block_I2_J,axiom,
! [A3: mat_a,B3: mat_a,C2: mat_a,D2: mat_a] :
( ( dim_row_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) )
= ( plus_plus_nat @ ( dim_row_a @ A3 ) @ ( dim_row_a @ D2 ) ) ) ).
% index_mat_four_block(2)
thf(fact_1217_left__mult__one__mat_H,axiom,
! [A3: mat_a,N: nat] :
( ( ( dim_row_a @ A3 )
= N )
=> ( ( times_times_mat_a @ ( one_mat_a @ N ) @ A3 )
= A3 ) ) ).
% left_mult_one_mat'
thf(fact_1218_dim__mult__mat__vec,axiom,
! [A3: mat_a,V: vec_a] :
( ( dim_vec_a @ ( mult_mat_vec_a @ A3 @ V ) )
= ( dim_row_a @ A3 ) ) ).
% dim_mult_mat_vec
thf(fact_1219_dim__col,axiom,
! [A3: mat_a,I: nat] :
( ( dim_vec_a @ ( col_a @ A3 @ I ) )
= ( dim_row_a @ A3 ) ) ).
% dim_col
thf(fact_1220_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_1221_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_1222_col__transpose,axiom,
! [I: nat,A3: mat_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A3 ) )
=> ( ( col_a @ ( transpose_mat_a @ A3 ) @ I )
= ( row_a @ A3 @ I ) ) ) ).
% col_transpose
thf(fact_1223_row__uminus,axiom,
! [I: nat,A3: mat_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A3 ) )
=> ( ( row_a @ ( uminus_uminus_mat_a @ A3 ) @ I )
= ( uminus_uminus_vec_a @ ( row_a @ A3 @ I ) ) ) ) ).
% row_uminus
thf(fact_1224_col__dim,axiom,
! [A3: mat_a,I: nat] : ( member_vec_a @ ( col_a @ A3 @ I ) @ ( carrier_vec_a @ ( dim_row_a @ A3 ) ) ) ).
% col_dim
thf(fact_1225_mat__of__row__dim__row__1,axiom,
! [A3: mat_a] :
( ( ( dim_row_a @ A3 )
= one_one_nat )
= ( A3
= ( mat_of_row_a @ ( row_a @ A3 @ zero_zero_nat ) ) ) ) ).
% mat_of_row_dim_row_1
thf(fact_1226_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_1227_mat__row__first__last__append,axiom,
! [A3: mat_a,M: nat,N: nat] :
( ( ( dim_row_a @ A3 )
= ( plus_plus_nat @ M @ N ) )
=> ( ( append_rows_a @ ( missin3040492613037353666irst_a @ A3 @ M ) @ ( missin5577565584678110354last_a @ A3 @ N ) )
= A3 ) ) ).
% mat_row_first_last_append
thf(fact_1228_addrow__mat,axiom,
! [A3: mat_a,N: nat,Nc: nat,L: nat,A: a,K: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A3 )
= ( times_times_mat_a @ ( gauss_8159914756388622152_mat_a @ N @ A @ K @ L ) @ A3 ) ) ) ) ).
% addrow_mat
thf(fact_1229_addrow__mat,axiom,
! [A3: mat_nat,N: nat,Nc: nat,L: nat,A: nat,K: nat] :
( ( member_mat_nat @ A3 @ ( carrier_mat_nat @ N @ Nc ) )
=> ( ( ord_less_nat @ L @ N )
=> ( ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A3 )
= ( times_times_mat_nat @ ( gauss_6496870380031412486at_nat @ N @ A @ K @ L ) @ A3 ) ) ) ) ).
% addrow_mat
thf(fact_1230_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_1231_row__add_I2_J,axiom,
! [I: nat,A3: mat_a,B3: mat_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A3 ) )
=> ( ( ( dim_row_a @ B3 )
= ( dim_row_a @ A3 ) )
=> ( ( ( dim_col_a @ B3 )
= ( dim_col_a @ A3 ) )
=> ( ( row_a @ ( plus_plus_mat_a @ A3 @ B3 ) @ I )
= ( plus_plus_vec_a @ ( row_a @ A3 @ I ) @ ( row_a @ B3 @ I ) ) ) ) ) ) ).
% row_add(2)
thf(fact_1232_mat__col__first__def,axiom,
( missin6964418172097587752irst_a
= ( ^ [A5: mat_a,N4: nat] : ( transpose_mat_a @ ( missin3040492613037353666irst_a @ ( transpose_mat_a @ A5 ) @ N4 ) ) ) ) ).
% mat_col_first_def
thf(fact_1233_carrier__matD_I2_J,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_col_a @ A3 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_1234_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_1235_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_a @ ( one_mat_a @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_1236_index__uminus__mat_I3_J,axiom,
! [A3: mat_a] :
( ( dim_col_a @ ( uminus_uminus_mat_a @ A3 ) )
= ( dim_col_a @ A3 ) ) ).
% index_uminus_mat(3)
thf(fact_1237_carrier__matI,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_a @ A3 )
= Nr )
=> ( ( ( dim_col_a @ A3 )
= Nc )
=> ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_1238_index__mat__four__block_I3_J,axiom,
! [A3: mat_a,B3: mat_a,C2: mat_a,D2: mat_a] :
( ( dim_col_a @ ( four_block_mat_a @ A3 @ B3 @ C2 @ D2 ) )
= ( plus_plus_nat @ ( dim_col_a @ A3 ) @ ( dim_col_a @ D2 ) ) ) ).
% index_mat_four_block(3)
thf(fact_1239_index__transpose__mat_I2_J,axiom,
! [A3: mat_a] :
( ( dim_row_a @ ( transpose_mat_a @ A3 ) )
= ( dim_col_a @ A3 ) ) ).
% index_transpose_mat(2)
thf(fact_1240_index__transpose__mat_I3_J,axiom,
! [A3: mat_a] :
( ( dim_col_a @ ( transpose_mat_a @ A3 ) )
= ( dim_row_a @ A3 ) ) ).
% index_transpose_mat(3)
thf(fact_1241_right__mult__one__mat_H,axiom,
! [A3: mat_a,N: nat] :
( ( ( dim_col_a @ A3 )
= N )
=> ( ( times_times_mat_a @ A3 @ ( one_mat_a @ N ) )
= A3 ) ) ).
% right_mult_one_mat'
thf(fact_1242_index__row_I2_J,axiom,
! [A3: mat_a,I: nat] :
( ( dim_vec_a @ ( row_a @ A3 @ I ) )
= ( dim_col_a @ A3 ) ) ).
% index_row(2)
thf(fact_1243_mat__of__col__dim_I2_J,axiom,
! [V: vec_a] :
( ( dim_col_a @ ( missing_mat_of_col_a @ V ) )
= one_one_nat ) ).
% mat_of_col_dim(2)
thf(fact_1244_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_1245_left__mult__zero__mat_H,axiom,
! [A3: mat_a,N: nat,Nr: nat] :
( ( ( dim_row_a @ A3 )
= N )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A3 )
= ( zero_mat_a @ Nr @ ( dim_col_a @ A3 ) ) ) ) ).
% left_mult_zero_mat'
thf(fact_1246_right__mult__zero__mat_H,axiom,
! [A3: mat_a,N: nat,Nc: nat] :
( ( ( dim_col_a @ A3 )
= N )
=> ( ( times_times_mat_a @ A3 @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ ( dim_row_a @ A3 ) @ Nc ) ) ) ).
% right_mult_zero_mat'
thf(fact_1247_uminus__mult__left__mat,axiom,
! [A3: mat_a,B3: mat_a] :
( ( ( dim_col_a @ A3 )
= ( dim_row_a @ B3 ) )
=> ( ( times_times_mat_a @ ( uminus_uminus_mat_a @ A3 ) @ B3 )
= ( uminus_uminus_mat_a @ ( times_times_mat_a @ A3 @ B3 ) ) ) ) ).
% uminus_mult_left_mat
thf(fact_1248_uminus__mult__right__mat,axiom,
! [A3: mat_a,B3: mat_a] :
( ( ( dim_col_a @ A3 )
= ( dim_row_a @ B3 ) )
=> ( ( times_times_mat_a @ A3 @ ( uminus_uminus_mat_a @ B3 ) )
= ( uminus_uminus_mat_a @ ( times_times_mat_a @ A3 @ B3 ) ) ) ) ).
% uminus_mult_right_mat
thf(fact_1249_row__transpose,axiom,
! [J: nat,A3: mat_a] :
( ( ord_less_nat @ J @ ( dim_col_a @ A3 ) )
=> ( ( row_a @ ( transpose_mat_a @ A3 ) @ J )
= ( col_a @ A3 @ J ) ) ) ).
% row_transpose
thf(fact_1250_col__uminus,axiom,
! [I: nat,A3: mat_a] :
( ( ord_less_nat @ I @ ( dim_col_a @ A3 ) )
=> ( ( col_a @ ( uminus_uminus_mat_a @ A3 ) @ I )
= ( uminus_uminus_vec_a @ ( col_a @ A3 @ I ) ) ) ) ).
% col_uminus
thf(fact_1251_uminus__mult__mat__vec,axiom,
! [V: vec_a,A3: mat_a] :
( ( ( dim_vec_a @ V )
= ( dim_col_a @ A3 ) )
=> ( ( mult_mat_vec_a @ ( uminus_uminus_mat_a @ A3 ) @ V )
= ( uminus_uminus_vec_a @ ( mult_mat_vec_a @ A3 @ V ) ) ) ) ).
% uminus_mult_mat_vec
thf(fact_1252_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_1253_mat__row__first__carrier,axiom,
! [A3: mat_a,N: nat] : ( member_mat_a @ ( missin3040492613037353666irst_a @ A3 @ N ) @ ( carrier_mat_a @ N @ ( dim_col_a @ A3 ) ) ) ).
% mat_row_first_carrier
thf(fact_1254_row__carrier,axiom,
! [A3: mat_a,I: nat] : ( member_vec_a @ ( row_a @ A3 @ I ) @ ( carrier_vec_a @ ( dim_col_a @ A3 ) ) ) ).
% row_carrier
thf(fact_1255_assoc__four__block__mat,axiom,
! [A3: mat_a,B3: mat_a,C2: mat_a] :
( ( four_block_mat_a @ A3 @ ( zero_mat_a @ ( dim_row_a @ A3 ) @ ( dim_col_a @ ( four_block_mat_a @ B3 @ ( zero_mat_a @ ( dim_row_a @ B3 ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ B3 ) ) @ C2 ) ) ) @ ( zero_mat_a @ ( dim_row_a @ ( four_block_mat_a @ B3 @ ( zero_mat_a @ ( dim_row_a @ B3 ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ B3 ) ) @ C2 ) ) @ ( dim_col_a @ A3 ) ) @ ( four_block_mat_a @ B3 @ ( zero_mat_a @ ( dim_row_a @ B3 ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ B3 ) ) @ C2 ) )
= ( four_block_mat_a @ ( four_block_mat_a @ A3 @ ( zero_mat_a @ ( dim_row_a @ A3 ) @ ( dim_col_a @ B3 ) ) @ ( zero_mat_a @ ( dim_row_a @ B3 ) @ ( dim_col_a @ A3 ) ) @ B3 ) @ ( zero_mat_a @ ( dim_row_a @ ( four_block_mat_a @ A3 @ ( zero_mat_a @ ( dim_row_a @ A3 ) @ ( dim_col_a @ B3 ) ) @ ( zero_mat_a @ ( dim_row_a @ B3 ) @ ( dim_col_a @ A3 ) ) @ B3 ) ) @ ( dim_col_a @ C2 ) ) @ ( zero_mat_a @ ( dim_row_a @ C2 ) @ ( dim_col_a @ ( four_block_mat_a @ A3 @ ( zero_mat_a @ ( dim_row_a @ A3 ) @ ( dim_col_a @ B3 ) ) @ ( zero_mat_a @ ( dim_row_a @ B3 ) @ ( dim_col_a @ A3 ) ) @ B3 ) ) ) @ C2 ) ) ).
% assoc_four_block_mat
thf(fact_1256_mat__row__last__carrier,axiom,
! [A3: mat_a,N: nat] : ( member_mat_a @ ( missin5577565584678110354last_a @ A3 @ N ) @ ( carrier_mat_a @ N @ ( dim_col_a @ A3 ) ) ) ).
% mat_row_last_carrier
thf(fact_1257_mat__of__col__dim__col__1,axiom,
! [A3: mat_a] :
( ( ( dim_col_a @ A3 )
= one_one_nat )
= ( A3
= ( missing_mat_of_col_a @ ( col_a @ A3 @ zero_zero_nat ) ) ) ) ).
% mat_of_col_dim_col_1
thf(fact_1258_mat__col__first__carrier,axiom,
! [A3: mat_a,N: nat] : ( member_mat_a @ ( missin6964418172097587752irst_a @ A3 @ N ) @ ( carrier_mat_a @ ( dim_row_a @ A3 ) @ N ) ) ).
% mat_col_first_carrier
thf(fact_1259_orthogonal__matI,axiom,
! [A3: mat_nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_col_nat @ A3 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ A3 ) )
=> ( ( ( scalar_prod_nat @ ( col_nat @ A3 @ I2 ) @ ( col_nat @ A3 @ J2 ) )
= zero_zero_nat )
= ( I2 != J2 ) ) ) )
=> ( orthogonal_mat_nat @ A3 ) ) ).
% orthogonal_matI
thf(fact_1260_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
thf(fact_1261_orthogonal__matD,axiom,
! [A3: mat_nat,I: nat,J: nat] :
( ( orthogonal_mat_nat @ A3 )
=> ( ( ord_less_nat @ I @ ( dim_col_nat @ A3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A3 ) )
=> ( ( ( scalar_prod_nat @ ( col_nat @ A3 @ I ) @ ( col_nat @ A3 @ J ) )
= zero_zero_nat )
= ( I != J ) ) ) ) ) ).
% orthogonal_matD
thf(fact_1262_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_1263_mat__col__first__last__append,axiom,
! [A3: mat_a,M: nat,N: nat] :
( ( ( dim_col_a @ A3 )
= ( plus_plus_nat @ M @ N ) )
=> ( ( missin386308114684349109cols_a @ ( missin6964418172097587752irst_a @ A3 @ M ) @ ( missin1527294172917009324last_a @ A3 @ N ) )
= A3 ) ) ).
% mat_col_first_last_append
thf(fact_1264_mat__col__last__carrier,axiom,
! [A3: mat_a,N: nat] : ( member_mat_a @ ( missin1527294172917009324last_a @ A3 @ N ) @ ( carrier_mat_a @ ( dim_row_a @ A3 ) @ N ) ) ).
% mat_col_last_carrier
thf(fact_1265_mat__col__last__def,axiom,
( missin1527294172917009324last_a
= ( ^ [A5: mat_a,N4: nat] : ( transpose_mat_a @ ( missin5577565584678110354last_a @ ( transpose_mat_a @ A5 ) @ N4 ) ) ) ) ).
% mat_col_last_def
thf(fact_1266_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_1267_mat__kernel__mult__eq,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,B3: mat_a,C2: mat_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ Nr @ Nr ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nr ) )
=> ( ( ( times_times_mat_a @ C2 @ B3 )
= ( one_mat_a @ Nr ) )
=> ( ( matrix_mat_kernel_a @ ( times_times_mat_a @ B3 @ A3 ) )
= ( matrix_mat_kernel_a @ A3 ) ) ) ) ) ) ).
% mat_kernel_mult_eq
thf(fact_1268_mat__kernel__carrier,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ord_le4791951621262958845_vec_a @ ( matrix_mat_kernel_a @ A3 ) @ ( carrier_vec_a @ Nc ) ) ) ).
% mat_kernel_carrier
% 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,
( ( plus_plus_vec_a @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) @ t )
= ( plus_plus_vec_a @ ( zero_vec_a @ nr ) @ t ) ) ).
%------------------------------------------------------------------------------